math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 2.4s

Alternatives: 8

Speedup: 1.3×

• 50.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\sin re \cdot \cosh im \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (sin re) (cosh im)))

C

double code(double re, double im) {
	return sin(re) * cosh(im);
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * cosh(im)
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * Math.cosh(im);
}

Python

def code(re, im):
	return math.sin(re) * math.cosh(im)

Julia

function code(re, im)
	return Float64(sin(re) * cosh(im))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * cosh(im);
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

      \[\leadsto \sin re \cdot \color{blue}{\frac{e^{0 - im} + e^{im}}{2}} \]

   8. lift-+.f64 N/A

      \[\leadsto \sin re \cdot \frac{\color{blue}{e^{0 - im} + e^{im}}}{2} \]

   9. +-commutative N/A

      \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im} + e^{0 - im}}}{2} \]

   10. lift-exp.f64 N/A

       \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im}} + e^{0 - im}}{2} \]

   11. lift-exp.f64 N/A

       \[\leadsto \sin re \cdot \frac{e^{im} + \color{blue}{e^{0 - im}}}{2} \]

   12. lift--.f64 N/A

       \[\leadsto \sin re \cdot \frac{e^{im} + e^{\color{blue}{0 - im}}}{2} \]

   13. sub0-neg N/A

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

   14. cosh-def N/A

       \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

   15. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   16. lower-cosh.f64 100.0%

       \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
4. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|re\right|\right)\\ t_1 := \left(0.5 \cdot t\_0\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left|re\right| \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left|re\right|, \left|re\right|, 1\right)\right) \cdot \cosh im\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|re\right| \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sin (fabs re)))
       (t_1 (* (* 0.5 t_0) (+ (exp (- 0.0 im)) (exp im)))))
  (*
   (copysign 1.0 re)
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fabs re)
       (fma (* -0.16666666666666666 (fabs re)) (fabs re) 1.0))
      (cosh im))
     (if (<= t_1 1.0)
       (* (fma (* im im) 0.5 1.0) t_0)
       (* (fabs re) (cosh im)))))))

C

double code(double re, double im) {
	double t_0 = sin(fabs(re));
	double t_1 = (0.5 * t_0) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fabs(re) * fma((-0.16666666666666666 * fabs(re)), fabs(re), 1.0)) * cosh(im);
	} else if (t_1 <= 1.0) {
		tmp = fma((im * im), 0.5, 1.0) * t_0;
	} else {
		tmp = fabs(re) * cosh(im);
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	t_0 = sin(abs(re))
	t_1 = Float64(Float64(0.5 * t_0) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(abs(re) * fma(Float64(-0.16666666666666666 * abs(re)), abs(re), 1.0)) * cosh(im));
	elseif (t_1 <= 1.0)
		tmp = Float64(fma(Float64(im * im), 0.5, 1.0) * t_0);
	else
		tmp = Float64(abs(re) * cosh(im));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[Abs[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Abs[re], $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

         \[\leadsto \sin re \cdot \color{blue}{\frac{e^{0 - im} + e^{im}}{2}} \]

      8. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \frac{\color{blue}{e^{0 - im} + e^{im}}}{2} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im} + e^{0 - im}}}{2} \]

      10. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im}} + e^{0 - im}}{2} \]

      11. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \frac{e^{im} + \color{blue}{e^{0 - im}}}{2} \]

      12. lift--.f64 N/A

          \[\leadsto \sin re \cdot \frac{e^{im} + e^{\color{blue}{0 - im}}}{2} \]

      13. sub0-neg N/A

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

      14. cosh-def N/A

          \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      16. lower-cosh.f64 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   4. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 63.2%

         \[\leadsto \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot \cosh im \]

   6. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \cdot \cosh im \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 63.2%

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

   8. Applied rewrites 63.2%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 76.0%

         \[\leadsto \sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right) \]

   4. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 76.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, 0.5, 1\right) \cdot \sin \color{blue}{re} \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.0%

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

   6. Applied rewrites 76.0%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.5%

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

   4. Applied rewrites 63.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{\color{blue}{2}} \]

      7. lift-+.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      8. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      9. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      10. lift-neg.f64 N/A

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

      11. cosh-def N/A

          \[\leadsto re \cdot \cosh im \]

      12. lift-cosh.f64 N/A

          \[\leadsto re \cdot \cosh im \]

      13. lower-*.f64 63.5%

          \[\leadsto re \cdot \color{blue}{\cosh im} \]

   6. Applied rewrites 63.5%

      \[\leadsto re \cdot \color{blue}{\cosh im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \sin \left(\left|re\right|\right)\\ t_1 := t\_0 \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left|re\right| \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left|re\right|, \left|re\right|, 1\right)\right) \cdot \cosh im\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left|re\right| \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* 0.5 (sin (fabs re))))
       (t_1 (* t_0 (+ (exp (- 0.0 im)) (exp im)))))
  (*
   (copysign 1.0 re)
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fabs re)
       (fma (* -0.16666666666666666 (fabs re)) (fabs re) 1.0))
      (cosh im))
     (if (<= t_1 1.0) (* t_0 2.0) (* (fabs re) (cosh im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(fabs(re));
	double t_1 = t_0 * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fabs(re) * fma((-0.16666666666666666 * fabs(re)), fabs(re), 1.0)) * cosh(im);
	} else if (t_1 <= 1.0) {
		tmp = t_0 * 2.0;
	} else {
		tmp = fabs(re) * cosh(im);
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(abs(re)))
	t_1 = Float64(t_0 * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(abs(re) * fma(Float64(-0.16666666666666666 * abs(re)), abs(re), 1.0)) * cosh(im));
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * 2.0);
	else
		tmp = Float64(abs(re) * cosh(im));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[Abs[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * 2.0), $MachinePrecision], N[(N[Abs[re], $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

         \[\leadsto \sin re \cdot \color{blue}{\frac{e^{0 - im} + e^{im}}{2}} \]

      8. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \frac{\color{blue}{e^{0 - im} + e^{im}}}{2} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im} + e^{0 - im}}}{2} \]

      10. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im}} + e^{0 - im}}{2} \]

      11. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \frac{e^{im} + \color{blue}{e^{0 - im}}}{2} \]

      12. lift--.f64 N/A

          \[\leadsto \sin re \cdot \frac{e^{im} + e^{\color{blue}{0 - im}}}{2} \]

      13. sub0-neg N/A

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

      14. cosh-def N/A

          \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      16. lower-cosh.f64 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   4. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 63.2%

         \[\leadsto \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot \cosh im \]

   6. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \cdot \cosh im \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 63.2%

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

   8. Applied rewrites 63.2%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.1%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.5%

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

   4. Applied rewrites 63.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{\color{blue}{2}} \]

      7. lift-+.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      8. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      9. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      10. lift-neg.f64 N/A

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

      11. cosh-def N/A

          \[\leadsto re \cdot \cosh im \]

      12. lift-cosh.f64 N/A

          \[\leadsto re \cdot \cosh im \]

      13. lower-*.f64 63.5%

          \[\leadsto re \cdot \color{blue}{\cosh im} \]

   6. Applied rewrites 63.5%

      \[\leadsto re \cdot \color{blue}{\cosh im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.0% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\left|re\right| \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left|re\right|, \left|re\right|, 1\right)\right) \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\left|re\right| \cdot \cosh im\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<=
      (* (* 0.5 (sin (fabs re))) (+ (exp (- 0.0 im)) (exp im)))
      0.0002)
   (*
    (*
     (fabs re)
     (fma (* -0.16666666666666666 (fabs re)) (fabs re) 1.0))
    (cosh im))
   (* (fabs re) (cosh im)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp((0.0 - im)) + exp(im))) <= 0.0002) {
		tmp = (fabs(re) * fma((-0.16666666666666666 * fabs(re)), fabs(re), 1.0)) * cosh(im);
	} else {
		tmp = fabs(re) * cosh(im);
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0002)
		tmp = Float64(Float64(abs(re) * fma(Float64(-0.16666666666666666 * abs(re)), abs(re), 1.0)) * cosh(im));
	else
		tmp = Float64(abs(re) * cosh(im));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[Abs[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Abs[re], $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

         \[\leadsto \sin re \cdot \color{blue}{\frac{e^{0 - im} + e^{im}}{2}} \]

      8. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \frac{\color{blue}{e^{0 - im} + e^{im}}}{2} \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im} + e^{0 - im}}}{2} \]

      10. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im}} + e^{0 - im}}{2} \]

      11. lift-exp.f64 N/A

          \[\leadsto \sin re \cdot \frac{e^{im} + \color{blue}{e^{0 - im}}}{2} \]

      12. lift--.f64 N/A

          \[\leadsto \sin re \cdot \frac{e^{im} + e^{\color{blue}{0 - im}}}{2} \]

      13. sub0-neg N/A

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

      14. cosh-def N/A

          \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      16. lower-cosh.f64 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   4. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 63.2%

         \[\leadsto \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot \cosh im \]

   6. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \cdot \cosh im \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 63.2%

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

   8. Applied rewrites 63.2%

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

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.5%

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

   4. Applied rewrites 63.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{\color{blue}{2}} \]

      7. lift-+.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      8. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      9. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      10. lift-neg.f64 N/A

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

      11. cosh-def N/A

          \[\leadsto re \cdot \cosh im \]

      12. lift-cosh.f64 N/A

          \[\leadsto re \cdot \cosh im \]

      13. lower-*.f64 63.5%

          \[\leadsto re \cdot \color{blue}{\cosh im} \]

   6. Applied rewrites 63.5%

      \[\leadsto re \cdot \color{blue}{\cosh im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.8% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{0 - \left|im\right|} + e^{\left|im\right|}\right) \leq -0.75:\\ \;\;\;\;0.5 \cdot \left(\left|re\right| \cdot \left(1 + \left(1 + \left|im\right| \cdot \left(\left|im\right| \cdot \left(0.5 + -0.16666666666666666 \cdot \left|im\right|\right) - 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|re\right| \cdot \cosh \left(\left|im\right|\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<=
      (*
       (* 0.5 (sin (fabs re)))
       (+ (exp (- 0.0 (fabs im))) (exp (fabs im))))
      -0.75)
   (*
    0.5
    (*
     (fabs re)
     (+
      1.0
      (+
       1.0
       (*
        (fabs im)
        (-
         (* (fabs im) (+ 0.5 (* -0.16666666666666666 (fabs im))))
         1.0))))))
   (* (fabs re) (cosh (fabs im))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp((0.0 - fabs(im))) + exp(fabs(im)))) <= -0.75) {
		tmp = 0.5 * (fabs(re) * (1.0 + (1.0 + (fabs(im) * ((fabs(im) * (0.5 + (-0.16666666666666666 * fabs(im)))) - 1.0)))));
	} else {
		tmp = fabs(re) * cosh(fabs(im));
	}
	return copysign(1.0, re) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(Math.abs(re))) * (Math.exp((0.0 - Math.abs(im))) + Math.exp(Math.abs(im)))) <= -0.75) {
		tmp = 0.5 * (Math.abs(re) * (1.0 + (1.0 + (Math.abs(im) * ((Math.abs(im) * (0.5 + (-0.16666666666666666 * Math.abs(im)))) - 1.0)))));
	} else {
		tmp = Math.abs(re) * Math.cosh(Math.abs(im));
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(math.fabs(re))) * (math.exp((0.0 - math.fabs(im))) + math.exp(math.fabs(im)))) <= -0.75:
		tmp = 0.5 * (math.fabs(re) * (1.0 + (1.0 + (math.fabs(im) * ((math.fabs(im) * (0.5 + (-0.16666666666666666 * math.fabs(im)))) - 1.0)))))
	else:
		tmp = math.fabs(re) * math.cosh(math.fabs(im))
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(Float64(0.0 - abs(im))) + exp(abs(im)))) <= -0.75)
		tmp = Float64(0.5 * Float64(abs(re) * Float64(1.0 + Float64(1.0 + Float64(abs(im) * Float64(Float64(abs(im) * Float64(0.5 + Float64(-0.16666666666666666 * abs(im)))) - 1.0))))));
	else
		tmp = Float64(abs(re) * cosh(abs(im)));
	end
	return Float64(copysign(1.0, re) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(abs(re))) * (exp((0.0 - abs(im))) + exp(abs(im)))) <= -0.75)
		tmp = 0.5 * (abs(re) * (1.0 + (1.0 + (abs(im) * ((abs(im) * (0.5 + (-0.16666666666666666 * abs(im)))) - 1.0)))));
	else
		tmp = abs(re) * cosh(abs(im));
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.75], N[(0.5 * N[(N[Abs[re], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[Abs[im], $MachinePrecision] * N[(N[(N[Abs[im], $MachinePrecision] * N[(0.5 + N[(-0.16666666666666666 * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[re], $MachinePrecision] * N[Cosh[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.75

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.5%

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

   4. Applied rewrites 63.5%

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

   5. Taylor expanded in im around 0

      \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + e^{\color{blue}{-im}}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 44.6%

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

   8. Taylor expanded in im around 0

      \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot im\right) - 1\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 44.4%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + im \cdot \left(im \cdot \left(0.5 + -0.16666666666666666 \cdot im\right) - 1\right)\right)\right)\right) \]

   10. Applied rewrites 44.4%

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

   if -0.75 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.5%

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

   4. Applied rewrites 63.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{\color{blue}{2}} \]

      7. lift-+.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      8. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      9. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      10. lift-neg.f64 N/A

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

      11. cosh-def N/A

          \[\leadsto re \cdot \cosh im \]

      12. lift-cosh.f64 N/A

          \[\leadsto re \cdot \cosh im \]

      13. lower-*.f64 63.5%

          \[\leadsto re \cdot \color{blue}{\cosh im} \]

   6. Applied rewrites 63.5%

      \[\leadsto re \cdot \color{blue}{\cosh im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.8% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{0 - \left|im\right|} + e^{\left|im\right|}\right) \leq -0.76:\\ \;\;\;\;0.5 \cdot \left(\left|re\right| \cdot \left(1 + \left(1 + -1 \cdot \left|im\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|re\right| \cdot \cosh \left(\left|im\right|\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<=
      (*
       (* 0.5 (sin (fabs re)))
       (+ (exp (- 0.0 (fabs im))) (exp (fabs im))))
      -0.76)
   (* 0.5 (* (fabs re) (+ 1.0 (+ 1.0 (* -1.0 (fabs im))))))
   (* (fabs re) (cosh (fabs im))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp((0.0 - fabs(im))) + exp(fabs(im)))) <= -0.76) {
		tmp = 0.5 * (fabs(re) * (1.0 + (1.0 + (-1.0 * fabs(im)))));
	} else {
		tmp = fabs(re) * cosh(fabs(im));
	}
	return copysign(1.0, re) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(Math.abs(re))) * (Math.exp((0.0 - Math.abs(im))) + Math.exp(Math.abs(im)))) <= -0.76) {
		tmp = 0.5 * (Math.abs(re) * (1.0 + (1.0 + (-1.0 * Math.abs(im)))));
	} else {
		tmp = Math.abs(re) * Math.cosh(Math.abs(im));
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(math.fabs(re))) * (math.exp((0.0 - math.fabs(im))) + math.exp(math.fabs(im)))) <= -0.76:
		tmp = 0.5 * (math.fabs(re) * (1.0 + (1.0 + (-1.0 * math.fabs(im)))))
	else:
		tmp = math.fabs(re) * math.cosh(math.fabs(im))
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(Float64(0.0 - abs(im))) + exp(abs(im)))) <= -0.76)
		tmp = Float64(0.5 * Float64(abs(re) * Float64(1.0 + Float64(1.0 + Float64(-1.0 * abs(im))))));
	else
		tmp = Float64(abs(re) * cosh(abs(im)));
	end
	return Float64(copysign(1.0, re) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(abs(re))) * (exp((0.0 - abs(im))) + exp(abs(im)))) <= -0.76)
		tmp = 0.5 * (abs(re) * (1.0 + (1.0 + (-1.0 * abs(im)))));
	else
		tmp = abs(re) * cosh(abs(im));
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.76], N[(0.5 * N[(N[Abs[re], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(-1.0 * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[re], $MachinePrecision] * N[Cosh[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.76000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.5%

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

   4. Applied rewrites 63.5%

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

   5. Taylor expanded in im around 0

      \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + e^{\color{blue}{-im}}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 44.6%

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

   8. Taylor expanded in im around 0

      \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + \color{blue}{-1 \cdot im}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 32.7%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + -1 \cdot im\right)\right)\right) \]

   10. Applied rewrites 32.7%

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

   if -0.76000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.5%

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

   4. Applied rewrites 63.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{\color{blue}{2}} \]

      7. lift-+.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      8. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      9. lift-exp.f64 N/A

         \[\leadsto re \cdot \frac{e^{im} + e^{-im}}{2} \]

      10. lift-neg.f64 N/A

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

      11. cosh-def N/A

          \[\leadsto re \cdot \cosh im \]

      12. lift-cosh.f64 N/A

          \[\leadsto re \cdot \cosh im \]

      13. lower-*.f64 63.5%

          \[\leadsto re \cdot \color{blue}{\cosh im} \]

   6. Applied rewrites 63.5%

      \[\leadsto re \cdot \color{blue}{\cosh im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 32.7% accurate, 4.5× speedup

Math

\[0.5 \cdot \left(re \cdot \left(1 + \left(1 + -1 \cdot im\right)\right)\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* 0.5 (* re (+ 1.0 (+ 1.0 (* -1.0 im))))))

C

double code(double re, double im) {
	return 0.5 * (re * (1.0 + (1.0 + (-1.0 * im))));
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (re * (1.0d0 + (1.0d0 + ((-1.0d0) * im))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (re * (1.0 + (1.0 + (-1.0 * im))));
}

Python

def code(re, im):
	return 0.5 * (re * (1.0 + (1.0 + (-1.0 * im))))

Julia

function code(re, im)
	return Float64(0.5 * Float64(re * Float64(1.0 + Float64(1.0 + Float64(-1.0 * im)))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (re * (1.0 + (1.0 + (-1.0 * im))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(1.0 + N[(1.0 + N[(-1.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 63.5%

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

4. Applied rewrites 63.5%

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

5. Taylor expanded in im around 0

   \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + e^{\color{blue}{-im}}\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 44.6%

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

8. Taylor expanded in im around 0

   \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + \color{blue}{-1 \cdot im}\right)\right)\right) \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 32.7%

      \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + -1 \cdot im\right)\right)\right) \]

10. Applied rewrites 32.7%

    \[\leadsto 0.5 \cdot \left(re \cdot \left(1 + \left(1 + \color{blue}{-1 \cdot im}\right)\right)\right) \]
11. Add Preprocessing

Alternative 8: 26.9% accurate, 9.7× speedup

Math

\[\left(0.5 \cdot re\right) \cdot 2 \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 re) 2.0))

C

double code(double re, double im) {
	return (0.5 * re) * 2.0;
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * 2.0d0
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * 2.0;
}

Python

def code(re, im):
	return (0.5 * re) * 2.0

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * 2.0)
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * 2.0;
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
3. Step-by-step derivation

4. Applied rewrites 51.1%

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

5. Taylor expanded in re around 0

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

   1. lower-*.f64 26.9%

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

7. Applied rewrites 26.9%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))