math.cos on complex, imaginary part

Percentage Accurate: 65.7% → 99.9%

Time: 3.6s

Alternatives: 7

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-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(-im) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-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(-im) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.3× speedup

Math

\[\sinh \left(-im\right) \cdot \sin re \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (sinh (- im)) (sin re)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. distribute-lft-neg-out N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. sinh-def N/A

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

   14. sinh-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sinh.f64 99.9%

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sinh \left(-im\right) \cdot \sin re} \]
4. Add Preprocessing

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := -\left|im\right|\\ t_1 := e^{\left|im\right|}\\ t_2 := \sin \left(\left|re\right|\right)\\ t_3 := \left(0.5 \cdot t\_2\right) \cdot \left(e^{t\_0} - t\_1\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \left(\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(1 - t\_1\right)\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sinh t\_0 \cdot \mathsf{fma}\left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot \left|re\right|, -0.16666666666666666, \left|re\right|\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (- (fabs im)))
       (t_1 (exp (fabs im)))
       (t_2 (sin (fabs re)))
       (t_3 (* (* 0.5 t_2) (- (exp t_0) t_1))))
  (*
   (copysign 1.0 re)
   (*
    (copysign 1.0 im)
    (if (<= t_3 (- INFINITY))
      (* (* 0.5 (fabs re)) (- 1.0 t_1))
      (if (<= t_3 4e-8)
        (* t_2 t_0)
        (*
         (sinh t_0)
         (fma
          (* (* (fabs re) (fabs re)) (fabs re))
          -0.16666666666666666
          (fabs re)))))))))

C

double code(double re, double im) {
	double t_0 = -fabs(im);
	double t_1 = exp(fabs(im));
	double t_2 = sin(fabs(re));
	double t_3 = (0.5 * t_2) * (exp(t_0) - t_1);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (0.5 * fabs(re)) * (1.0 - t_1);
	} else if (t_3 <= 4e-8) {
		tmp = t_2 * t_0;
	} else {
		tmp = sinh(t_0) * fma(((fabs(re) * fabs(re)) * fabs(re)), -0.16666666666666666, fabs(re));
	}
	return copysign(1.0, re) * (copysign(1.0, im) * tmp);
}

Julia

function code(re, im)
	t_0 = Float64(-abs(im))
	t_1 = exp(abs(im))
	t_2 = sin(abs(re))
	t_3 = Float64(Float64(0.5 * t_2) * Float64(exp(t_0) - t_1))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(1.0 - t_1));
	elseif (t_3 <= 4e-8)
		tmp = Float64(t_2 * t_0);
	else
		tmp = Float64(sinh(t_0) * fma(Float64(Float64(abs(re) * abs(re)) * abs(re)), -0.16666666666666666, abs(re)));
	end
	return Float64(copysign(1.0, re) * Float64(copysign(1.0, im) * tmp))
end

Wolfram

code[re_, im_] := Block[{t$95$0 = (-N[Abs[im], $MachinePrecision])}, Block[{t$95$1 = N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * t$95$2), $MachinePrecision] * N[(N[Exp[t$95$0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e-8], N[(t$95$2 * t$95$0), $MachinePrecision], N[(N[Sinh[t$95$0], $MachinePrecision] * N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.7%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 34.4%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 4.0000000000000001e-8

   1. Initial program 65.7%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.8%

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

   4. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sin re \cdot im\right) \]

      5. distribute-rgt-neg-in N/A

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

      6. lift-neg.f64 N/A

         \[\leadsto \sin re \cdot \left(-im\right) \]

      7. lower-*.f64 51.8%

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

   6. Applied rewrites 51.8%

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

   if 4.0000000000000001e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. distribute-lft-neg-out N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-def N/A

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

      14. sinh-neg N/A

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

      15. lift-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 63.2%

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

   6. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. cube-unmult N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. lift-pow.f64 N/A

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

      18. lower-*.f64 63.2%

          \[\leadsto \sinh \left(-im\right) \cdot \mathsf{fma}\left({re}^{2} \cdot re, -0.16666666666666666, re\right) \]

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 63.2%

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

   8. Applied rewrites 63.2%

      \[\leadsto \sinh \left(-im\right) \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot re, \color{blue}{-0.16666666666666666}, re\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.8% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \left(\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq 0.0001:\\ \;\;\;\;\sinh \left(-\left|im\right|\right) \cdot \mathsf{fma}\left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot \left|re\right|, -0.16666666666666666, \left|re\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(1 - e^{\left|im\right|}\right)\\ \end{array}\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (*
  (copysign 1.0 im)
  (if (<= (* 0.5 (sin (fabs re))) 0.0001)
    (*
     (sinh (- (fabs im)))
     (fma
      (* (* (fabs re) (fabs re)) (fabs re))
      -0.16666666666666666
      (fabs re)))
    (* (* 0.5 (fabs re)) (- 1.0 (exp (fabs im))))))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= 0.0001) {
		tmp = sinh(-fabs(im)) * fma(((fabs(re) * fabs(re)) * fabs(re)), -0.16666666666666666, fabs(re));
	} else {
		tmp = (0.5 * fabs(re)) * (1.0 - exp(fabs(im)));
	}
	return copysign(1.0, re) * (copysign(1.0, im) * tmp);
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= 0.0001)
		tmp = Float64(sinh(Float64(-abs(im))) * fma(Float64(Float64(abs(re) * abs(re)) * abs(re)), -0.16666666666666666, abs(re)));
	else
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(1.0 - exp(abs(im))));
	end
	return Float64(copysign(1.0, re) * Float64(copysign(1.0, im) * tmp))
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[Sinh[(-N[Abs[im], $MachinePrecision])], $MachinePrecision] * N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 1e-4

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. distribute-lft-neg-out N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-def N/A

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

      14. sinh-neg N/A

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

      15. lift-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 63.2%

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

   6. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. cube-unmult N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. lift-pow.f64 N/A

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

      18. lower-*.f64 63.2%

          \[\leadsto \sinh \left(-im\right) \cdot \mathsf{fma}\left({re}^{2} \cdot re, -0.16666666666666666, re\right) \]

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 63.2%

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

   8. Applied rewrites 63.2%

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

   if 1e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.7%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 34.4%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.0% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq -0.004:\\ \;\;\;\;\left|re\right| \cdot \left(0.16666666666666666 \cdot \left(im \cdot {\left(\left|re\right|\right)}^{2}\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot \left(\left|re\right| \cdot 1\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<= (* 0.5 (sin (fabs re))) -0.004)
   (*
    (fabs re)
    (- (* 0.16666666666666666 (* im (pow (fabs re) 2.0))) im))
   (* (sinh (- im)) (* (fabs re) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= -0.004) {
		tmp = fabs(re) * ((0.16666666666666666 * (im * pow(fabs(re), 2.0))) - im);
	} else {
		tmp = sinh(-im) * (fabs(re) * 1.0);
	}
	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))) <= -0.004) {
		tmp = Math.abs(re) * ((0.16666666666666666 * (im * Math.pow(Math.abs(re), 2.0))) - im);
	} else {
		tmp = Math.sinh(-im) * (Math.abs(re) * 1.0);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= -0.004:
		tmp = math.fabs(re) * ((0.16666666666666666 * (im * math.pow(math.fabs(re), 2.0))) - im)
	else:
		tmp = math.sinh(-im) * (math.fabs(re) * 1.0)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= -0.004)
		tmp = Float64(abs(re) * Float64(Float64(0.16666666666666666 * Float64(im * (abs(re) ^ 2.0))) - im));
	else
		tmp = Float64(sinh(Float64(-im)) * Float64(abs(re) * 1.0));
	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))) <= -0.004)
		tmp = abs(re) * ((0.16666666666666666 * (im * (abs(re) ^ 2.0))) - im);
	else
		tmp = sinh(-im) * (abs(re) * 1.0);
	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[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[Abs[re], $MachinePrecision] * N[(N[(0.16666666666666666 * N[(im * N[Power[N[Abs[re], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[(-im)], $MachinePrecision] * N[(N[Abs[re], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.7%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.8%

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 33.6%

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

   7. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]

      3. lower-neg.f64 33.6%

         \[\leadsto -im \cdot re \]

   9. Applied rewrites 33.6%

      \[\leadsto -im \cdot re \]

   10. Taylor expanded in re around 0

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 36.8%

          \[\leadsto re \cdot \left(0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right) - im\right) \]

   12. Applied rewrites 36.8%

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

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

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. distribute-lft-neg-out N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-def N/A

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

      14. sinh-neg N/A

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

      15. lift-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 63.2%

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in re around 0

      \[\leadsto \sinh \left(-im\right) \cdot \left(re \cdot 1\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 63.7%

      \[\leadsto \sinh \left(-im\right) \cdot \left(re \cdot 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.7% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (sinh (- im)) (* re 1.0)))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return Math.sinh(-im) * (re * 1.0);
}

Python

def code(re, im):
	return math.sinh(-im) * (re * 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. distribute-lft-neg-out N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. sinh-def N/A

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

   14. sinh-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sinh.f64 99.9%

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 63.2%

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

6. Applied rewrites 63.2%

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

7. Taylor expanded in re around 0

   \[\leadsto \sinh \left(-im\right) \cdot \left(re \cdot 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 63.7%

   \[\leadsto \sinh \left(-im\right) \cdot \left(re \cdot 1\right) \]
10. Add Preprocessing

Alternative 6: 63.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := e^{\left|im\right|}\\ \mathsf{copysign}\left(1, re\right) \cdot \left(\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{-\left|im\right|} - t\_0\right) \leq -0.0002:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|im\right| \cdot \left|re\right|\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (exp (fabs im))))
  (*
   (copysign 1.0 re)
   (*
    (copysign 1.0 im)
    (if (<=
         (* (* 0.5 (sin (fabs re))) (- (exp (- (fabs im))) t_0))
         -0.0002)
      (* (* 0.5 (fabs re)) (- 1.0 t_0))
      (- (* (fabs im) (fabs re))))))))

C

double code(double re, double im) {
	double t_0 = exp(fabs(im));
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp(-fabs(im)) - t_0)) <= -0.0002) {
		tmp = (0.5 * fabs(re)) * (1.0 - t_0);
	} else {
		tmp = -(fabs(im) * fabs(re));
	}
	return copysign(1.0, re) * (copysign(1.0, im) * tmp);
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(Math.abs(im));
	double tmp;
	if (((0.5 * Math.sin(Math.abs(re))) * (Math.exp(-Math.abs(im)) - t_0)) <= -0.0002) {
		tmp = (0.5 * Math.abs(re)) * (1.0 - t_0);
	} else {
		tmp = -(Math.abs(im) * Math.abs(re));
	}
	return Math.copySign(1.0, re) * (Math.copySign(1.0, im) * tmp);
}

Python

def code(re, im):
	t_0 = math.exp(math.fabs(im))
	tmp = 0
	if ((0.5 * math.sin(math.fabs(re))) * (math.exp(-math.fabs(im)) - t_0)) <= -0.0002:
		tmp = (0.5 * math.fabs(re)) * (1.0 - t_0)
	else:
		tmp = -(math.fabs(im) * math.fabs(re))
	return math.copysign(1.0, re) * (math.copysign(1.0, im) * tmp)

Julia

function code(re, im)
	t_0 = exp(abs(im))
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(Float64(-abs(im))) - t_0)) <= -0.0002)
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(1.0 - t_0));
	else
		tmp = Float64(-Float64(abs(im) * abs(re)));
	end
	return Float64(copysign(1.0, re) * Float64(copysign(1.0, im) * tmp))
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(abs(im));
	tmp = 0.0;
	if (((0.5 * sin(abs(re))) * (exp(-abs(im)) - t_0)) <= -0.0002)
		tmp = (0.5 * abs(re)) * (1.0 - t_0);
	else
		tmp = -(abs(im) * abs(re));
	end
	tmp_2 = (sign(re) * abs(1.0)) * ((sign(im) * abs(1.0)) * tmp);
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, 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[Abs[im], $MachinePrecision])], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], -0.0002], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], (-N[(N[Abs[im], $MachinePrecision] * N[Abs[re], $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 (neg.f64 im)) (exp.f64 im))) < -2.0000000000000001e-4

   1. Initial program 65.7%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 34.4%

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

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

   1. Initial program 65.7%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.8%

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 33.6%

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

   7. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]

      3. lower-neg.f64 33.6%

         \[\leadsto -im \cdot re \]

   9. Applied rewrites 33.6%

      \[\leadsto -im \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 33.6% accurate, 13.2× speedup

Math

\[-im \cdot re \]

FPCore

(FPCore (re im)
  :precision binary64
  (- (* im re)))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -(im * re)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

2. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 51.8%

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

4. Applied rewrites 51.8%

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

5. Taylor expanded in re around 0

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

   1. lower-*.f64 33.6%

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

7. Applied rewrites 33.6%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]

   3. lower-neg.f64 33.6%

      \[\leadsto -im \cdot re \]

9. Applied rewrites 33.6%

   \[\leadsto -im \cdot re \]
10. Add Preprocessing

Reproduce

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