math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.5s

Alternatives: 21

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. lift-exp.f64 N/A

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

   2. lift--.f64 N/A

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

   3. exp-diff N/A

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

   4. lift-exp.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. 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 rewrites 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

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

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 -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 = cosh(im_m) * fma(-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(cosh(im_m) * fma(-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(Float64(im_m * im_m) * fma(im_m, Float64(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[(-0.16666666666666666 * 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[(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]), $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-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      17. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 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-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 73.7

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

   7. Applied rewrites 73.7%

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

   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-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

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

   5. Applied rewrites 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

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

   5. Applied rewrites 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

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

   8. Applied rewrites 69.4%

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

4. Final simplification 87.0%

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

Alternative 3: 82.6% accurate, 0.4× speedup

Math

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

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))
     (*
      (fma
       (* im_m im_m)
       (fma
        im_m
        (* im_m (fma im_m (* im_m 0.001388888888888889) 0.041666666666666664))
        0.5)
       1.0)
      (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 = fma((im_m * im_m), fma(im_m, (im_m * fma(im_m, (im_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) * 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(fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(im_m, Float64(im_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) * 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(Float64(im_m * im_m) * fma(im_m, Float64(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[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $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[(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]), $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

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

   5. Applied rewrites 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 re around 0

      \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right)}, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 61.9%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right)}, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \left(re \cdot \color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \]

   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-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

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

   5. Applied rewrites 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

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

   5. Applied rewrites 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

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

   8. Applied rewrites 69.4%

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

4. Final simplification 84.4%

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

Alternative 4: 82.4% accurate, 0.4× speedup

Math

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

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))
     (*
      (fma
       (* im_m im_m)
       (fma
        im_m
        (* im_m (fma im_m (* im_m 0.001388888888888889) 0.041666666666666664))
        0.5)
       1.0)
      (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 = fma((im_m * im_m), fma(im_m, (im_m * fma(im_m, (im_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) * 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(fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(im_m, Float64(im_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) * 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(Float64(im_m * im_m) * fma(im_m, Float64(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[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $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[(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]), $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

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

   5. Applied rewrites 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 re around 0

      \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right)}, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 61.9%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right)}, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \left(re \cdot \color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \]

   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

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

      1. lower-sin.f64 99.1

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

   5. Applied rewrites 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

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

   5. Applied rewrites 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

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

   8. Applied rewrites 69.4%

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

4. Final simplification 84.1%

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

Alternative 5: 58.6% accurate, 0.8× speedup

Math

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

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)
   (*
    (fma
     (* im_m im_m)
     (fma
      im_m
      (* im_m (fma im_m (* im_m 0.001388888888888889) 0.041666666666666664))
      0.5)
     1.0)
    (fma
     (fma
      (* re re)
      (fma (* re re) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     (* re (* re re))
     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 * fma(im_m, (im_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) * fma(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), 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 * fma(im_m, Float64(im_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) * fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	else
		tmp = fma(re, Float64(Float64(im_m * im_m) * fma(im_m, Float64(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 * N[(im$95$m * N[(im$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $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], N[(re * 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]), $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

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right)}, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 66.1%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right)}, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 66.1%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \left(re \cdot \color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 66.1%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \]

   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

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

   5. Applied rewrites 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

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

   8. Applied rewrites 46.9%

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

4. Final simplification 59.5%

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

Alternative 6: 57.6% accurate, 0.8× speedup

Math

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

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)
   (*
    (fma (* im_m im_m) (fma (* im_m im_m) 0.041666666666666664 0.5) 1.0)
    (fma
     (fma
      (* re re)
      (fma (* re re) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     (* re (* re re))
     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(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), 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(Float64(im_m * im_m), 0.041666666666666664, 0.5), 1.0) * fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	else
		tmp = fma(re, Float64(Float64(im_m * im_m) * fma(im_m, Float64(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[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $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], N[(re * 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]), $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-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 65.0%

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

   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

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

   5. Applied rewrites 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

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

   8. Applied rewrites 46.9%

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

4. Final simplification 58.8%

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

Alternative 7: 58.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq 0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, t\_0, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im\_m \cdot im\_m\right) \cdot t\_0, re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0
         (fma
          im_m
          (*
           im_m
           (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664))
          0.5)))
   (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) 0.004)
     (*
      re
      (* (fma (* im_m im_m) t_0 1.0) (fma -0.16666666666666666 (* re re) 1.0)))
     (fma re (* (* im_m im_m) t_0) re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = fma(im_m, (im_m * fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664)), 0.5);
	double tmp;
	if (((0.5 * sin(re)) * (exp(im_m) + exp(-im_m))) <= 0.004) {
		tmp = re * (fma((im_m * im_m), t_0, 1.0) * fma(-0.16666666666666666, (re * re), 1.0));
	} else {
		tmp = fma(re, ((im_m * im_m) * t_0), re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664)), 0.5)
	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(im_m * im_m), t_0, 1.0) * fma(-0.16666666666666666, Float64(re * re), 1.0)));
	else
		tmp = fma(re, Float64(Float64(im_m * im_m) * t_0), re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = 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]}, 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[(im$95$m * im$95$m), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * t$95$0), $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

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.1%

      \[\leadsto re \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)} \]

   if 0.0040000000000000001 < (*.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

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

   5. Applied rewrites 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

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

   8. Applied rewrites 46.9%

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

4. Final simplification 59.5%

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

Alternative 8: 57.4% accurate, 0.9× speedup

Math

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

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)
   (*
    (fma -0.16666666666666666 (* re (* re re)) re)
    (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 = fma(-0.16666666666666666, (re * (re * re)), re) * 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(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = fma(re, Float64(Float64(im_m * im_m) * fma(im_m, Float64(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[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * 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]), $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-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 65.0%

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

   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

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

   5. Applied rewrites 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

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

   8. Applied rewrites 46.9%

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

4. Final simplification 58.8%

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

Alternative 9: 47.2% accurate, 0.9× speedup

Math

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

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)
   (fma
    (fma
     (* re re)
     (fma (* re re) -0.0001984126984126984 0.008333333333333333)
     -0.16666666666666666)
    (* re (* re re))
    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(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), 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(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re);
	else
		tmp = fma(re, Float64(Float64(im_m * im_m) * fma(im_m, Float64(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[(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], N[(re * 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]), $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. Applied rewrites 66.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 52.6%

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

   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

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

   5. Applied rewrites 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

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

   8. Applied rewrites 46.9%

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

4. Final simplification 50.7%

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

Alternative 10: 44.5% accurate, 0.9× speedup

Math

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

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)
   (fma
    (fma
     (* re re)
     (fma (* re re) -0.0001984126984126984 0.008333333333333333)
     -0.16666666666666666)
    (* re (* re re))
    re)
   (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))) <= 0.004) {
		tmp = fma(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	} 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)))) <= 0.004)
		tmp = fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re);
	else
		tmp = fma(Float64(im_m * im_m), Float64(re * fma(Float64(im_m * 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], 0.004], 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], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $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. Applied rewrites 66.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 52.6%

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

   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-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.4%

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

4. Final simplification 48.4%

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

Alternative 11: 51.8% accurate, 0.9× speedup

Math

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

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 -0.16666666666666666 (* re re) 1.0) (fma im_m (* 0.5 im_m) 1.0)))
   (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))) <= 0.004) {
		tmp = re * (fma(-0.16666666666666666, (re * re), 1.0) * fma(im_m, (0.5 * im_m), 1.0));
	} 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)))) <= 0.004)
		tmp = Float64(re * Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * fma(im_m, Float64(0.5 * im_m), 1.0)));
	else
		tmp = fma(Float64(im_m * im_m), Float64(re * fma(Float64(im_m * 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], 0.004], N[(re * N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * N[(0.5 * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $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-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 83.3

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

   5. Applied rewrites 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

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

   8. Applied rewrites 61.5%

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

   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-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.4%

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

4. Final simplification 54.2%

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

Alternative 12: 44.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + e^{-im\_m}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(0.5, im\_m \cdot im\_m, 1\right) \cdot \left(-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) -0.02)
   (* (fma 0.5 (* im_m im_m) 1.0) (* -0.16666666666666666 (* re (* re re))))
   (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 = fma(0.5, (im_m * im_m), 1.0) * (-0.16666666666666666 * (re * (re * re)));
	} 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(fma(0.5, Float64(im_m * im_m), 1.0) * Float64(-0.16666666666666666 * Float64(re * Float64(re * re))));
	else
		tmp = fma(Float64(im_m * im_m), Float64(re * fma(Float64(im_m * 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[(N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $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-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 69.9

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

   5. Applied rewrites 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

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

   8. Applied rewrites 30.6%

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

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {re}^{3}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 15.2%

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

   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

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

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.3%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \left(-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.7% accurate, 0.9× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im_m) (exp (- im_m)))) -0.02)
   (* (* -0.16666666666666666 (* re (* re re))) (* 0.5 (* im_m im_m)))
   (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 = (-0.16666666666666666 * (re * (re * re))) * (0.5 * (im_m * im_m));
	} 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(Float64(-0.16666666666666666 * Float64(re * Float64(re * re))) * Float64(0.5 * Float64(im_m * im_m)));
	else
		tmp = fma(Float64(im_m * im_m), Float64(re * fma(Float64(im_m * 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[(N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $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-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 69.9

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

   5. Applied rewrites 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

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

   8. Applied rewrites 30.6%

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

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {re}^{3}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 15.2%

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

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 14.8%

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

   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

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

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.3%

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

4. Final simplification 48.2%

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

Alternative 14: 43.4% accurate, 0.9× speedup

Math

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

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)
   (fma -0.16666666666666666 (* re (* re re)) re)
   (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))) <= 0.004) {
		tmp = fma(-0.16666666666666666, (re * (re * re)), re);
	} 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)))) <= 0.004)
		tmp = fma(-0.16666666666666666, Float64(re * Float64(re * re)), re);
	else
		tmp = fma(Float64(im_m * im_m), Float64(re * fma(Float64(im_m * 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], 0.004], N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $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. Applied rewrites 66.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.8%

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

   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-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.4%

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

4. Final simplification 46.6%

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

Alternative 15: 41.1% accurate, 0.9× speedup

Math

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

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 -0.16666666666666666 (* re (* re re)) 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(-0.16666666666666666, (re * (re * re)), 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(-0.16666666666666666, Float64(re * Float64(re * re)), 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[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $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. Applied rewrites 66.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.8%

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

   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-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 73.9

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

   5. Applied rewrites 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

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

   8. Applied rewrites 38.1%

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

4. Final simplification 45.8%

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

Alternative 16: 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-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub0-neg N/A

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

   13. cosh-undef N/A

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

   14. associate-*r* N/A

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

   15. metadata-eval N/A

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

   16. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   17. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   18. exp-0 N/A

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

   19. lower-cosh.f64 100.0

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

4. Applied rewrites 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 17: 97.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right)\\ \mathbf{if}\;im\_m \leq 0.88:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot t\_0, 0.5\right), 1\right)\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(t\_0 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)))
   (if (<= im_m 0.88)
     (* (sin re) (fma (* im_m im_m) (fma im_m (* im_m t_0) 0.5) 1.0))
     (if (<= im_m 7.2e+51)
       (* (cosh im_m) (fma -0.16666666666666666 (* re (* re re)) re))
       (* (sin re) (* t_0 (* (* im_m im_m) (* im_m im_m))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if (im_m <= 0.88) {
		tmp = sin(re) * fma((im_m * im_m), fma(im_m, (im_m * t_0), 0.5), 1.0);
	} else if (im_m <= 7.2e+51) {
		tmp = cosh(im_m) * fma(-0.16666666666666666, (re * (re * re)), re);
	} else {
		tmp = sin(re) * (t_0 * ((im_m * im_m) * (im_m * im_m)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (im_m <= 0.88)
		tmp = Float64(sin(re) * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * t_0), 0.5), 1.0));
	elseif (im_m <= 7.2e+51)
		tmp = Float64(cosh(im_m) * fma(-0.16666666666666666, Float64(re * Float64(re * re)), re));
	else
		tmp = Float64(sin(re) * Float64(t_0 * 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_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[im$95$m, 0.88], N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * t$95$0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 7.2e+51], N[(N[Cosh[im$95$m], $MachinePrecision] * N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $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. Taylor expanded in im around 0

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 95.4%

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

   if 0.880000000000000004 < im < 7.20000000000000022e51

   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-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      17. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 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-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 90.0

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

   7. Applied rewrites 90.0%

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

   if 7.20000000000000022e51 < 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

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

   5. Applied rewrites 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

      \[\leadsto \sin re \cdot \left({im}^{6} \cdot \color{blue}{\left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)}\right) \]

   8. Applied rewrites 100.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 97.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 0.225:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \]

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 7.2e+51)
     (* (cosh im_m) (fma -0.16666666666666666 (* re (* re re)) re))
     (*
      (sin re)
      (*
       (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
       (* (* 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 <= 7.2e+51) {
		tmp = cosh(im_m) * fma(-0.16666666666666666, (re * (re * re)), re);
	} else {
		tmp = sin(re) * (fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664) * ((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(Float64(im_m * im_m), 0.041666666666666664, 0.5), 1.0));
	elseif (im_m <= 7.2e+51)
		tmp = Float64(cosh(im_m) * fma(-0.16666666666666666, Float64(re * Float64(re * re)), re));
	else
		tmp = Float64(sin(re) * Float64(fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664) * 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[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 7.2e+51], N[(N[Cosh[im$95$m], $MachinePrecision] * N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $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-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 92.8%

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

   if 0.225000000000000006 < im < 7.20000000000000022e51

   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-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      17. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 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-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 90.0

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

   7. Applied rewrites 90.0%

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

   if 7.20000000000000022e51 < 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

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

   5. Applied rewrites 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

      \[\leadsto \sin re \cdot \left({im}^{6} \cdot \color{blue}{\left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)}\right) \]

   8. Applied rewrites 100.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.225:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 96.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{if}\;im\_m \leq 0.225:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;\cosh im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -0.16666666666666666 (* re (* re re)) 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(-0.16666666666666666, (re * (re * re)), 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(Float64(im_m * 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(-0.16666666666666666, Float64(re * Float64(re * re)), 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[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 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[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $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-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 94.4%

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

   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-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      17. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 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-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 86.7

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

   7. Applied rewrites 86.7%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.225:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 34.0% accurate, 18.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (fma -0.16666666666666666 (* re (* re re)) re))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return fma(-0.16666666666666666, (re * (re * re)), re);
}

Julia

im_m = abs(im)
function code(re, im_m)
	return fma(-0.16666666666666666, Float64(re * Float64(re * re)), re)
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(-0.16666666666666666 * N[(re * N[(re * re), $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

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

   1. lower-sin.f64 55.8

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

5. Applied rewrites 55.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 37.5%

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

Alternative 21: 10.2% accurate, 19.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ -0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* -0.16666666666666666 (* re (* re re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return -0.16666666666666666 * (re * (re * re));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = (-0.16666666666666666d0) * (re * (re * re))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return -0.16666666666666666 * (re * (re * re));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return -0.16666666666666666 * (re * (re * re))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(-0.16666666666666666 * Float64(re * Float64(re * re)))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = -0.16666666666666666 * (re * (re * re));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(-0.16666666666666666 * N[(re * N[(re * re), $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

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

   1. lower-sin.f64 55.8

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

5. Applied rewrites 55.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 37.5%

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

9. Taylor expanded in re around inf

   \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
10. Step-by-step derivation

11. Applied rewrites 9.6%

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