math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 12

Speedup: 1.0×

• 75.9% 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: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot e^{re}\\ t_1 := \sin \left(\left|im\right|\right)\\ t_2 := e^{re} \cdot t\_1\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\left|im\right| \cdot \left|im\right|, -0.16666666666666666 \cdot \left|im\right|, \left|im\right|\right)\\ \mathbf{elif}\;t\_2 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) (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) (fabs im))
       (* -0.16666666666666666 (fabs im))
       (fabs im)))
     (if (<= t_2 -0.02)
       t_1
       (if (<= t_2 0.0)
         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) * 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) * fabs(im)), (-0.16666666666666666 * fabs(im)), fabs(im));
	} else if (t_2 <= -0.02) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		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) * 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(abs(im) * abs(im)), Float64(-0.16666666666666666 * abs(im)), abs(im)));
	elseif (t_2 <= -0.02)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		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[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[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[im], $MachinePrecision]), $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.02], t$95$1, If[LessEqual[t$95$2, 0.0], 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 60.3%

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

   4. Applied rewrites 60.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.3%

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

   6. Applied rewrites 60.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.02

   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 -0.02 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0 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 68.6%

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

   4. Applied rewrites 68.6%

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

   if 0.0 < (*.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 52.0%

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

   4. Applied rewrites 52.0%

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

Alternative 2: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot e^{re}\\ t_1 := \sin \left(\left|im\right|\right)\\ t_2 := e^{re} \cdot t\_1\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\left|im\right| \cdot \left|im\right|, -0.16666666666666666 \cdot \left|im\right|, \left|im\right|\right)\\ \mathbf{elif}\;t\_2 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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) (fabs im))
       (* -0.16666666666666666 (fabs im))
       (fabs im)))
     (if (<= t_2 -0.02)
       t_1
       (if (<= t_2 0.0) 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) * fabs(im)), (-0.16666666666666666 * fabs(im)), fabs(im));
	} else if (t_2 <= -0.02) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		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(abs(im) * abs(im)), Float64(-0.16666666666666666 * abs(im)), abs(im)));
	elseif (t_2 <= -0.02)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		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[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[im], $MachinePrecision]), $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.02], t$95$1, If[LessEqual[t$95$2, 0.0], 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 60.3%

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

   4. Applied rewrites 60.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.3%

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

   6. Applied rewrites 60.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.02 or 0.0 < (*.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 -0.02 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0 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 68.6%

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

   4. Applied rewrites 68.6%

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

Alternative 3: 75.1% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq -0.02:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\left|im\right| \cdot \left|im\right|, -0.16666666666666666 \cdot \left|im\right|, \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<= (* (exp re) (sin (fabs im))) -0.02)
   (*
    (exp 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))) <= -0.02) {
		tmp = exp(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))) <= -0.02)
		tmp = Float64(exp(re) * fma(Float64(abs(im) * 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], -0.02], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[im], $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)) < -0.02

   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 60.3%

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

   4. Applied rewrites 60.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.3%

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

   6. Applied rewrites 60.3%

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

   if -0.02 < (*.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 68.6%

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

   4. Applied rewrites 68.6%

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

Alternative 4: 74.0% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left|im\right| \cdot \left|im\right|, -0.16666666666666666 \cdot \left|im\right|, \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<= (* (exp re) (sin (fabs im))) -0.02)
   (*
    (+ 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))) <= -0.02) {
		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))) <= -0.02)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(abs(im) * 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], -0.02], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[im], $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)) < -0.02

   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 60.3%

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

   4. Applied rewrites 60.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.3%

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

   6. Applied rewrites 60.3%

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

   7. Taylor expanded in re around 0

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

      1. lower-+.f64 31.3%

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

   9. Applied rewrites 31.3%

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

   if -0.02 < (*.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 68.6%

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

   4. Applied rewrites 68.6%

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

Alternative 5: 73.0% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin \left(\left|im\right|\right) \leq -0.02:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\left|im\right| \cdot \left|im\right|, -0.16666666666666666 \cdot \left|im\right|, \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right| \cdot e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<= (* (exp re) (sin (fabs im))) -0.02)
   (*
    1.0
    (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))) <= -0.02) {
		tmp = 1.0 * 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))) <= -0.02)
		tmp = Float64(1.0 * fma(Float64(abs(im) * 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], -0.02], N[(1.0 * N[(N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[im], $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)) < -0.02

   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 60.3%

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

   4. Applied rewrites 60.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.3%

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

   6. Applied rewrites 60.3%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 30.2%

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

   if -0.02 < (*.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 68.6%

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

   4. Applied rewrites 68.6%

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

Alternative 6: 68.6% 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 68.6%

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

4. Applied rewrites 68.6%

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

Alternative 7: 39.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(im, re, im\right) \cdot im}{im}\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+137}:\\ \;\;\;\;im \cdot \left(im \cdot \frac{re - -1}{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(re, re, re\right) \cdot im}{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= re -3.4e+18)
  (/ (* (fma im re im) im) im)
  (if (<= re 1.35e+137)
    (* im (* im (/ (- re -1.0) im)))
    (/ (* (fma re re re) im) re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e+18) {
		tmp = (fma(im, re, im) * im) / im;
	} else if (re <= 1.35e+137) {
		tmp = im * (im * ((re - -1.0) / im));
	} else {
		tmp = (fma(re, re, re) * im) / re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e+18)
		tmp = Float64(Float64(fma(im, re, im) * im) / im);
	elseif (re <= 1.35e+137)
		tmp = Float64(im * Float64(im * Float64(Float64(re - -1.0) / im)));
	else
		tmp = Float64(Float64(fma(re, re, re) * im) / re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e+18], N[(N[(N[(im * re + im), $MachinePrecision] * im), $MachinePrecision] / im), $MachinePrecision], If[LessEqual[re, 1.35e+137], N[(im * N[(im * N[(N[(re - -1.0), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re + re), $MachinePrecision] * im), $MachinePrecision] / re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.4e18

   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 68.6%

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

   4. Applied rewrites 68.6%

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

   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 29.7%

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

   7. Applied rewrites 29.7%

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

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

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

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

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

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

      6. lower-unsound-/.f64 18.1%

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

   9. Applied rewrites 18.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-+.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. add-to-fraction N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. frac-2neg N/A

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

       10. frac-2neg N/A

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

       11. *-lft-identity N/A

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

       12. lift-*.f64 N/A

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

       13. add-to-fraction-rev N/A

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

       14. sum-to-mult-rev N/A

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

       15. associate-/r* N/A

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

       16. *-commutative N/A

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

       17. lift-*.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. lift-+.f64 N/A

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

   11. Applied rewrites 22.5%

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

   if -3.4e18 < re < 1.3500000000000001e137

   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 68.6%

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 29.7%

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

   7. Applied rewrites 29.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sub-to-mult N/A

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

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

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

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

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

      8. lower-unsound-/.f64 29.6%

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

   9. Applied rewrites 29.6%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. sub-to-mult-rev N/A

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

       5. metadata-eval N/A

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

       6. add-flip N/A

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

       7. *-lft-identity N/A

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

       8. *-inverses N/A

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

       9. *-lft-identity N/A

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

       10. add-to-fraction-rev N/A

           \[\leadsto im \cdot \frac{re \cdot im + im}{im} \]

       11. *-commutative N/A

           \[\leadsto im \cdot \frac{im \cdot re + im}{im} \]

       12. lift-*.f64 N/A

           \[\leadsto im \cdot \frac{im \cdot re + im}{im} \]

       13. *-lft-identity N/A

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

       14. *-rgt-identity N/A

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

       15. *-lft-identity N/A

           \[\leadsto im \cdot \frac{im \cdot re + im \cdot 1}{im} \]

       16. lift-*.f64 N/A

           \[\leadsto im \cdot \frac{im \cdot re + im \cdot 1}{im} \]

       17. distribute-lft-in N/A

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

       18. associate-/l* N/A

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

       19. lower-*.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. add-flip N/A

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

       22. metadata-eval N/A

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

       23. lower--.f64 35.0%

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

   11. Applied rewrites 35.0%

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

   if 1.3500000000000001e137 < re

   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 68.6%

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

   4. Applied rewrites 68.6%

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

   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 29.7%

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

   7. Applied rewrites 29.7%

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

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

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

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

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

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

      6. lower-unsound-/.f64 18.1%

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

   9. Applied rewrites 18.1%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. sum-to-mult-rev N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. distribute-lft1-in N/A

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

       8. add-flip N/A

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

       9. metadata-eval N/A

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

       10. sub-to-mult-rev N/A

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

       11. lift-/.f64 N/A

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

       12. lift--.f64 N/A

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

       13. *-commutative N/A

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

       14. lift--.f64 N/A

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

       15. lift-/.f64 N/A

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

       16. sub-to-fraction N/A

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

       17. associate-*r/ N/A

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

       18. associate-*l/ N/A

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

       19. lower-/.f64 N/A

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

   11. Applied rewrites 25.1%

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

Alternative 8: 38.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -1.68 \cdot 10^{+19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(im, re, im\right) \cdot im}{im}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-38}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(re, re, re\right) \cdot im}{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= re -1.68e+19)
  (/ (* (fma im re im) im) im)
  (if (<= re 5e-38) im (/ (* (fma re re re) im) re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.68e+19) {
		tmp = (fma(im, re, im) * im) / im;
	} else if (re <= 5e-38) {
		tmp = im;
	} else {
		tmp = (fma(re, re, re) * im) / re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.68e+19)
		tmp = Float64(Float64(fma(im, re, im) * im) / im);
	elseif (re <= 5e-38)
		tmp = im;
	else
		tmp = Float64(Float64(fma(re, re, re) * im) / re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.68e+19], N[(N[(N[(im * re + im), $MachinePrecision] * im), $MachinePrecision] / im), $MachinePrecision], If[LessEqual[re, 5e-38], im, N[(N[(N[(re * re + re), $MachinePrecision] * im), $MachinePrecision] / re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.68e19

   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 68.6%

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

   4. Applied rewrites 68.6%

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

   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 29.7%

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

   7. Applied rewrites 29.7%

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

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

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

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

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

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

      6. lower-unsound-/.f64 18.1%

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

   9. Applied rewrites 18.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-+.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. add-to-fraction N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. frac-2neg N/A

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

       10. frac-2neg N/A

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

       11. *-lft-identity N/A

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

       12. lift-*.f64 N/A

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

       13. add-to-fraction-rev N/A

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

       14. sum-to-mult-rev N/A

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

       15. associate-/r* N/A

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

       16. *-commutative N/A

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

       17. lift-*.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. lift-+.f64 N/A

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

   11. Applied rewrites 22.5%

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

   if -1.68e19 < re < 5.0000000000000003e-38

   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 68.6%

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 26.7%

      \[\leadsto im \]

   if 5.0000000000000003e-38 < re

   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 68.6%

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

   4. Applied rewrites 68.6%

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

   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 29.7%

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

   7. Applied rewrites 29.7%

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

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

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

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

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

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

      6. lower-unsound-/.f64 18.1%

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

   9. Applied rewrites 18.1%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. sum-to-mult-rev N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. distribute-lft1-in N/A

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

       8. add-flip N/A

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

       9. metadata-eval N/A

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

       10. sub-to-mult-rev N/A

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

       11. lift-/.f64 N/A

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

       12. lift--.f64 N/A

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

       13. *-commutative N/A

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

       14. lift--.f64 N/A

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

       15. lift-/.f64 N/A

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

       16. sub-to-fraction N/A

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

       17. associate-*r/ N/A

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

       18. associate-*l/ N/A

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

       19. lower-/.f64 N/A

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

   11. Applied rewrites 25.1%

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

Alternative 9: 32.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(im, re, im\right) \cdot im}{im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im, im\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= re -3.4e+18) (/ (* (fma im re im) im) im) (fma re im im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e+18) {
		tmp = (fma(im, re, im) * im) / im;
	} else {
		tmp = fma(re, im, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e+18)
		tmp = Float64(Float64(fma(im, re, im) * im) / im);
	else
		tmp = fma(re, im, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e+18], N[(N[(N[(im * re + im), $MachinePrecision] * im), $MachinePrecision] / im), $MachinePrecision], N[(re * im + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.4e18

   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 68.6%

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

   4. Applied rewrites 68.6%

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

   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 29.7%

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

   7. Applied rewrites 29.7%

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

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

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

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

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

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

      6. lower-unsound-/.f64 18.1%

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

   9. Applied rewrites 18.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-+.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. add-to-fraction N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. frac-2neg N/A

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

       10. frac-2neg N/A

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

       11. *-lft-identity N/A

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

       12. lift-*.f64 N/A

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

       13. add-to-fraction-rev N/A

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

       14. sum-to-mult-rev N/A

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

       15. associate-/r* N/A

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

       16. *-commutative N/A

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

       17. lift-*.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. lift-+.f64 N/A

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

   11. Applied rewrites 22.5%

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

   if -3.4e18 < re

   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 68.6%

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

   4. Applied rewrites 68.6%

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

   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 29.7%

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

   7. Applied rewrites 29.7%

      \[\leadsto im + \color{blue}{im \cdot re} \]
   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 29.7%

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

   9. Applied rewrites 29.7%

      \[\leadsto \mathsf{fma}\left(re, im, im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 29.7% 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 68.6%

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

4. Applied rewrites 68.6%

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

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 29.7%

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

7. Applied rewrites 29.7%

   \[\leadsto im + \color{blue}{im \cdot re} \]
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 29.7%

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

9. Applied rewrites 29.7%

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

Alternative 11: 26.7% 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 68.6%

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

4. Applied rewrites 68.6%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 26.7%

   \[\leadsto im \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025238 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))