math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.3s

Alternatives: 28

Speedup: 1.5×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + \frac{1}{e^{im\_m}}\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp im_m) (/ 1.0 (exp im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return (0.5 * sin(re)) * (exp(im_m) + (1.0 / exp(im_m)));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = (0.5d0 * sin(re)) * (exp(im_m) + (1.0d0 / exp(im_m)))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return (0.5 * Math.sin(re)) * (Math.exp(im_m) + (1.0 / Math.exp(im_m)));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return (0.5 * math.sin(re)) * (math.exp(im_m) + (1.0 / math.exp(im_m)))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + Float64(1.0 / exp(im_m))))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = (0.5 * sin(re)) * (exp(im_m) + (1.0 / exp(im_m)));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[(1.0 / N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. exp-diff N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\frac{e^{0}}{e^{im}}} + e^{im}\right) \]

   2. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\frac{e^{0}}{\color{blue}{e^{im}}} + e^{im}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\frac{e^{0}}{e^{im}}} + e^{im}\right) \]

   4. exp-0 100.0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\frac{\color{blue}{1}}{e^{im}} + e^{im}\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\frac{1}{e^{im}}} + e^{im}\right) \]

5. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + \frac{1}{e^{im}}\right) \]
6. Add Preprocessing

Alternative 2: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh im\_m \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im\_m \cdot im\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m))))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im_m) (fma re (* (* re re) -0.16666666666666666) re))
     (if (<= t_0 1.0)
       (* (sin re) (fma 0.5 (* im_m im_m) 1.0))
       (fma
        re
        (*
         im_m
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
           0.5)))
        re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(im_m) + exp(-im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = cosh(im_m) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * fma(0.5, (im_m * im_m), 1.0);
	} else {
		tmp = fma(re, (im_m * (im_m * fma((im_m * im_m), fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cosh(im_m) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(0.5, Float64(im_m * im_m), 1.0));
	else
		tmp = fma(re, Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Cosh[im$95$m], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 73.7

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Simplified 73.7%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 99.7

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 83.6%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   11. Simplified 69.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.001388888888888889 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im\_m \cdot im\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m))))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       0.001388888888888889
       (* (* im_m im_m) (* (* im_m im_m) (* im_m im_m))))
      (fma
       (fma
        (* re re)
        (fma (* re re) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* re (* re re))
       re))
     (if (<= t_0 1.0)
       (* (sin re) (fma 0.5 (* im_m im_m) 1.0))
       (fma
        re
        (*
         im_m
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
           0.5)))
        re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(im_m) + exp(-im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.001388888888888889 * ((im_m * im_m) * ((im_m * im_m) * (im_m * im_m)))) * fma(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * fma(0.5, (im_m * im_m), 1.0);
	} else {
		tmp = fma(re, (im_m * (im_m * fma((im_m * im_m), fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.001388888888888889 * Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(im_m * im_m)))) * fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(0.5, Float64(im_m * im_m), 1.0));
	else
		tmp = fma(re, Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.001388888888888889 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 84.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{720} \cdot \left({im}^{6} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {im}^{6}\right) \cdot \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{720} \cdot {im}^{6}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      5. cube-prod N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

      6. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {\color{blue}{\left({im}^{2}\right)}}^{3}\right) \]

      7. cube-unmult N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right)}\right) \]

      8. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{\left(2 \cdot 2\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{\color{blue}{4}}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{4}\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      17. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      18. cube-unmult N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{\left({im}^{2}\right)}^{3}}\right) \]

      19. unpow2 N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {\color{blue}{\left(im \cdot im\right)}}^{3}\right) \]

      20. cube-prod N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      21. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{im}^{\left(2 \cdot 3\right)}}\right) \]

      22. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{6}}\right) \]

      23. lower-*.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{6}\right)} \]

      24. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      25. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

   8. Simplified 84.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot re\right) + 1 \cdot re\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       6. unpow3 N/A

          \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + 1 \cdot re\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       7. *-lft-identity N/A

          \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{3} + \color{blue}{re}\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, {re}^{3}, re\right)} \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

   11. Simplified 61.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \cdot \left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 99.7

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 83.6%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   11. Simplified 69.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -\infty:\\ \;\;\;\;\left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.001388888888888889 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m))))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       0.001388888888888889
       (* (* im_m im_m) (* (* im_m im_m) (* im_m im_m))))
      (fma
       (fma
        (* re re)
        (fma (* re re) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* re (* re re))
       re))
     (if (<= t_0 1.0)
       (sin re)
       (fma
        re
        (*
         im_m
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
           0.5)))
        re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(im_m) + exp(-im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.001388888888888889 * ((im_m * im_m) * ((im_m * im_m) * (im_m * im_m)))) * fma(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = fma(re, (im_m * (im_m * fma((im_m * im_m), fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.001388888888888889 * Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(im_m * im_m)))) * fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = fma(re, Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.001388888888888889 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 84.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{720} \cdot \left({im}^{6} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {im}^{6}\right) \cdot \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{720} \cdot {im}^{6}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      5. cube-prod N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

      6. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {\color{blue}{\left({im}^{2}\right)}}^{3}\right) \]

      7. cube-unmult N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right)}\right) \]

      8. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{\left(2 \cdot 2\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{\color{blue}{4}}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{4}\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      17. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      18. cube-unmult N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{\left({im}^{2}\right)}^{3}}\right) \]

      19. unpow2 N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {\color{blue}{\left(im \cdot im\right)}}^{3}\right) \]

      20. cube-prod N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      21. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{im}^{\left(2 \cdot 3\right)}}\right) \]

      22. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{6}}\right) \]

      23. lower-*.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{6}\right)} \]

      24. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      25. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

   8. Simplified 84.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot re\right) + 1 \cdot re\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       6. unpow3 N/A

          \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + 1 \cdot re\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       7. *-lft-identity N/A

          \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{3} + \color{blue}{re}\right) \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, {re}^{3}, re\right)} \cdot \left(\frac{1}{720} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

   11. Simplified 61.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \cdot \left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 99.1

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 83.6%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   11. Simplified 69.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -\infty:\\ \;\;\;\;\left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.912:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m))))))
   (if (<= t_0 -0.02)
     (*
      re
      (*
       (* re re)
       (fma (* im_m im_m) -0.08333333333333333 -0.16666666666666666)))
     (if (<= t_0 0.912)
       (fma
        (* re re)
        (* re (fma (* re re) 0.008333333333333333 -0.16666666666666666))
        re)
       (*
        re
        (* im_m (* im_m (fma (* im_m im_m) 0.041666666666666664 0.5))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(im_m) + exp(-im_m));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = re * ((re * re) * fma((im_m * im_m), -0.08333333333333333, -0.16666666666666666));
	} else if (t_0 <= 0.912) {
		tmp = fma((re * re), (re * fma((re * re), 0.008333333333333333, -0.16666666666666666)), re);
	} else {
		tmp = re * (im_m * (im_m * fma((im_m * im_m), 0.041666666666666664, 0.5)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m))))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(re * Float64(Float64(re * re) * fma(Float64(im_m * im_m), -0.08333333333333333, -0.16666666666666666)));
	elseif (t_0 <= 0.912)
		tmp = fma(Float64(re * re), Float64(re * fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666)), re);
	else
		tmp = Float64(re * Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), 0.041666666666666664, 0.5))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.912], N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 69.9

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{6}, 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)}\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right)\right) \]

      16. lower-*.f64 30.6

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \]

   8. Simplified 30.6%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \cdot \frac{-1}{6}\right)} \]

       2. associate-*r* N/A

          \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{6}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto re \cdot \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{-1}{6}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{-1}{6}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left({im}^{2} \cdot \frac{-1}{12} + \color{blue}{\frac{-1}{6}}\right)\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{12}, \frac{-1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{12}, \frac{-1}{6}\right)\right) \]

       15. lower-*.f64 15.2

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.08333333333333333, -0.16666666666666666\right)\right) \]

   11. Simplified 15.2%

       \[\leadsto re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)} \]

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.912000000000000033

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 99.5

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right)} + 1 \cdot re \]

      4. *-lft-identity N/A

         \[\leadsto {re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right) + \color{blue}{re} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re}, re\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot re, re\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot re, re\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right) \cdot re, re\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)} \cdot re, re\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right) \cdot re, re\right) \]

      14. lower-*.f64 77.0

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.008333333333333333, -0.16666666666666666\right) \cdot re, re\right) \]

   8. Simplified 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot re, re\right)} \]

   if 0.912000000000000033 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2} \cdot re, re\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2} \cdot re, re\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot \frac{1}{24}\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot im\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, \frac{1}{24} \cdot im, \frac{1}{2}\right)}, {im}^{2} \cdot re, re\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{{im}^{2} \cdot re}, re\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

      17. lower-*.f64 53.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. Simplified 53.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) + {im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right)} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot {im}^{4}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       4. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot re}{{im}^{2}}} \cdot {im}^{4} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       5. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot re\right) \cdot {im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       6. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       7. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       8. pow-sqr N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       9. *-lft-identity N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{2} \cdot {im}^{2}}{\color{blue}{1 \cdot {im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       10. times-frac N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(\frac{{im}^{2}}{1} \cdot \frac{{im}^{2}}{{im}^{2}}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       11. /-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \frac{{im}^{2}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       12. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       13. associate-*r/ N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       14. rgt-mult-inverse N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{1}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       15. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       16. *-commutative N/A

           \[\leadsto \color{blue}{\left(re \cdot \frac{1}{2}\right)} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       17. associate-*r* N/A

           \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       18. *-commutative N/A

           \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{1}{24} \cdot re\right) \cdot {im}^{4}} \]

   11. Simplified 56.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.912:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right)\\ \mathbf{if}\;t\_0 \leq -0.95:\\ \;\;\;\;\left(re \cdot \left(im\_m \cdot im\_m\right)\right) \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m))))))
   (if (<= t_0 -0.95)
     (* (* re (* im_m im_m)) (fma -0.08333333333333333 (* re re) 0.5))
     (if (<= t_0 0.004)
       (fma re (* (* re re) -0.16666666666666666) re)
       (*
        re
        (* im_m (* im_m (fma (* im_m im_m) 0.041666666666666664 0.5))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(im_m) + exp(-im_m));
	double tmp;
	if (t_0 <= -0.95) {
		tmp = (re * (im_m * im_m)) * fma(-0.08333333333333333, (re * re), 0.5);
	} else if (t_0 <= 0.004) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = re * (im_m * (im_m * fma((im_m * im_m), 0.041666666666666664, 0.5)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m))))
	tmp = 0.0
	if (t_0 <= -0.95)
		tmp = Float64(Float64(re * Float64(im_m * im_m)) * fma(-0.08333333333333333, Float64(re * re), 0.5));
	elseif (t_0 <= 0.004)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(re * Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), 0.041666666666666664, 0.5))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.95], N[(N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.004], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.94999999999999996

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 55.6

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{6}, 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)}\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right)\right) \]

      16. lower-*.f64 44.0

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)\right) \cdot \frac{1}{2}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot re\right) \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \frac{1}{2} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot re\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \frac{1}{2}\right)} \]

       4. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot re\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\left(re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \left(re \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right)\right)} \]

       11. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2} \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \left(re \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{\left(\frac{1}{2} \cdot \frac{-1}{6}\right) \cdot {re}^{2}}\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12}} \cdot {re}^{2}\right) \]

       14. +-commutative N/A

           \[\leadsto \left(re \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \]

       15. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{12}, {re}^{2}, \frac{1}{2}\right)} \]

       16. unpow2 N/A

           \[\leadsto \left(re \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{12}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \]

       17. lower-*.f64 44.0

           \[\leadsto \left(re \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(-0.08333333333333333, \color{blue}{re \cdot re}, 0.5\right) \]

   11. Simplified 44.0%

       \[\leadsto \color{blue}{\left(re \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)} \]

   if -0.94999999999999996 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 99.6

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 71.8

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   8. Simplified 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 83.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2} \cdot re, re\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2} \cdot re, re\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot \frac{1}{24}\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot im\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, \frac{1}{24} \cdot im, \frac{1}{2}\right)}, {im}^{2} \cdot re, re\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{{im}^{2} \cdot re}, re\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

      17. lower-*.f64 40.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. Simplified 40.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) + {im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right)} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot {im}^{4}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       4. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot re}{{im}^{2}}} \cdot {im}^{4} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       5. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot re\right) \cdot {im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       6. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       7. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       8. pow-sqr N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       9. *-lft-identity N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{2} \cdot {im}^{2}}{\color{blue}{1 \cdot {im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       10. times-frac N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(\frac{{im}^{2}}{1} \cdot \frac{{im}^{2}}{{im}^{2}}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       11. /-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \frac{{im}^{2}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       12. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       13. associate-*r/ N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       14. rgt-mult-inverse N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{1}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       15. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       16. *-commutative N/A

           \[\leadsto \color{blue}{\left(re \cdot \frac{1}{2}\right)} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       17. associate-*r* N/A

           \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       18. *-commutative N/A

           \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{1}{24} \cdot re\right) \cdot {im}^{4}} \]

   11. Simplified 42.5%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.95:\\ \;\;\;\;\left(re \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.92:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im\_m \cdot im\_m\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m))))))
   (if (<= t_0 -0.02)
     (*
      re
      (*
       (* re re)
       (fma (* im_m im_m) -0.08333333333333333 -0.16666666666666666)))
     (if (<= t_0 0.92)
       (fma 0.5 (* re (* im_m im_m)) re)
       (*
        re
        (* im_m (* im_m (fma (* im_m im_m) 0.041666666666666664 0.5))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(im_m) + exp(-im_m));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = re * ((re * re) * fma((im_m * im_m), -0.08333333333333333, -0.16666666666666666));
	} else if (t_0 <= 0.92) {
		tmp = fma(0.5, (re * (im_m * im_m)), re);
	} else {
		tmp = re * (im_m * (im_m * fma((im_m * im_m), 0.041666666666666664, 0.5)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m))))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(re * Float64(Float64(re * re) * fma(Float64(im_m * im_m), -0.08333333333333333, -0.16666666666666666)));
	elseif (t_0 <= 0.92)
		tmp = fma(0.5, Float64(re * Float64(im_m * im_m)), re);
	else
		tmp = Float64(re * Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), 0.041666666666666664, 0.5))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.92], N[(0.5 * N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 69.9

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{6}, 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)}\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right)\right) \]

      16. lower-*.f64 30.6

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \]

   8. Simplified 30.6%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \cdot \frac{-1}{6}\right)} \]

       2. associate-*r* N/A

          \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{6}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto re \cdot \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{-1}{6}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{-1}{6}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left({im}^{2} \cdot \frac{-1}{12} + \color{blue}{\frac{-1}{6}}\right)\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{12}, \frac{-1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{12}, \frac{-1}{6}\right)\right) \]

       15. lower-*.f64 15.2

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.08333333333333333, -0.16666666666666666\right)\right) \]

   11. Simplified 15.2%

       \[\leadsto re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)} \]

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.92000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 100.0

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + 1 \cdot re} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

      4. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2} \cdot re, re\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{im}^{2} \cdot re}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

      8. lower-*.f64 76.6

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. Simplified 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left(im \cdot im\right) \cdot re, re\right)} \]

   if 0.92000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2} \cdot re, re\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2} \cdot re, re\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot \frac{1}{24}\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot im\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, \frac{1}{24} \cdot im, \frac{1}{2}\right)}, {im}^{2} \cdot re, re\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{{im}^{2} \cdot re}, re\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

      17. lower-*.f64 53.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. Simplified 53.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) + {im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right)} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot {im}^{4}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       4. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot re}{{im}^{2}}} \cdot {im}^{4} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       5. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot re\right) \cdot {im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       6. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       7. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       8. pow-sqr N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       9. *-lft-identity N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{2} \cdot {im}^{2}}{\color{blue}{1 \cdot {im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       10. times-frac N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(\frac{{im}^{2}}{1} \cdot \frac{{im}^{2}}{{im}^{2}}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       11. /-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \frac{{im}^{2}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       12. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       13. associate-*r/ N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       14. rgt-mult-inverse N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{1}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       15. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       16. *-commutative N/A

           \[\leadsto \color{blue}{\left(re \cdot \frac{1}{2}\right)} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       17. associate-*r* N/A

           \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       18. *-commutative N/A

           \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{1}{24} \cdot re\right) \cdot {im}^{4}} \]

   11. Simplified 56.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.92:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (*
    (fma (* im_m im_m) (fma im_m (* im_m 0.041666666666666664) 0.5) 1.0)
    (fma
     (* re re)
     (*
      re
      (fma
       re
       (* re (fma (* re re) -0.0001984126984126984 0.008333333333333333))
       -0.16666666666666666))
     re))
   (fma
    re
    (*
     im_m
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
       0.5)))
    re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = fma((im_m * im_m), fma(im_m, (im_m * 0.041666666666666664), 0.5), 1.0) * fma((re * re), (re * fma(re, (re * fma((re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re);
	} else {
		tmp = fma(re, (im_m * (im_m * fma((im_m * im_m), fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = Float64(fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * 0.041666666666666664), 0.5), 1.0) * fma(Float64(re * re), Float64(re * fma(re, Float64(re * fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re));
	else
		tmp = fma(re, Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot re\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left({re}^{2} \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot re\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 46.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   11. Simplified 46.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot 0.041666666666666664, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (*
    re
    (*
     (fma (* re re) -0.16666666666666666 1.0)
     (fma im_m (* im_m (fma im_m (* im_m 0.041666666666666664) 0.5)) 1.0)))
   (fma
    re
    (*
     im_m
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
       0.5)))
    re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = re * (fma((re * re), -0.16666666666666666, 1.0) * fma(im_m, (im_m * fma(im_m, (im_m * 0.041666666666666664), 0.5)), 1.0));
	} else {
		tmp = fma(re, (im_m * (im_m * fma((im_m * im_m), fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = Float64(re * Float64(fma(Float64(re * re), -0.16666666666666666, 1.0) * fma(im_m, Float64(im_m * fma(im_m, Float64(im_m * 0.041666666666666664), 0.5)), 1.0)));
	else
		tmp = fma(re, Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(re * N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 46.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   11. Simplified 46.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (fma
    (* re re)
    (*
     re
     (fma
      re
      (* re (fma (* re re) -0.0001984126984126984 0.008333333333333333))
      -0.16666666666666666))
    re)
   (fma
    re
    (*
     im_m
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
       0.5)))
    re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = fma((re * re), (re * fma(re, (re * fma((re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re);
	} else {
		tmp = fma(re, (im_m * (im_m * fma((im_m * im_m), fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = fma(Float64(re * re), Float64(re * fma(re, Float64(re * fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re);
	else
		tmp = fma(re, Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.4

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 66.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot re\right)} + 1 \cdot re \]

      4. *-lft-identity N/A

         \[\leadsto {re}^{2} \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot re\right) + \color{blue}{re} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot re, re\right)} \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right) \cdot re, re\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 46.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   11. Simplified 46.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im\_m \cdot im\_m, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (*
    re
    (* (fma 0.5 (* im_m im_m) 1.0) (fma (* re re) -0.16666666666666666 1.0)))
   (fma
    re
    (*
     im_m
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
       0.5)))
    re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = re * (fma(0.5, (im_m * im_m), 1.0) * fma((re * re), -0.16666666666666666, 1.0));
	} else {
		tmp = fma(re, (im_m * (im_m * fma((im_m * im_m), fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = Float64(re * Float64(fma(0.5, Float64(im_m * im_m), 1.0) * fma(Float64(re * re), -0.16666666666666666, 1.0)));
	else
		tmp = fma(re, Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(re * N[(N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 83.3

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{6}, 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)}\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right)\right) \]

      16. lower-*.f64 61.5

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 46.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   11. Simplified 46.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im\_m \cdot im\_m, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im\_m \cdot im\_m\right) \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889\right), 0.5\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (*
    re
    (* (fma 0.5 (* im_m im_m) 1.0) (fma (* re re) -0.16666666666666666 1.0)))
   (fma
    (*
     (* im_m im_m)
     (fma im_m (* im_m (* (* im_m im_m) 0.001388888888888889)) 0.5))
    re
    re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = re * (fma(0.5, (im_m * im_m), 1.0) * fma((re * re), -0.16666666666666666, 1.0));
	} else {
		tmp = fma(((im_m * im_m) * fma(im_m, (im_m * ((im_m * im_m) * 0.001388888888888889)), 0.5)), re, re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = Float64(re * Float64(fma(0.5, Float64(im_m * im_m), 1.0) * fma(Float64(re * re), -0.16666666666666666, 1.0)));
	else
		tmp = fma(Float64(Float64(im_m * im_m) * fma(im_m, Float64(im_m * Float64(Float64(im_m * im_m) * 0.001388888888888889)), 0.5)), re, re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(re * N[(N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 83.3

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{6}, 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)}\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right)\right) \]

      16. lower-*.f64 61.5

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 46.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right), re\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{720}\right)\right), re\right) \]

       4. lower-*.f64 46.9

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.001388888888888889\right)\right), re\right) \]

   11. Simplified 46.9%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)}\right), re\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right) + re \]

       2. lift-*.f64 N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot im\right) + \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right) + re \]

       3. lift-*.f64 N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot im\right) + \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right) + re \]

       4. lift-*.f64 N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot im\right) + \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right) + re \]

       5. lift-*.f64 N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot im\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{720}\right)\right) + re \]

       6. lift-*.f64 N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot im\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)}\right) + re \]

       7. lift-*.f64 N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot im\right) + \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)}\right) + re \]

       8. lift-fma.f64 N/A

          \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right)} + re \]

       9. *-commutative N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720}\right)\right) \cdot re} + re \]

       10. lower-fma.f64 46.9

           \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right), re, re\right)} \]

   13. Applied egg-rr 46.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right), 0.5\right), re, re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right), 0.5\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im\_m \cdot im\_m, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im\_m \cdot im\_m\right), \mathsf{fma}\left(im\_m \cdot im\_m, \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889, 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (*
    re
    (* (fma 0.5 (* im_m im_m) 1.0) (fma (* re re) -0.16666666666666666 1.0)))
   (fma
    (* re (* im_m im_m))
    (fma (* im_m im_m) (* (* im_m im_m) 0.001388888888888889) 0.5)
    re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = re * (fma(0.5, (im_m * im_m), 1.0) * fma((re * re), -0.16666666666666666, 1.0));
	} else {
		tmp = fma((re * (im_m * im_m)), fma((im_m * im_m), ((im_m * im_m) * 0.001388888888888889), 0.5), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = Float64(re * Float64(fma(0.5, Float64(im_m * im_m), 1.0) * fma(Float64(re * re), -0.16666666666666666, 1.0)));
	else
		tmp = fma(Float64(re * Float64(im_m * im_m)), fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * 0.001388888888888889), 0.5), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(re * N[(N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.5), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 83.3

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{6}, 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)}\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right)\right) \]

      16. lower-*.f64 61.5

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 46.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right), re\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{720}\right)\right), re\right) \]

       4. lower-*.f64 46.9

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.001388888888888889\right)\right), re\right) \]

   11. Simplified 46.9%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)}\right), re\right) \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right) + \frac{1}{2} \cdot re\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right) + \frac{1}{2} \cdot re\right) + re} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right)} + re \]

       3. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right)\right) \cdot {im}^{2}} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       4. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot re\right)} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       5. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left(re \cdot {im}^{2}\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \color{blue}{\left({im}^{2} \cdot re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left({im}^{2} \cdot re\right) + {im}^{2} \cdot \color{blue}{\left(re \cdot \frac{1}{2}\right)}\right) + re \]

       8. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{\left({im}^{2} \cdot re\right) \cdot \frac{1}{2}}\right) + re \]

       9. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)}\right) + re \]

       10. distribute-rgt-out N/A

           \[\leadsto \color{blue}{\left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{720} \cdot {im}^{4} + \frac{1}{2}\right)} + re \]

       11. metadata-eval N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}} + \frac{1}{2}\right) + re \]

       12. pow-sqr N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} + \frac{1}{2}\right) + re \]

       13. associate-*l* N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}\right) + re \]

       14. *-commutative N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)} + \frac{1}{2}\right) + re \]

       15. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2} \cdot re, {im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}, re\right)} \]

   14. Simplified 42.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \left(im \cdot im\right), \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.001388888888888889, 0.5\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im \cdot im\right), \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.001388888888888889, 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im\_m \cdot im\_m, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im\_m \cdot im\_m, im\_m \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot 0.041666666666666664\right)\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (*
    re
    (* (fma 0.5 (* im_m im_m) 1.0) (fma (* re re) -0.16666666666666666 1.0)))
   (fma
    re
    (fma
     0.5
     (* im_m im_m)
     (* im_m (* im_m (* (* im_m im_m) 0.041666666666666664))))
    re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = re * (fma(0.5, (im_m * im_m), 1.0) * fma((re * re), -0.16666666666666666, 1.0));
	} else {
		tmp = fma(re, fma(0.5, (im_m * im_m), (im_m * (im_m * ((im_m * im_m) * 0.041666666666666664)))), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = Float64(re * Float64(fma(0.5, Float64(im_m * im_m), 1.0) * fma(Float64(re * re), -0.16666666666666666, 1.0)));
	else
		tmp = fma(re, fma(0.5, Float64(im_m * im_m), Float64(im_m * Float64(im_m * Float64(Float64(im_m * im_m) * 0.041666666666666664)))), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(re * N[(N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 83.3

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{6}, 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)}\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right)\right) \]

      16. lower-*.f64 61.5

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 46.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{4}}\right), re\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \frac{1}{24} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right), re\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right), re\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}}\right), re\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right), re\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right), re\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right), re\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right), re\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right), re\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, im \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)}\right)\right), re\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, im \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)}\right)\right), re\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, im \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24}\right)\right)\right), re\right) \]

       12. lower-*.f64 42.5

           \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, im \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.041666666666666664\right)\right)\right), re\right) \]

   11. Simplified 42.5%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)}\right), re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im\_m \cdot im\_m, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (*
    re
    (* (fma 0.5 (* im_m im_m) 1.0) (fma (* re re) -0.16666666666666666 1.0)))
   (* re (* im_m (* im_m (fma (* im_m im_m) 0.041666666666666664 0.5))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = re * (fma(0.5, (im_m * im_m), 1.0) * fma((re * re), -0.16666666666666666, 1.0));
	} else {
		tmp = re * (im_m * (im_m * fma((im_m * im_m), 0.041666666666666664, 0.5)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = Float64(re * Float64(fma(0.5, Float64(im_m * im_m), 1.0) * fma(Float64(re * re), -0.16666666666666666, 1.0)));
	else
		tmp = Float64(re * Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), 0.041666666666666664, 0.5))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(re * N[(N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 83.3

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{6}, 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)}\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right)\right) \]

      16. lower-*.f64 61.5

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 83.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2} \cdot re, re\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2} \cdot re, re\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot \frac{1}{24}\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot im\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, \frac{1}{24} \cdot im, \frac{1}{2}\right)}, {im}^{2} \cdot re, re\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{{im}^{2} \cdot re}, re\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

      17. lower-*.f64 40.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. Simplified 40.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) + {im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right)} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot {im}^{4}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       4. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot re}{{im}^{2}}} \cdot {im}^{4} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       5. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot re\right) \cdot {im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       6. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       7. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       8. pow-sqr N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       9. *-lft-identity N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{2} \cdot {im}^{2}}{\color{blue}{1 \cdot {im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       10. times-frac N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(\frac{{im}^{2}}{1} \cdot \frac{{im}^{2}}{{im}^{2}}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       11. /-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \frac{{im}^{2}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       12. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       13. associate-*r/ N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       14. rgt-mult-inverse N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{1}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       15. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       16. *-commutative N/A

           \[\leadsto \color{blue}{\left(re \cdot \frac{1}{2}\right)} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       17. associate-*r* N/A

           \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       18. *-commutative N/A

           \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{1}{24} \cdot re\right) \cdot {im}^{4}} \]

   11. Simplified 42.5%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot 0.041666666666666664, 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) -0.02)
   (*
    re
    (*
     (* re re)
     (fma (* im_m im_m) -0.08333333333333333 -0.16666666666666666)))
   (fma im_m (* im_m (* re (fma im_m (* im_m 0.041666666666666664) 0.5))) re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= -0.02) {
		tmp = re * ((re * re) * fma((im_m * im_m), -0.08333333333333333, -0.16666666666666666));
	} else {
		tmp = fma(im_m, (im_m * (re * fma(im_m, (im_m * 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= -0.02)
		tmp = Float64(re * Float64(Float64(re * re) * fma(Float64(im_m * im_m), -0.08333333333333333, -0.16666666666666666)));
	else
		tmp = fma(im_m, Float64(im_m * Float64(re * fma(im_m, Float64(im_m * 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(im$95$m * N[(re * N[(im$95$m * N[(im$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 69.9

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{6}, 1\right)} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6}, 1\right) \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)}\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right)\right) \]

      16. lower-*.f64 30.6

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \]

   8. Simplified 30.6%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left({re}^{2} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \cdot \frac{-1}{6}\right)} \]

       2. associate-*r* N/A

          \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{6}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto re \cdot \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{-1}{6}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{-1}{6}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left({im}^{2} \cdot \frac{-1}{12} + \color{blue}{\frac{-1}{6}}\right)\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{12}, \frac{-1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{12}, \frac{-1}{6}\right)\right) \]

       15. lower-*.f64 15.2

           \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.08333333333333333, -0.16666666666666666\right)\right) \]

   11. Simplified 15.2%

       \[\leadsto re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)} \]

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 94.1%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 70.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) + re} \]

       2. unpow2 N/A

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) + re \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right)\right)} + re \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right), re\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right)}, re\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{2} \cdot re + \frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)}, re\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \left(\frac{1}{2} \cdot re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re}\right), re\right) \]

       8. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}, re\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}, re\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(im, im \cdot \left(re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right), re\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(im, im \cdot \left(re \cdot \left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right)\right), re\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(im, im \cdot \left(re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24} + \frac{1}{2}\right)\right), re\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(im, im \cdot \left(re \cdot \left(\color{blue}{im \cdot \left(im \cdot \frac{1}{24}\right)} + \frac{1}{2}\right)\right), re\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(im, im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)}\right), re\right) \]

       15. lower-*.f64 67.3

           \[\leadsto \mathsf{fma}\left(im, im \cdot \left(re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot 0.041666666666666664}, 0.5\right)\right), re\right) \]

   11. Simplified 67.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \left(re \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \left(re \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (fma re (* (* re re) -0.16666666666666666) re)
   (* re (* im_m (* im_m (fma (* im_m im_m) 0.041666666666666664 0.5))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = re * (im_m * (im_m * fma((im_m * im_m), 0.041666666666666664, 0.5)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(re * Float64(im_m * Float64(im_m * fma(Float64(im_m * im_m), 0.041666666666666664, 0.5))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.4

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 66.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 49.8

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   8. Simplified 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 83.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2} \cdot re, re\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2} \cdot re, re\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot \frac{1}{24}\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot im\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, \frac{1}{24} \cdot im, \frac{1}{2}\right)}, {im}^{2} \cdot re, re\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{{im}^{2} \cdot re}, re\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

      17. lower-*.f64 40.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. Simplified 40.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) + {im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right)} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot {im}^{4}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       4. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot re}{{im}^{2}}} \cdot {im}^{4} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       5. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot re\right) \cdot {im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       6. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       7. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       8. pow-sqr N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       9. *-lft-identity N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \frac{{im}^{2} \cdot {im}^{2}}{\color{blue}{1 \cdot {im}^{2}}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       10. times-frac N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(\frac{{im}^{2}}{1} \cdot \frac{{im}^{2}}{{im}^{2}}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       11. /-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \frac{{im}^{2}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       12. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       13. associate-*r/ N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       14. rgt-mult-inverse N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{2} \cdot \color{blue}{1}\right) + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       15. *-rgt-identity N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{{im}^{2}} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       16. *-commutative N/A

           \[\leadsto \color{blue}{\left(re \cdot \frac{1}{2}\right)} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       17. associate-*r* N/A

           \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)} + {im}^{4} \cdot \left(\frac{1}{24} \cdot re\right) \]

       18. *-commutative N/A

           \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{1}{24} \cdot re\right) \cdot {im}^{4}} \]

   11. Simplified 42.5%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(re \cdot \left(\left(im\_m \cdot im\_m\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (fma re (* (* re re) -0.16666666666666666) re)
   (* im_m (* im_m (* re (* (* im_m im_m) 0.041666666666666664))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = im_m * (im_m * (re * ((im_m * im_m) * 0.041666666666666664)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(im_m * Float64(im_m * Float64(re * Float64(Float64(im_m * im_m) * 0.041666666666666664))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(im$95$m * N[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.4

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 66.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 49.8

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   8. Simplified 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 83.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2} \cdot re, re\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2} \cdot re, re\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot \frac{1}{24}\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot im\right)} + \frac{1}{2}, {im}^{2} \cdot re, re\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, \frac{1}{24} \cdot im, \frac{1}{2}\right)}, {im}^{2} \cdot re, re\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{1}{2}\right), {im}^{2} \cdot re, re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{{im}^{2} \cdot re}, re\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

      17. lower-*.f64 40.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. Simplified 40.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot re} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot re \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \cdot re \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot re \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right) \]

       8. associate-*r* N/A

          \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)} \]

       9. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]

       12. associate-*r* N/A

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

       15. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)}\right)\right) \]

       16. lower-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto im \cdot \left(im \cdot \left(re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24}\right)\right)\right) \]

       18. lower-*.f64 40.2

           \[\leadsto im \cdot \left(im \cdot \left(re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.041666666666666664\right)\right)\right) \]

   11. Simplified 40.2%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 41.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im\_m \cdot im\_m\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
   (fma re (* (* re re) -0.16666666666666666) re)
   (fma 0.5 (* re (* im_m im_m)) re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = fma(0.5, (re * (im_m * im_m)), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + exp(Float64(-im_m)))) <= 0.004)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = fma(0.5, Float64(re * Float64(im_m * im_m)), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(0.5 * N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.4

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 66.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 49.8

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   8. Simplified 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. unpow2 N/A

         \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      12. lower-*.f64 73.9

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 73.9%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + 1 \cdot re} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

      4. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2} \cdot re, re\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{im}^{2} \cdot re}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

      8. lower-*.f64 38.1

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. Simplified 38.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left(im \cdot im\right) \cdot re, re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sin re \cdot \cosh im\_m \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (sin re) (cosh im_m)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return sin(re) * cosh(im_m);
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = sin(re) * cosh(im_m)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return Math.sin(re) * Math.cosh(im_m);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return math.sin(re) * math.cosh(im_m)

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(sin(re) * cosh(im_m))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = sin(re) * cosh(im_m);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Sin[re], $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

   4. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

   5. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

   6. lift-+.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

5. Final simplification 100.0%

   \[\leadsto \sin re \cdot \cosh im \]
6. Add Preprocessing

Alternative 21: 58.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im\_m, im\_m \cdot 0.001388888888888889, 0.041666666666666664\right), im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), \mathsf{fma}\left(0.5, im\_m \cdot im\_m, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im\_m \cdot im\_m\right), \mathsf{fma}\left(im\_m \cdot im\_m, \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889, 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) 0.004)
   (*
    (fma
     (* re re)
     (*
      re
      (fma
       re
       (* re (fma (* re re) -0.0001984126984126984 0.008333333333333333))
       -0.16666666666666666))
     re)
    (fma
     (fma im_m (* im_m 0.001388888888888889) 0.041666666666666664)
     (* im_m (* im_m (* im_m im_m)))
     (fma 0.5 (* im_m im_m) 1.0)))
   (fma
    (* re (* im_m im_m))
    (fma (* im_m im_m) (* (* im_m im_m) 0.001388888888888889) 0.5)
    re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (sin(re) <= 0.004) {
		tmp = fma((re * re), (re * fma(re, (re * fma((re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re) * fma(fma(im_m, (im_m * 0.001388888888888889), 0.041666666666666664), (im_m * (im_m * (im_m * im_m))), fma(0.5, (im_m * im_m), 1.0));
	} else {
		tmp = fma((re * (im_m * im_m)), fma((im_m * im_m), ((im_m * im_m) * 0.001388888888888889), 0.5), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (sin(re) <= 0.004)
		tmp = Float64(fma(Float64(re * re), Float64(re * fma(re, Float64(re * fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re) * fma(fma(im_m, Float64(im_m * 0.001388888888888889), 0.041666666666666664), Float64(im_m * Float64(im_m * Float64(im_m * im_m))), fma(0.5, Float64(im_m * im_m), 1.0)));
	else
		tmp = fma(Float64(re * Float64(im_m * im_m)), fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * 0.001388888888888889), 0.5), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sin[re], $MachinePrecision], 0.004], N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.5), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot re\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left({re}^{2} \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot re\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

   8. Simplified 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   if 0.0040000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 87.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right), re\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{720}\right)\right), re\right) \]

       4. lower-*.f64 26.6

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.001388888888888889\right)\right), re\right) \]

   11. Simplified 26.6%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)}\right), re\right) \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right) + \frac{1}{2} \cdot re\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right) + \frac{1}{2} \cdot re\right) + re} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right)} + re \]

       3. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right)\right) \cdot {im}^{2}} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       4. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot re\right)} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       5. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left(re \cdot {im}^{2}\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \color{blue}{\left({im}^{2} \cdot re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left({im}^{2} \cdot re\right) + {im}^{2} \cdot \color{blue}{\left(re \cdot \frac{1}{2}\right)}\right) + re \]

       8. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{\left({im}^{2} \cdot re\right) \cdot \frac{1}{2}}\right) + re \]

       9. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)}\right) + re \]

       10. distribute-rgt-out N/A

           \[\leadsto \color{blue}{\left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{720} \cdot {im}^{4} + \frac{1}{2}\right)} + re \]

       11. metadata-eval N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}} + \frac{1}{2}\right) + re \]

       12. pow-sqr N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} + \frac{1}{2}\right) + re \]

       13. associate-*l* N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}\right) + re \]

       14. *-commutative N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)} + \frac{1}{2}\right) + re \]

       15. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2} \cdot re, {im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}, re\right)} \]

   14. Simplified 26.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \left(im \cdot im\right), \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.001388888888888889, 0.5\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im \cdot im\right), \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.001388888888888889, 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im\_m \cdot im\_m\right), \mathsf{fma}\left(im\_m \cdot im\_m, \left(im\_m \cdot im\_m\right) \cdot 0.001388888888888889, 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) 0.004)
   (*
    re
    (*
     (fma re (* re -0.16666666666666666) 1.0)
     (fma
      (* im_m im_m)
      (fma
       im_m
       (* im_m (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664))
       0.5)
      1.0)))
   (fma
    (* re (* im_m im_m))
    (fma (* im_m im_m) (* (* im_m im_m) 0.001388888888888889) 0.5)
    re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (sin(re) <= 0.004) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma((im_m * im_m), fma(im_m, (im_m * fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0));
	} else {
		tmp = fma((re * (im_m * im_m)), fma((im_m * im_m), ((im_m * im_m) * 0.001388888888888889), 0.5), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (sin(re) <= 0.004)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0)));
	else
		tmp = fma(Float64(re * Float64(im_m * im_m)), fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * 0.001388888888888889), 0.5), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sin[re], $MachinePrecision], 0.004], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.5), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right) \cdot \sin re \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{im \cdot \left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} + 1\right) \cdot \sin re \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right), 1\right)} \cdot \sin re \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}, 1\right) \cdot \sin re \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}\right)}, 1\right) \cdot \sin re \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \sin re \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \sin re \]

      14. lower-*.f64 94.6

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \sin re \]

   7. Simplified 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

   10. Simplified 69.8%

       \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)} \]

   if 0.0040000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 87.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right), re\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{2}, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{720}\right)\right), re\right) \]

       4. lower-*.f64 26.6

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.001388888888888889\right)\right), re\right) \]

   11. Simplified 26.6%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)}\right), re\right) \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right) + \frac{1}{2} \cdot re\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right) + \frac{1}{2} \cdot re\right) + re} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right)} + re \]

       3. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{720} \cdot \left({im}^{4} \cdot re\right)\right) \cdot {im}^{2}} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       4. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot re\right)} \cdot {im}^{2} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       5. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left(re \cdot {im}^{2}\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \color{blue}{\left({im}^{2} \cdot re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)\right) + re \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left({im}^{2} \cdot re\right) + {im}^{2} \cdot \color{blue}{\left(re \cdot \frac{1}{2}\right)}\right) + re \]

       8. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{\left({im}^{2} \cdot re\right) \cdot \frac{1}{2}}\right) + re \]

       9. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{720} \cdot {im}^{4}\right) \cdot \left({im}^{2} \cdot re\right) + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)}\right) + re \]

       10. distribute-rgt-out N/A

           \[\leadsto \color{blue}{\left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{720} \cdot {im}^{4} + \frac{1}{2}\right)} + re \]

       11. metadata-eval N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}} + \frac{1}{2}\right) + re \]

       12. pow-sqr N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} + \frac{1}{2}\right) + re \]

       13. associate-*l* N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}\right) + re \]

       14. *-commutative N/A

           \[\leadsto \left({im}^{2} \cdot re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)} + \frac{1}{2}\right) + re \]

       15. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2} \cdot re, {im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}, re\right)} \]

   14. Simplified 26.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \left(im \cdot im\right), \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.001388888888888889, 0.5\right), re\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 97.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 0.88:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{elif}\;im\_m \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\_m \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.001388888888888889 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 0.88)
   (*
    (sin re)
    (fma
     im_m
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
       0.5))
     1.0))
   (if (<= im_m 2.4e+51)
     (* (cosh im_m) (fma re (* (* re re) -0.16666666666666666) re))
     (*
      (sin re)
      (*
       0.001388888888888889
       (* (* im_m im_m) (* (* im_m im_m) (* im_m im_m))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 0.88) {
		tmp = sin(re) * fma(im_m, (im_m * fma((im_m * im_m), fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0);
	} else if (im_m <= 2.4e+51) {
		tmp = cosh(im_m) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = sin(re) * (0.001388888888888889 * ((im_m * im_m) * ((im_m * im_m) * (im_m * im_m))));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 0.88)
		tmp = Float64(sin(re) * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0));
	elseif (im_m <= 2.4e+51)
		tmp = Float64(cosh(im_m) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	else
		tmp = Float64(sin(re) * Float64(0.001388888888888889 * Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(im_m * im_m)))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 0.88], N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.4e+51], N[(N[Cosh[im$95$m], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.001388888888888889 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.880000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right) \cdot \sin re \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{im \cdot \left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} + 1\right) \cdot \sin re \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right), 1\right)} \cdot \sin re \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}, 1\right) \cdot \sin re \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}\right)}, 1\right) \cdot \sin re \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \sin re \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \sin re \]

      14. lower-*.f64 95.4

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \sin re \]

   7. Simplified 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \sin re \]

   if 0.880000000000000004 < im < 2.3999999999999999e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 90.0

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Simplified 90.0%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 2.3999999999999999e51 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{720} \cdot \left({im}^{6} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {im}^{6}\right) \cdot \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{720} \cdot {im}^{6}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      5. cube-prod N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

      6. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {\color{blue}{\left({im}^{2}\right)}}^{3}\right) \]

      7. cube-unmult N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right)}\right) \]

      8. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{\left(2 \cdot 2\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{\color{blue}{4}}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{4}\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      17. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      18. cube-unmult N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{\left({im}^{2}\right)}^{3}}\right) \]

      19. unpow2 N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {\color{blue}{\left(im \cdot im\right)}}^{3}\right) \]

      20. cube-prod N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      21. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{im}^{\left(2 \cdot 3\right)}}\right) \]

      22. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{6}}\right) \]

      23. lower-*.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{6}\right)} \]

      24. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      25. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 97.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 0.225:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;im\_m \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\_m \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.001388888888888889 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 0.225)
   (*
    (sin re)
    (fma (* im_m im_m) (fma im_m (* im_m 0.041666666666666664) 0.5) 1.0))
   (if (<= im_m 2.4e+51)
     (* (cosh im_m) (fma re (* (* re re) -0.16666666666666666) re))
     (*
      (sin re)
      (*
       0.001388888888888889
       (* (* im_m im_m) (* (* im_m im_m) (* im_m im_m))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 0.225) {
		tmp = sin(re) * fma((im_m * im_m), fma(im_m, (im_m * 0.041666666666666664), 0.5), 1.0);
	} else if (im_m <= 2.4e+51) {
		tmp = cosh(im_m) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = sin(re) * (0.001388888888888889 * ((im_m * im_m) * ((im_m * im_m) * (im_m * im_m))));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 0.225)
		tmp = Float64(sin(re) * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * 0.041666666666666664), 0.5), 1.0));
	elseif (im_m <= 2.4e+51)
		tmp = Float64(cosh(im_m) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	else
		tmp = Float64(sin(re) * Float64(0.001388888888888889 * Float64(Float64(im_m * im_m) * Float64(Float64(im_m * im_m) * Float64(im_m * im_m)))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 0.225], N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.4e+51], N[(N[Cosh[im$95$m], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.001388888888888889 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.225000000000000006

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 92.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   if 0.225000000000000006 < im < 2.3999999999999999e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 90.0

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Simplified 90.0%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 2.3999999999999999e51 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\sin re + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{720} \cdot \left({im}^{6} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {im}^{6}\right) \cdot \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{720} \cdot {im}^{6}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      5. cube-prod N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

      6. unpow2 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {\color{blue}{\left({im}^{2}\right)}}^{3}\right) \]

      7. cube-unmult N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot {im}^{2}\right)\right)}\right) \]

      8. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{\left(2 \cdot 2\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{\color{blue}{4}}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left({im}^{4} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{4}\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{4}\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      17. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      18. cube-unmult N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{\left({im}^{2}\right)}^{3}}\right) \]

      19. unpow2 N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {\color{blue}{\left(im \cdot im\right)}}^{3}\right) \]

      20. cube-prod N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      21. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{{im}^{\left(2 \cdot 3\right)}}\right) \]

      22. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{6}}\right) \]

      23. lower-*.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{6}\right)} \]

      24. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      25. pow-sqr N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.225:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.001388888888888889 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 96.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{if}\;im\_m \leq 0.225:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\_m \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0
         (*
          (sin re)
          (fma
           (* im_m im_m)
           (fma im_m (* im_m 0.041666666666666664) 0.5)
           1.0))))
   (if (<= im_m 0.225)
     t_0
     (if (<= im_m 1.12e+77)
       (* (cosh im_m) (fma re (* (* re re) -0.16666666666666666) re))
       t_0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = sin(re) * fma((im_m * im_m), fma(im_m, (im_m * 0.041666666666666664), 0.5), 1.0);
	double tmp;
	if (im_m <= 0.225) {
		tmp = t_0;
	} else if (im_m <= 1.12e+77) {
		tmp = cosh(im_m) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(sin(re) * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * 0.041666666666666664), 0.5), 1.0))
	tmp = 0.0
	if (im_m <= 0.225)
		tmp = t_0;
	elseif (im_m <= 1.12e+77)
		tmp = Float64(cosh(im_m) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 0.225], t$95$0, If[LessEqual[im$95$m, 1.12e+77], N[(N[Cosh[im$95$m], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.225000000000000006 or 1.1199999999999999e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right) + \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + {im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)}\right) + \sin re \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}}\right) + \sin re \]

      7. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)}\right) + \sin re \]

      8. associate-*l* N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re}\right) + \sin re \]

      9. unpow2 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re\right) + \sin re \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re\right) + \sin re \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right)} + \sin re \]

      12. *-rgt-identity N/A

          \[\leadsto \sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + \color{blue}{\sin re \cdot 1} \]

      13. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\frac{1}{2} \cdot im\right) \cdot im\right) + 1\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)\right)} \]

   5. Simplified 94.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   if 0.225000000000000006 < im < 1.1199999999999999e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 86.7

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Simplified 86.7%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.225:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 49.1% accurate, 18.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \mathsf{fma}\left(0.5, re \cdot \left(im\_m \cdot im\_m\right), re\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (fma 0.5 (* re (* im_m im_m)) re))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return fma(0.5, (re * (im_m * im_m)), re);
}

Julia

im_m = abs(im)
function code(re, im_m)
	return fma(0.5, Float64(re * Float64(im_m * im_m)), re)
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

   2. unpow2 N/A

      \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

   3. associate-*r* N/A

      \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

   4. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

   8. associate-*r* N/A

      \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

   9. unpow2 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

   12. lower-*.f64 80.1

       \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

5. Simplified 80.1%

   \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + 1 \cdot re} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

   4. *-lft-identity N/A

      \[\leadsto \frac{1}{2} \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2} \cdot re, re\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{im}^{2} \cdot re}, re\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. lower-*.f64 53.6

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

8. Simplified 53.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left(im \cdot im\right) \cdot re, re\right)} \]

9. Final simplification 53.6%

   \[\leadsto \mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right) \]
10. Add Preprocessing

Alternative 27: 25.9% accurate, 19.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re \cdot \left(im\_m \cdot \left(0.5 \cdot im\_m\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* re (* im_m (* 0.5 im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return re * (im_m * (0.5 * im_m));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = re * (im_m * (0.5d0 * im_m))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return re * (im_m * (0.5 * im_m));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return re * (im_m * (0.5 * im_m))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(re * Float64(im_m * Float64(0.5 * im_m)))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = re * (im_m * (0.5 * im_m));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(re * N[(im$95$m * N[(0.5 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

   2. unpow2 N/A

      \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

   3. associate-*r* N/A

      \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

   4. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

   8. associate-*r* N/A

      \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

   9. unpow2 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

   12. lower-*.f64 80.1

       \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

5. Simplified 80.1%

   \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + 1 \cdot re} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

   4. *-lft-identity N/A

      \[\leadsto \frac{1}{2} \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2} \cdot re, re\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{im}^{2} \cdot re}, re\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. lower-*.f64 53.6

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

8. Simplified 53.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left(im \cdot im\right) \cdot re, re\right)} \]

9. Taylor expanded in im around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot {im}^{2}\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot {im}^{2}\right)} \]

    4. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

    5. lower-*.f64 26.5

       \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

11. Simplified 26.5%

    \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)} \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot re\right)} \]

    3. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot im\right)\right) \cdot re} \]

    4. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot im\right)\right) \cdot re} \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot re \]

    6. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot re \]

    7. *-commutative N/A

       \[\leadsto \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)} \cdot re \]

    8. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right)\right)} \cdot re \]

    9. *-commutative N/A

       \[\leadsto \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{2}\right)}\right) \cdot re \]

    10. lower-*.f64 26.5

        \[\leadsto \left(im \cdot \color{blue}{\left(im \cdot 0.5\right)}\right) \cdot re \]

13. Applied egg-rr 26.5%

    \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot 0.5\right)\right) \cdot re} \]

14. Final simplification 26.5%

    \[\leadsto re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right) \]
15. Add Preprocessing

Alternative 28: 25.9% accurate, 19.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.5 \cdot \left(re \cdot \left(im\_m \cdot im\_m\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* 0.5 (* re (* im_m im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.5 * (re * (im_m * im_m));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.5d0 * (re * (im_m * im_m))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.5 * (re * (im_m * im_m));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.5 * (re * (im_m * im_m))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(0.5 * Float64(re * Float64(im_m * im_m)))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.5 * (re * (im_m * im_m));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

   2. unpow2 N/A

      \[\leadsto \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

   3. associate-*r* N/A

      \[\leadsto \sin re + \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im\right)} \cdot \sin re \]

   4. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \cdot \sin re} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin re} \cdot \left(\left(\frac{1}{2} \cdot im\right) \cdot im + 1\right) \]

   8. associate-*r* N/A

      \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

   9. unpow2 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

   12. lower-*.f64 80.1

       \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

5. Simplified 80.1%

   \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + 1 \cdot re} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} + 1 \cdot re \]

   4. *-lft-identity N/A

      \[\leadsto \frac{1}{2} \cdot \left({im}^{2} \cdot re\right) + \color{blue}{re} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2} \cdot re, re\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{im}^{2} \cdot re}, re\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

   8. lower-*.f64 53.6

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(im \cdot im\right)} \cdot re, re\right) \]

8. Simplified 53.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left(im \cdot im\right) \cdot re, re\right)} \]

9. Taylor expanded in im around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot {im}^{2}\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot {im}^{2}\right)} \]

    4. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

    5. lower-*.f64 26.5

       \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

11. Simplified 26.5%

    \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))