math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 1.9s

Alternatives: 10

Speedup: 1.0×

• 75.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[e^{re} \cdot \sin im \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[e^{re} \cdot \sin im \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|im\right|\right)\\ t_1 := \left(1 + re\right) \cdot t\_0\\ t_2 := e^{re} \cdot t\_0\\ t_3 := \left|im\right| \cdot e^{re}\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\left(\left|im\right| \cdot -0.16666666666666666\right) \cdot \left|im\right|, \left|im\right|, \left|im\right|\right)\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-165}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sin (fabs im)))
       (t_1 (* (+ 1.0 re) t_0))
       (t_2 (* (exp re) t_0))
       (t_3 (* (fabs im) (exp re))))
  (*
   (copysign 1.0 im)
   (if (<= t_2 (- INFINITY))
     (*
      (exp re)
      (fma
       (* (* (fabs im) -0.16666666666666666) (fabs im))
       (fabs im)
       (fabs im)))
     (if (<= t_2 -0.05)
       t_1
       (if (<= t_2 2e-165) t_3 (if (<= t_2 1.0) t_1 t_3)))))))

C

double code(double re, double im) {
	double t_0 = sin(fabs(im));
	double t_1 = (1.0 + re) * t_0;
	double t_2 = exp(re) * t_0;
	double t_3 = fabs(im) * exp(re);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = exp(re) * fma(((fabs(im) * -0.16666666666666666) * fabs(im)), fabs(im), fabs(im));
	} else if (t_2 <= -0.05) {
		tmp = t_1;
	} else if (t_2 <= 2e-165) {
		tmp = t_3;
	} else if (t_2 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return copysign(1.0, im) * tmp;
}

Julia

function code(re, im)
	t_0 = sin(abs(im))
	t_1 = Float64(Float64(1.0 + re) * t_0)
	t_2 = Float64(exp(re) * t_0)
	t_3 = Float64(abs(im) * exp(re))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(exp(re) * fma(Float64(Float64(abs(im) * -0.16666666666666666) * abs(im)), abs(im), abs(im)));
	elseif (t_2 <= -0.05)
		tmp = t_1;
	elseif (t_2 <= 2e-165)
		tmp = t_3;
	elseif (t_2 <= 1.0)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return Float64(copysign(1.0, im) * tmp)
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + re), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(N[Abs[im], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$1, If[LessEqual[t$95$2, 2e-165], t$95$3, If[LessEqual[t$95$2, 1.0], t$95$1, t$95$3]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 60.1%

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

   4. Applied rewrites 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.1%

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

   6. Applied rewrites 60.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 2e-165 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 51.9%

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

   4. Applied rewrites 51.9%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-165 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.2%

         \[\leadsto im \cdot e^{re} \]

   4. Applied rewrites 69.2%

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

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot e^{re}\\ t_1 := \sin \left(\left|im\right|\right)\\ t_2 := e^{re} \cdot t\_1\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\left(\left|im\right| \cdot -0.16666666666666666\right) \cdot \left|im\right|, \left|im\right|, \left|im\right|\right)\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) (exp re)))
       (t_1 (sin (fabs im)))
       (t_2 (* (exp re) t_1)))
  (*
   (copysign 1.0 im)
   (if (<= t_2 (- INFINITY))
     (*
      (exp re)
      (fma
       (* (* (fabs im) -0.16666666666666666) (fabs im))
       (fabs im)
       (fabs im)))
     (if (<= t_2 -0.05)
       t_1
       (if (<= t_2 2e-73) t_0 (if (<= t_2 1.0) t_1 t_0)))))))

C

double code(double re, double im) {
	double t_0 = fabs(im) * exp(re);
	double t_1 = sin(fabs(im));
	double t_2 = exp(re) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = exp(re) * fma(((fabs(im) * -0.16666666666666666) * fabs(im)), fabs(im), fabs(im));
	} else if (t_2 <= -0.05) {
		tmp = t_1;
	} else if (t_2 <= 2e-73) {
		tmp = t_0;
	} else if (t_2 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return copysign(1.0, im) * tmp;
}

Julia

function code(re, im)
	t_0 = Float64(abs(im) * exp(re))
	t_1 = sin(abs(im))
	t_2 = Float64(exp(re) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(exp(re) * fma(Float64(Float64(abs(im) * -0.16666666666666666) * abs(im)), abs(im), abs(im)));
	elseif (t_2 <= -0.05)
		tmp = t_1;
	elseif (t_2 <= 2e-73)
		tmp = t_0;
	elseif (t_2 <= 1.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, im) * tmp)
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Abs[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(N[Abs[im], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$1, If[LessEqual[t$95$2, 2e-73], t$95$0, If[LessEqual[t$95$2, 1.0], t$95$1, t$95$0]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 60.1%

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

   4. Applied rewrites 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.1%

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

   6. Applied rewrites 60.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 2e-73 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   3. Step-by-step derivation

      1. lower-sin.f64 51.4%

         \[\leadsto \sin im \]

   4. Applied rewrites 51.4%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-73 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.2%

         \[\leadsto im \cdot e^{re} \]

   4. Applied rewrites 69.2%

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

Alternative 3: 75.4% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq -0.05:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\left(\left|im\right| \cdot -0.16666666666666666\right) \cdot \left|im\right|, \left|im\right|, \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<= (* (exp re) (sin (fabs im))) -0.05)
   (*
    (exp re)
    (fma
     (* (* (fabs im) -0.16666666666666666) (fabs im))
     (fabs im)
     (fabs im)))
   (* (fabs im) (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(fabs(im))) <= -0.05) {
		tmp = exp(re) * fma(((fabs(im) * -0.16666666666666666) * fabs(im)), fabs(im), fabs(im));
	} else {
		tmp = fabs(im) * exp(re);
	}
	return copysign(1.0, im) * tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(abs(im))) <= -0.05)
		tmp = Float64(exp(re) * fma(Float64(Float64(abs(im) * -0.16666666666666666) * abs(im)), abs(im), abs(im)));
	else
		tmp = Float64(abs(im) * exp(re));
	end
	return Float64(copysign(1.0, im) * tmp)
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.05], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(N[Abs[im], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 60.1%

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

   4. Applied rewrites 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.1%

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

   6. Applied rewrites 60.1%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.2%

         \[\leadsto im \cdot e^{re} \]

   4. Applied rewrites 69.2%

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

Alternative 4: 74.5% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq -0.05:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left|im\right|, \left|im\right| \cdot \left(-0.16666666666666666 \cdot \left|im\right|\right), \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<= (* (exp re) (sin (fabs im))) -0.05)
   (*
    (+ 1.0 re)
    (fma
     (fabs im)
     (* (fabs im) (* -0.16666666666666666 (fabs im)))
     (fabs im)))
   (* (fabs im) (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(fabs(im))) <= -0.05) {
		tmp = (1.0 + re) * fma(fabs(im), (fabs(im) * (-0.16666666666666666 * fabs(im))), fabs(im));
	} else {
		tmp = fabs(im) * exp(re);
	}
	return copysign(1.0, im) * tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(abs(im))) <= -0.05)
		tmp = Float64(Float64(1.0 + re) * fma(abs(im), Float64(abs(im) * Float64(-0.16666666666666666 * abs(im))), abs(im)));
	else
		tmp = Float64(abs(im) * exp(re));
	end
	return Float64(copysign(1.0, im) * tmp)
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(1.0 + re), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 60.1%

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

   4. Applied rewrites 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. cube-unmult N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. lift-pow.f64 N/A

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

      18. lower-*.f64 60.1%

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 60.1%

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

   6. Applied rewrites 60.1%

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

   7. Taylor expanded in re around 0

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

      1. lower-+.f64 31.6%

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

   9. Applied rewrites 31.6%

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

       1. lift-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 31.6%

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

   11. Applied rewrites 31.6%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.2%

         \[\leadsto im \cdot e^{re} \]

   4. Applied rewrites 69.2%

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

Alternative 5: 73.6% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq -0.05:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\left(\left|im\right| \cdot \left|im\right|\right) \cdot \left|im\right|, -0.16666666666666666, \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<= (* (exp re) (sin (fabs im))) -0.05)
   (*
    1.0
    (fma
     (* (* (fabs im) (fabs im)) (fabs im))
     -0.16666666666666666
     (fabs im)))
   (* (fabs im) (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(fabs(im))) <= -0.05) {
		tmp = 1.0 * fma(((fabs(im) * fabs(im)) * fabs(im)), -0.16666666666666666, fabs(im));
	} else {
		tmp = fabs(im) * exp(re);
	}
	return copysign(1.0, im) * tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(abs(im))) <= -0.05)
		tmp = Float64(1.0 * fma(Float64(Float64(abs(im) * abs(im)) * abs(im)), -0.16666666666666666, abs(im)));
	else
		tmp = Float64(abs(im) * exp(re));
	end
	return Float64(copysign(1.0, im) * tmp)
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.05], N[(1.0 * N[(N[(N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 60.1%

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

   4. Applied rewrites 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. cube-unmult N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. lift-pow.f64 N/A

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

      18. lower-*.f64 60.1%

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 60.1%

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

   6. Applied rewrites 60.1%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 30.5%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

      2. lower-exp.f64 69.2%

         \[\leadsto im \cdot e^{re} \]

   4. Applied rewrites 69.2%

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

Alternative 6: 69.2% accurate, 3.2× speedup

Math

\[im \cdot e^{re} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

   2. lower-exp.f64 69.2%

      \[\leadsto im \cdot e^{re} \]

4. Applied rewrites 69.2%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
5. Add Preprocessing

Alternative 7: 37.2% accurate, 3.4× speedup

Math

\[im \cdot \sqrt{\left(re - -1\right) \cdot \left(re - -1\right)} \]

FPCore

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

C

double code(double re, double im) {
	return im * sqrt(((re - -1.0) * (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 = im * sqrt(((re - (-1.0d0)) * (re - (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

   2. lower-exp.f64 69.2%

      \[\leadsto im \cdot e^{re} \]

4. Applied rewrites 69.2%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

5. Taylor expanded in re around 0

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

   1. lower-+.f64 30.3%

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

7. Applied rewrites 30.3%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

      \[\leadsto im \cdot \sqrt{\left(1 + re\right) \cdot \left(1 + re\right)} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto im \cdot \sqrt{\left(1 + re\right) \cdot \left(1 + re\right)} \]

   4. lower-*.f64 37.2%

      \[\leadsto im \cdot \sqrt{\left(1 + re\right) \cdot \left(1 + re\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto im \cdot \sqrt{\left(1 + re\right) \cdot \left(1 + re\right)} \]

   6. +-commutative N/A

      \[\leadsto im \cdot \sqrt{\left(re + 1\right) \cdot \left(1 + re\right)} \]

   7. add-flip N/A

      \[\leadsto im \cdot \sqrt{\left(re - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(1 + re\right)} \]

   8. lower--.f64 N/A

      \[\leadsto im \cdot \sqrt{\left(re - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(1 + re\right)} \]

   9. metadata-eval 37.2%

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

   10. metadata-eval N/A

       \[\leadsto im \cdot \sqrt{\left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + re\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto im \cdot \sqrt{\left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(+-commutative, \left(re + 1\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto im \cdot \sqrt{\left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(add-flip, \left(re - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

   13. metadata-eval N/A

       \[\leadsto im \cdot \sqrt{\left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(lower--.f64, \left(re - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

   14. metadata-eval N/A

       \[\leadsto im \cdot \sqrt{\left(re - -1\right) \cdot \left(re - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)\right)} \]

9. Applied rewrites 37.2%

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

Alternative 8: 30.3% accurate, 8.1× speedup

Math

\[\mathsf{fma}\left(re, im, im\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (fma re im im))

C

double code(double re, double im) {
	return fma(re, im, im);
}

Julia

function code(re, im)
	return fma(re, im, im)
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

   2. lower-exp.f64 69.2%

      \[\leadsto im \cdot e^{re} \]

4. Applied rewrites 69.2%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

5. Taylor expanded in re around 0

   \[\leadsto im + \color{blue}{im \cdot re} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 30.3%

      \[\leadsto im + im \cdot re \]

7. Applied rewrites 30.3%

   \[\leadsto im + \color{blue}{im \cdot re} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

      \[\leadsto im \cdot re + im \]

   3. lift-*.f64 N/A

      \[\leadsto im \cdot re + im \]

   4. *-commutative N/A

      \[\leadsto re \cdot im + im \]

   5. lower-fma.f64 30.3%

      \[\leadsto \mathsf{fma}\left(re, im, im\right) \]

9. Applied rewrites 30.3%

   \[\leadsto \mathsf{fma}\left(re, im, im\right) \]
10. Add Preprocessing

Alternative 9: 27.0% accurate, 48.6× speedup

Math

\[im \]

FPCore

(FPCore (re im)
  :precision binary64
  im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

   2. lower-exp.f64 69.2%

      \[\leadsto im \cdot e^{re} \]

4. Applied rewrites 69.2%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

5. Taylor expanded in re around 0

   \[\leadsto im \]
6. Step-by-step derivation

7. Applied rewrites 27.0%

   \[\leadsto im \]
8. Add Preprocessing

Reproduce

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