math.cos on complex, real part

Percentage Accurate: 100.0% → 99.2%

Time: 8.2s

Alternatives: 16

Speedup: 1.0×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)) (t_1 (* (+ (exp im_m) (exp (- im_m))) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma im_m im_m 2.0)
      (fma
       (*
        (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
        re)
       (* re re)
       0.5))
     (if (<= t_1 2.0)
       (* (fma im_m im_m 2.0) t_0)
       (* 0.5 (+ 1.0 (exp im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = cos(re) * 0.5;
	double t_1 = (exp(im_m) + exp(-im_m)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(im_m, im_m, 2.0) * fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re), (re * re), 0.5);
	} else if (t_1 <= 2.0) {
		tmp = fma(im_m, im_m, 2.0) * t_0;
	} else {
		tmp = 0.5 * (1.0 + exp(im_m));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(cos(re) * 0.5)
	t_1 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(im_m, im_m, 2.0) * fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re), Float64(re * re), 0.5));
	elseif (t_1 <= 2.0)
		tmp = Float64(fma(im_m, im_m, 2.0) * t_0);
	else
		tmp = Float64(0.5 * Float64(1.0 + exp(im_m)));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(0.5 * N[(1.0 + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 3.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot 2 \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot 2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot 2 \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot 2 \]

      15. lower-*.f64 86.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot 2 \]

   8. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot 2 \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left({re}^{4} \cdot \left(\frac{1}{48} \cdot \frac{1}{{re}^{2}} - \frac{1}{1440}\right), \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot 2 \]
   10. Step-by-step derivation

   11. Applied rewrites 86.0%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, \color{blue}{re} \cdot re, 0.5\right) \cdot 2 \]

   12. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

       3. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   14. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 51.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(1 + e^{im}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma im_m im_m 2.0)
      (fma
       (*
        (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
        re)
       (* re re)
       0.5))
     (if (<= t_0 2.0) (cos re) (* 0.5 (+ 1.0 (exp im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im_m, im_m, 2.0) * fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re), (re * re), 0.5);
	} else if (t_0 <= 2.0) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (1.0 + exp(im_m));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im_m, im_m, 2.0) * fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re), Float64(re * re), 0.5));
	elseif (t_0 <= 2.0)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(1.0 + exp(im_m)));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[Cos[re], $MachinePrecision], N[(0.5 * N[(1.0 + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 3.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot 2 \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot 2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot 2 \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot 2 \]

      15. lower-*.f64 86.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot 2 \]

   8. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot 2 \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left({re}^{4} \cdot \left(\frac{1}{48} \cdot \frac{1}{{re}^{2}} - \frac{1}{1440}\right), \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot 2 \]
   10. Step-by-step derivation

   11. Applied rewrites 86.0%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, \color{blue}{re} \cdot re, 0.5\right) \cdot 2 \]

   12. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

       3. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   14. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 99.6

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

   8. Applied rewrites 99.6%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 51.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(1 + e^{im}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im\_m, 0.5\right), im\_m, 1\right), im\_m, 1\right) + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma im_m im_m 2.0)
      (fma
       (*
        (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
        re)
       (* re re)
       0.5))
     (if (<= t_0 2.0)
       (cos re)
       (*
        (+
         (fma (fma (fma 0.16666666666666666 im_m 0.5) im_m 1.0) im_m 1.0)
         1.0)
        0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im_m, im_m, 2.0) * fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re), (re * re), 0.5);
	} else if (t_0 <= 2.0) {
		tmp = cos(re);
	} else {
		tmp = (fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im_m, im_m, 2.0) * fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re), Float64(re * re), 0.5));
	elseif (t_0 <= 2.0)
		tmp = cos(re);
	else
		tmp = Float64(Float64(fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[Cos[re], $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * im$95$m + 0.5), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 3.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot 2 \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot 2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot 2 \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot 2 \]

      15. lower-*.f64 86.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot 2 \]

   8. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot 2 \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left({re}^{4} \cdot \left(\frac{1}{48} \cdot \frac{1}{{re}^{2}} - \frac{1}{1440}\right), \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot 2 \]
   10. Step-by-step derivation

   11. Applied rewrites 86.0%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, \color{blue}{re} \cdot re, 0.5\right) \cdot 2 \]

   12. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

       3. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   14. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 99.6

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

   8. Applied rewrites 99.6%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 51.6%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 32.7

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

   11. Applied rewrites 32.7%

       \[\leadsto 0.5 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5))))
   (if (<= t_0 -0.2)
     (fma -0.5 (* re re) 1.0)
     (if (<= t_0 2.0) 1.0 (* (* im_m im_m) 0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = fma(-0.5, (re * re), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (im_m * im_m) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = fma(-0.5, Float64(re * re), 1.0);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(im_m * im_m) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.2], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 0.8%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 50.8

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

   8. Applied rewrites 50.8%

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

   9. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.3%

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

   if -0.20000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in re around 0

      \[\leadsto 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 67.9%

       \[\leadsto 1 \]

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 44.1

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 44.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.1%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.1%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im\_m, 0.5\right), im\_m, 1\right), im\_m, 1\right) + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.2)
   (*
    (fma
     (fma
      (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
      (* re re)
      -0.25)
     (* re re)
     0.5)
    (fma im_m im_m 2.0))
   (*
    (+ (fma (fma (fma 0.16666666666666666 im_m 0.5) im_m 1.0) im_m 1.0) 1.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= -0.2) {
		tmp = fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5) * fma(im_m, im_m, 2.0);
	} else {
		tmp = (fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= -0.2)
		tmp = Float64(fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * im$95$m + 0.5), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.8

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 71.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      15. lower-*.f64 52.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.20000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.3%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 50.6

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

   11. Applied rewrites 50.6%

       \[\leadsto 0.5 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im\_m, 0.5\right), im\_m, 1\right), im\_m, 1\right) + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.2)
   (*
    (fma im_m im_m 2.0)
    (fma
     (* (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re) re)
     (* re re)
     0.5))
   (*
    (+ (fma (fma (fma 0.16666666666666666 im_m 0.5) im_m 1.0) im_m 1.0) 1.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= -0.2) {
		tmp = fma(im_m, im_m, 2.0) * fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re), (re * re), 0.5);
	} else {
		tmp = (fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= -0.2)
		tmp = Float64(fma(im_m, im_m, 2.0) * fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re), Float64(re * re), 0.5));
	else
		tmp = Float64(Float64(fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * im$95$m + 0.5), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot 2 \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot 2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot 2 \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot 2 \]

      15. lower-*.f64 45.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot 2 \]

   8. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot 2 \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left({re}^{4} \cdot \left(\frac{1}{48} \cdot \frac{1}{{re}^{2}} - \frac{1}{1440}\right), \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot 2 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.4%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, \color{blue}{re} \cdot re, 0.5\right) \cdot 2 \]

   12. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re, re \cdot re, \frac{1}{2}\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

       3. lower-fma.f64 52.5

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   14. Applied rewrites 52.5%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if -0.20000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.3%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 50.6

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

   11. Applied rewrites 50.6%

       \[\leadsto 0.5 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im\_m, 0.5\right), im\_m, 1\right), im\_m, 1\right) + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.2)
   (fma
    (fma
     (fma -0.001388888888888889 (* re re) 0.041666666666666664)
     (* re re)
     -0.5)
    (* re re)
    1.0)
   (*
    (+ (fma (fma (fma 0.16666666666666666 im_m 0.5) im_m 1.0) im_m 1.0) 1.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= -0.2) {
		tmp = fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0);
	} else {
		tmp = (fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= -0.2)
		tmp = fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0);
	else
		tmp = Float64(Float64(fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * im$95$m + 0.5), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 0.8%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 50.8

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

   8. Applied rewrites 50.8%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 45.7%

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

   if -0.20000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.3%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 50.6

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

   11. Applied rewrites 50.6%

       \[\leadsto 0.5 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im\_m, 0.5\right), im\_m, 1\right), im\_m, 1\right) + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.2)
   (* (fma (* re re) -0.25 0.5) (fma im_m im_m 2.0))
   (*
    (+ (fma (fma (fma 0.16666666666666666 im_m 0.5) im_m 1.0) im_m 1.0) 1.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= -0.2) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im_m, im_m, 2.0);
	} else {
		tmp = (fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= -0.2)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.16666666666666666, im_m, 0.5), im_m, 1.0), im_m, 1.0) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * im$95$m + 0.5), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.8

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 71.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 43.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.20000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.3%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 50.6

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

   11. Applied rewrites 50.6%

       \[\leadsto 0.5 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) 2.0)
   1.0
   (* (* im_m im_m) 0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (im_m * im_m) * 0.5;
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5d0)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (im_m * im_m) * 0.5d0
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (((Math.exp(im_m) + Math.exp(-im_m)) * (Math.cos(re) * 0.5)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (im_m * im_m) * 0.5;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if ((math.exp(im_m) + math.exp(-im_m)) * (math.cos(re) * 0.5)) <= 2.0:
		tmp = 1.0
	else:
		tmp = (im_m * im_m) * 0.5
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(im_m * im_m) * 0.5);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= 2.0)
		tmp = 1.0;
	else
		tmp = (im_m * im_m) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], 1.0, N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 41.2%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 80.1

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

   8. Applied rewrites 80.1%

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

   9. Taylor expanded in re around 0

      \[\leadsto 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 41.2%

       \[\leadsto 1 \]

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 44.1

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 44.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.1%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.1%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 100.0% accurate, 1.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return (exp(im_m) + exp(-im_m)) * (cos(re) * 0.5);
}

Fortran

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

Java

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

Python

im_m = math.fabs(im)
def code(re, im_m):
	return (math.exp(im_m) + math.exp(-im_m)) * (math.cos(re) * 0.5)

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = (exp(im_m) + exp(-im_m)) * (cos(re) * 0.5);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \]
4. Add Preprocessing

Alternative 11: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.87:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, -0.5\right), re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.02)
   (* (fma (* re re) -0.25 0.5) (fma im_m im_m 2.0))
   (if (<= (cos re) 0.87)
     (fma (fma 0.041666666666666664 (* re re) -0.5) (* re re) 1.0)
     (* 0.5 (fma im_m im_m 2.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im_m, im_m, 2.0);
	} else if (cos(re) <= 0.87) {
		tmp = fma(fma(0.041666666666666664, (re * re), -0.5), (re * re), 1.0);
	} else {
		tmp = 0.5 * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im_m, im_m, 2.0));
	elseif (cos(re) <= 0.87)
		tmp = fma(fma(0.041666666666666664, Float64(re * re), -0.5), Float64(re * re), 1.0);
	else
		tmp = Float64(0.5 * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.87], N[(N[(0.041666666666666664 * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.8

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 71.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 43.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.0200000000000000004 < (cos.f64 re) < 0.869999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 52.9

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 40.2%

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

   if 0.869999999999999996 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.1

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 75.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 64.0%

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

Alternative 12: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.87:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, -0.5\right), re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.02)
   (* (* (* re re) -0.25) (fma im_m im_m 2.0))
   (if (<= (cos re) 0.87)
     (fma (fma 0.041666666666666664 (* re re) -0.5) (* re re) 1.0)
     (* 0.5 (fma im_m im_m 2.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = ((re * re) * -0.25) * fma(im_m, im_m, 2.0);
	} else if (cos(re) <= 0.87) {
		tmp = fma(fma(0.041666666666666664, (re * re), -0.5), (re * re), 1.0);
	} else {
		tmp = 0.5 * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(Float64(Float64(re * re) * -0.25) * fma(im_m, im_m, 2.0));
	elseif (cos(re) <= 0.87)
		tmp = fma(fma(0.041666666666666664, Float64(re * re), -0.5), Float64(re * re), 1.0);
	else
		tmp = Float64(0.5 * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.87], N[(N[(0.041666666666666664 * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.8

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 71.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 43.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 43.3%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{-0.25}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.0200000000000000004 < (cos.f64 re) < 0.869999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 52.9

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 40.2%

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

   if 0.869999999999999996 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.1

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 75.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 64.0%

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

Alternative 13: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Applied rewrites 73.2%

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

6. Final simplification 73.2%

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

Alternative 14: 58.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.02)
   (* (* (* re re) -0.25) (fma im_m im_m 2.0))
   (* 0.5 (fma im_m im_m 2.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = ((re * re) * -0.25) * fma(im_m, im_m, 2.0);
	} else {
		tmp = 0.5 * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(Float64(Float64(re * re) * -0.25) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(0.5 * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.8

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 71.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 43.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 43.3%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{-0.25}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.0

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 74.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.9%

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

Alternative 15: 54.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.02) (fma -0.5 (* re re) 1.0) (* 0.5 (fma im_m im_m 2.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(-0.5, (re * re), 1.0);
	} else {
		tmp = 0.5 * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = fma(-0.5, Float64(re * re), 1.0);
	else
		tmp = Float64(0.5 * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 0.8%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 50.8

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

   8. Applied rewrites 50.8%

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

   9. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.3%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.0

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 74.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.9%

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

Alternative 16: 28.4% accurate, 316.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 1 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 1.0)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 1.0;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 1.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 1.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 1.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 1.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 1.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

5. Applied rewrites 61.4%

   \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\cos re} \]
7. Step-by-step derivation

   1. lower-cos.f64 53.6

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

8. Applied rewrites 53.6%

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

9. Taylor expanded in re around 0

   \[\leadsto 1 \]
10. Step-by-step derivation

11. Applied rewrites 28.1%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024244 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))