math.cos on complex, imaginary part

Percentage Accurate: 65.5% → 99.8%

Time: 14.5s

Alternatives: 22

Speedup: 0.7×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)
    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: 65.5% accurate, 1.0× speedup

Math

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

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)
    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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - e^{im\_m} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-e^{im\_m} \cdot \sin re, 0.5, 0.5 \cdot \left(t\_0 \cdot \sin re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (exp (- im_m))))
   (*
    im_s
    (if (<= (- t_0 (exp im_m)) -0.1)
      (fma (- (* (exp im_m) (sin re))) 0.5 (* 0.5 (* t_0 (sin re))))
      (* (sin re) (* im_m (fma -0.16666666666666666 (* im_m im_m) -1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m);
	double tmp;
	if ((t_0 - exp(im_m)) <= -0.1) {
		tmp = fma(-(exp(im_m) * sin(re)), 0.5, (0.5 * (t_0 * sin(re))));
	} else {
		tmp = sin(re) * (im_m * fma(-0.16666666666666666, (im_m * im_m), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = exp(Float64(-im_m))
	tmp = 0.0
	if (Float64(t_0 - exp(im_m)) <= -0.1)
		tmp = fma(Float64(-Float64(exp(im_m) * sin(re))), 0.5, Float64(0.5 * Float64(t_0 * sin(re))));
	else
		tmp = Float64(sin(re) * Float64(im_m * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Exp[(-im$95$m)], $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -0.1], N[((-N[(N[Exp[im$95$m], $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]) * 0.5 + N[(0.5 * N[(t$95$0 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.10000000000000001

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

   4. Applied rewrites 100.0%

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

   if -0.10000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 34.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-e^{im} \cdot \sin re, 0.5, 0.5 \cdot \left(e^{-im} \cdot \sin re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right)\\ t_1 := \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \left(im\_m \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), t\_1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5)))
        (t_1 (fma -0.16666666666666666 (* im_m im_m) -1.0)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (- 1.0 (exp im_m)) (* re 0.5))
      (if (<= t_0 4e-5)
        (* (sin re) (* im_m t_1))
        (*
         im_m
         (*
          (fma re (* -0.16666666666666666 (* re re)) re)
          (fma
           (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333)
           (* im_m (* im_m (* im_m im_m)))
           t_1))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (sin(re) * 0.5);
	double t_1 = fma(-0.16666666666666666, (im_m * im_m), -1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 - exp(im_m)) * (re * 0.5);
	} else if (t_0 <= 4e-5) {
		tmp = sin(re) * (im_m * t_1);
	} else {
		tmp = im_m * (fma(re, (-0.16666666666666666 * (re * re)), re) * fma(fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333), (im_m * (im_m * (im_m * im_m))), t_1));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5))
	t_1 = fma(-0.16666666666666666, Float64(im_m * im_m), -1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(re * 0.5));
	elseif (t_0 <= 4e-5)
		tmp = Float64(sin(re) * Float64(im_m * t_1));
	else
		tmp = Float64(im_m * Float64(fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re) * fma(fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), Float64(im_m * Float64(im_m * Float64(im_m * im_m))), t_1)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-5], N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * t$95$1), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $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 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 74.2

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

   8. Applied rewrites 74.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 4.00000000000000033e-5

   1. Initial program 31.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 4.00000000000000033e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.0%

      \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right)}, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024226 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))