math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 9

Speedup: 1.0×

• 75.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot \sqrt{e^{re + re}}\\ t_1 := \sin \left(\left|im\right|\right)\\ t_2 := e^{re} \cdot t\_1\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left|im\right| \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left|im\right|, \left|im\right|, 1\right)\right)\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) (sqrt (exp (+ re re)))))
       (t_1 (sin (fabs im)))
       (t_2 (* (exp re) t_1)))
  (*
   (copysign 1.0 im)
   (if (<= t_2 (- INFINITY))
     (*
      (exp re)
      (*
       (fabs im)
       (fma (* -0.16666666666666666 (fabs im)) (fabs im) 1.0)))
     (if (<= t_2 -0.05)
       t_1
       (if (<= t_2 5e-57)
         t_0
         (if (<= t_2 1.0) (* (+ 1.0 re) t_1) t_0)))))))

C

double code(double re, double im) {
	double t_0 = fabs(im) * sqrt(exp((re + re)));
	double t_1 = sin(fabs(im));
	double t_2 = exp(re) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = exp(re) * (fabs(im) * fma((-0.16666666666666666 * fabs(im)), fabs(im), 1.0));
	} else if (t_2 <= -0.05) {
		tmp = t_1;
	} else if (t_2 <= 5e-57) {
		tmp = t_0;
	} else if (t_2 <= 1.0) {
		tmp = (1.0 + re) * t_1;
	} else {
		tmp = t_0;
	}
	return copysign(1.0, im) * tmp;
}

Julia

function code(re, im)
	t_0 = Float64(abs(im) * sqrt(exp(Float64(re + 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) * Float64(abs(im) * fma(Float64(-0.16666666666666666 * abs(im)), abs(im), 1.0)));
	elseif (t_2 <= -0.05)
		tmp = t_1;
	elseif (t_2 <= 5e-57)
		tmp = t_0;
	elseif (t_2 <= 1.0)
		tmp = Float64(Float64(1.0 + re) * 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[Sqrt[N[Exp[N[(re + re), $MachinePrecision]], $MachinePrecision]], $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[Abs[im], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$1, If[LessEqual[t$95$2, 5e-57], t$95$0, If[LessEqual[t$95$2, 1.0], N[(N[(1.0 + re), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$0]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 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 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 60.4%

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

   6. Applied rewrites 60.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   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 50.8%

         \[\leadsto \sin im \]

   4. Applied rewrites 50.8%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000002e-57 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

         \[\leadsto \color{blue}{\left|e^{re}\right|} \cdot \sin im \]

      3. lift-exp.f64 N/A

         \[\leadsto \left|\color{blue}{e^{re}}\right| \cdot \sin im \]

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. exp-lft-sqr-rev N/A

         \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

      9. lower-exp.f64 N/A

         \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

      10. *-commutative N/A

          \[\leadsto \sqrt{e^{\color{blue}{2 \cdot re}}} \cdot \sin im \]

      11. count-2 N/A

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

      12. lower-+.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{\sqrt{e^{2 \cdot re}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      3. lower-exp.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      4. lower-*.f64 69.1%

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

   6. Applied rewrites 69.1%

      \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      2. count-2-rev N/A

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

      3. lower-+.f64 69.1%

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

   8. Applied rewrites 69.1%

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

   if 5.0000000000000002e-57 < (*.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 51.3%

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

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot \sqrt{e^{re + re}}\\ t_1 := \sin \left(\left|im\right|\right)\\ t_2 := e^{re} \cdot t\_1\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left|im\right| \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left|im\right|, \left|im\right|, 1\right)\right)\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) (sqrt (exp (+ re re)))))
       (t_1 (sin (fabs im)))
       (t_2 (* (exp re) t_1)))
  (*
   (copysign 1.0 im)
   (if (<= t_2 (- INFINITY))
     (*
      (exp re)
      (*
       (fabs im)
       (fma (* -0.16666666666666666 (fabs im)) (fabs im) 1.0)))
     (if (<= t_2 -0.05)
       t_1
       (if (<= t_2 1e-54) t_0 (if (<= t_2 1.0) t_1 t_0)))))))

C

double code(double re, double im) {
	double t_0 = fabs(im) * sqrt(exp((re + re)));
	double t_1 = sin(fabs(im));
	double t_2 = exp(re) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = exp(re) * (fabs(im) * fma((-0.16666666666666666 * fabs(im)), fabs(im), 1.0));
	} else if (t_2 <= -0.05) {
		tmp = t_1;
	} else if (t_2 <= 1e-54) {
		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) * sqrt(exp(Float64(re + 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) * Float64(abs(im) * fma(Float64(-0.16666666666666666 * abs(im)), abs(im), 1.0)));
	elseif (t_2 <= -0.05)
		tmp = t_1;
	elseif (t_2 <= 1e-54)
		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[Sqrt[N[Exp[N[(re + re), $MachinePrecision]], $MachinePrecision]], $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[Abs[im], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$1, If[LessEqual[t$95$2, 1e-54], 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 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 60.4%

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

   6. Applied rewrites 60.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1e-54 < (*.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 50.8%

         \[\leadsto \sin im \]

   4. Applied rewrites 50.8%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-54 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

         \[\leadsto \color{blue}{\left|e^{re}\right|} \cdot \sin im \]

      3. lift-exp.f64 N/A

         \[\leadsto \left|\color{blue}{e^{re}}\right| \cdot \sin im \]

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. exp-lft-sqr-rev N/A

         \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

      9. lower-exp.f64 N/A

         \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

      10. *-commutative N/A

          \[\leadsto \sqrt{e^{\color{blue}{2 \cdot re}}} \cdot \sin im \]

      11. count-2 N/A

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

      12. lower-+.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{\sqrt{e^{2 \cdot re}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      3. lower-exp.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      4. lower-*.f64 69.1%

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

   6. Applied rewrites 69.1%

      \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      2. count-2-rev N/A

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

      3. lower-+.f64 69.1%

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

   8. Applied rewrites 69.1%

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

Alternative 3: 75.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(fabs(im))) <= -0.05) {
		tmp = exp(re) * (fabs(im) * fma((-0.16666666666666666 * fabs(im)), fabs(im), 1.0));
	} else {
		tmp = fabs(im) * sqrt(exp((re + 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) * Float64(abs(im) * fma(Float64(-0.16666666666666666 * abs(im)), abs(im), 1.0)));
	else
		tmp = Float64(abs(im) * sqrt(exp(Float64(re + 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[Abs[im], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] * N[Sqrt[N[Exp[N[(re + re), $MachinePrecision]], $MachinePrecision]], $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 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 60.4%

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

   6. Applied rewrites 60.4%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

         \[\leadsto \color{blue}{\left|e^{re}\right|} \cdot \sin im \]

      3. lift-exp.f64 N/A

         \[\leadsto \left|\color{blue}{e^{re}}\right| \cdot \sin im \]

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. exp-lft-sqr-rev N/A

         \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

      9. lower-exp.f64 N/A

         \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

      10. *-commutative N/A

          \[\leadsto \sqrt{e^{\color{blue}{2 \cdot re}}} \cdot \sin im \]

      11. count-2 N/A

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

      12. lower-+.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{\sqrt{e^{2 \cdot re}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      3. lower-exp.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      4. lower-*.f64 69.1%

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

   6. Applied rewrites 69.1%

      \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      2. count-2-rev N/A

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

      3. lower-+.f64 69.1%

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

   8. Applied rewrites 69.1%

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

Alternative 4: 71.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) re)))
  (*
   (copysign 1.0 im)
   (if (<= (* (exp re) (sin (fabs im))) (- INFINITY))
     (/ (- (* t_0 t_0) (* (fabs im) (fabs im))) (* -1.0 (fabs im)))
     (* (fabs im) (sqrt (exp (+ re re))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Abs[im], $MachinePrecision] * re), $MachinePrecision]}, 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], (-Infinity)], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[im], $MachinePrecision] * N[Sqrt[N[Exp[N[(re + re), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 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

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 69.1%

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

   4. Applied rewrites 69.1%

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

   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 29.6%

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

   7. Applied rewrites 29.6%

      \[\leadsto im + \color{blue}{im \cdot re} \]
   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. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. lower-unsound--.f64 19.3%

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in re around 0

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

       1. lower-*.f64 18.0%

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

   12. Applied rewrites 18.0%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

         \[\leadsto \color{blue}{\left|e^{re}\right|} \cdot \sin im \]

      3. lift-exp.f64 N/A

         \[\leadsto \left|\color{blue}{e^{re}}\right| \cdot \sin im \]

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. exp-lft-sqr-rev N/A

         \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

      9. lower-exp.f64 N/A

         \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

      10. *-commutative N/A

          \[\leadsto \sqrt{e^{\color{blue}{2 \cdot re}}} \cdot \sin im \]

      11. count-2 N/A

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

      12. lower-+.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{\sqrt{e^{2 \cdot re}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      3. lower-exp.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      4. lower-*.f64 69.1%

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

   6. Applied rewrites 69.1%

      \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

      2. count-2-rev N/A

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

      3. lower-+.f64 69.1%

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

   8. Applied rewrites 69.1%

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

Alternative 5: 69.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. exp-fabs N/A

      \[\leadsto \color{blue}{\left|e^{re}\right|} \cdot \sin im \]

   3. lift-exp.f64 N/A

      \[\leadsto \left|\color{blue}{e^{re}}\right| \cdot \sin im \]

   4. rem-sqrt-square-rev N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. exp-lft-sqr-rev N/A

      \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

   9. lower-exp.f64 N/A

      \[\leadsto \sqrt{\color{blue}{e^{re \cdot 2}}} \cdot \sin im \]

   10. *-commutative N/A

       \[\leadsto \sqrt{e^{\color{blue}{2 \cdot re}}} \cdot \sin im \]

   11. count-2 N/A

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

   12. lower-+.f64 99.9%

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto im \cdot \color{blue}{\sqrt{e^{2 \cdot re}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

   3. lower-exp.f64 N/A

      \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

   4. lower-*.f64 69.1%

      \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

6. Applied rewrites 69.1%

   \[\leadsto \color{blue}{im \cdot \sqrt{e^{2 \cdot re}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto im \cdot \sqrt{e^{2 \cdot re}} \]

   2. count-2-rev N/A

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

   3. lower-+.f64 69.1%

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

8. Applied rewrites 69.1%

   \[\leadsto im \cdot \sqrt{e^{re + re}} \]
9. Add Preprocessing

Alternative 6: 69.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 69.1%

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

4. Applied rewrites 69.1%

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

Alternative 7: 29.6% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(im * re + 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 69.1%

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

4. Applied rewrites 69.1%

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

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 29.6%

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

7. Applied rewrites 29.6%

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

   1. exp-fabs 29.6%

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

   2. lift-exp.f64 N/A

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

   3. rem-sqrt-square-rev N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-exp.f64 29.6%

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

   6. exp-sum 29.6%

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 29.6%

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

9. Applied rewrites 29.6%

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

Alternative 8: 26.6% accurate, 48.6× speedup

Math

\[im \]

FPCore

(FPCore (re im)
  :precision binary64
  im)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 69.1%

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

4. Applied rewrites 69.1%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 26.6%

   \[\leadsto im \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))