math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

Alternatives: 12

Speedup: 0.2×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

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

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

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: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ t_1 := e^{re} \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\sin im\_m \cdot \frac{1}{1 - re}\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))) (t_1 (* (exp re) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (exp re) (* (* (* im_m im_m) im_m) -0.16666666666666666))
      (if (<= t_0 -0.05)
        (* (sin im_m) (/ 1.0 (- 1.0 re)))
        (if (<= t_0 0.004)
          t_1
          (if (<= t_0 1.0) (* (- re -1.0) (sin im_m)) t_1)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double t_1 = exp(re) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	} else if (t_0 <= -0.05) {
		tmp = sin(im_m) * (1.0 / (1.0 - re));
	} else if (t_0 <= 0.004) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = (re - -1.0) * sin(im_m);
	} else {
		tmp = t_1;
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(re) * Math.sin(im_m);
	double t_1 = Math.exp(re) * im_m;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	} else if (t_0 <= -0.05) {
		tmp = Math.sin(im_m) * (1.0 / (1.0 - re));
	} else if (t_0 <= 0.004) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = (re - -1.0) * Math.sin(im_m);
	} else {
		tmp = t_1;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(re) * math.sin(im_m)
	t_1 = math.exp(re) * im_m
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666)
	elif t_0 <= -0.05:
		tmp = math.sin(im_m) * (1.0 / (1.0 - re))
	elif t_0 <= 0.004:
		tmp = t_1
	elif t_0 <= 1.0:
		tmp = (re - -1.0) * math.sin(im_m)
	else:
		tmp = t_1
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(re) * sin(im_m))
	t_1 = Float64(exp(re) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(Float64(im_m * im_m) * im_m) * -0.16666666666666666));
	elseif (t_0 <= -0.05)
		tmp = Float64(sin(im_m) * Float64(1.0 / Float64(1.0 - re)));
	elseif (t_0 <= 0.004)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(Float64(re - -1.0) * sin(im_m));
	else
		tmp = t_1;
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(re) * sin(im_m);
	t_1 = exp(re) * im_m;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	elseif (t_0 <= -0.05)
		tmp = sin(im_m) * (1.0 / (1.0 - re));
	elseif (t_0 <= 0.004)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = (re - -1.0) * sin(im_m);
	else
		tmp = t_1;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[Sin[im$95$m], $MachinePrecision] * N[(1.0 / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.004], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[(re - -1.0), $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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

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

      1. +-commutative N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

         \[\leadsto \frac{1 \cdot 1 - re \cdot re}{\color{blue}{1 - re}} \cdot \sin im \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 \cdot 1 - re \cdot re}{\color{blue}{1 - re}} \cdot \sin im \]

      7. metadata-eval N/A

         \[\leadsto \frac{1 - re \cdot re}{1 - re} \cdot \sin im \]

      8. unpow2 N/A

         \[\leadsto \frac{1 - {re}^{2}}{1 - re} \cdot \sin im \]

      9. lower--.f64 N/A

         \[\leadsto \frac{1 - {re}^{2}}{\color{blue}{1} - re} \cdot \sin im \]

      10. unpow2 N/A

          \[\leadsto \frac{1 - re \cdot re}{1 - re} \cdot \sin im \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1 - re \cdot re}{1 - re} \cdot \sin im \]

      12. lower--.f64 98.1

          \[\leadsto \frac{1 - re \cdot re}{1 - \color{blue}{re}} \cdot \sin im \]

   6. Applied rewrites 98.1%

      \[\leadsto \frac{1 - re \cdot re}{\color{blue}{1 - re}} \cdot \sin im \]

   7. Taylor expanded in re around 0

      \[\leadsto \frac{1}{\color{blue}{1} - re} \cdot \sin im \]
   8. Step-by-step derivation

   9. Applied rewrites 98.0%

      \[\leadsto \frac{1}{\color{blue}{1} - re} \cdot \sin im \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{1 - re} \cdot \sin im} \]

       2. lift-sin.f64 N/A

          \[\leadsto \frac{1}{1 - re} \cdot \color{blue}{\sin im} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\sin im \cdot \frac{1}{1 - re}} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin im \cdot \frac{1}{1 - re}} \]

       5. lift-sin.f64 98.0

          \[\leadsto \color{blue}{\sin im} \cdot \frac{1}{1 - re} \]

   11. Applied rewrites 98.0%

       \[\leadsto \color{blue}{\sin im \cdot \frac{1}{1 - re}} \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0040000000000000001 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

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

   4. Applied rewrites 99.0%

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

   if 0.0040000000000000001 < (*.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. +-commutative N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval 98.9

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

   4. Applied rewrites 98.9%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(e^{re} \cdot \sin im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* (exp re) (sin im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (exp(re) * sin(im_m));
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (exp(re) * sin(im_m))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (Math.exp(re) * Math.sin(im_m));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (math.exp(re) * math.sin(im_m))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(exp(re) * sin(im_m)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (exp(re) * sin(im_m));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

Alternative 3: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(re - -1\right) \cdot \sin im\_m\\ t_1 := e^{re} \cdot \sin im\_m\\ t_2 := e^{re} \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.004:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- re -1.0) (sin im_m)))
        (t_1 (* (exp re) (sin im_m)))
        (t_2 (* (exp re) im_m)))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* (exp re) (* (* (* im_m im_m) im_m) -0.16666666666666666))
      (if (<= t_1 -0.05)
        t_0
        (if (<= t_1 0.004) t_2 (if (<= t_1 1.0) t_0 t_2)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (re - -1.0) * sin(im_m);
	double t_1 = exp(re) * sin(im_m);
	double t_2 = exp(re) * im_m;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 0.004) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = (re - -1.0) * Math.sin(im_m);
	double t_1 = Math.exp(re) * Math.sin(im_m);
	double t_2 = Math.exp(re) * im_m;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 0.004) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = (re - -1.0) * math.sin(im_m)
	t_1 = math.exp(re) * math.sin(im_m)
	t_2 = math.exp(re) * im_m
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666)
	elif t_1 <= -0.05:
		tmp = t_0
	elif t_1 <= 0.004:
		tmp = t_2
	elif t_1 <= 1.0:
		tmp = t_0
	else:
		tmp = t_2
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(re - -1.0) * sin(im_m))
	t_1 = Float64(exp(re) * sin(im_m))
	t_2 = Float64(exp(re) * im_m)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(Float64(im_m * im_m) * im_m) * -0.16666666666666666));
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 0.004)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = (re - -1.0) * sin(im_m);
	t_1 = exp(re) * sin(im_m);
	t_2 = exp(re) * im_m;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 0.004)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(re - -1.0), $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$0, If[LessEqual[t$95$1, 0.004], t$95$2, If[LessEqual[t$95$1, 1.0], t$95$0, t$95$2]]]]), $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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 0.0040000000000000001 < (*.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. +-commutative N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0040000000000000001 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

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

   4. Applied rewrites 99.0%

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

Alternative 4: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ t_1 := e^{re} \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\sin im\_m\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))) (t_1 (* (exp re) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (exp re) (* (* (* im_m im_m) im_m) -0.16666666666666666))
      (if (<= t_0 -0.05)
        (sin im_m)
        (if (<= t_0 0.004) t_1 (if (<= t_0 1.0) (sin im_m) t_1)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double t_1 = exp(re) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	} else if (t_0 <= -0.05) {
		tmp = sin(im_m);
	} else if (t_0 <= 0.004) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im_m);
	} else {
		tmp = t_1;
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(re) * Math.sin(im_m);
	double t_1 = Math.exp(re) * im_m;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	} else if (t_0 <= -0.05) {
		tmp = Math.sin(im_m);
	} else if (t_0 <= 0.004) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(im_m);
	} else {
		tmp = t_1;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(re) * math.sin(im_m)
	t_1 = math.exp(re) * im_m
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666)
	elif t_0 <= -0.05:
		tmp = math.sin(im_m)
	elif t_0 <= 0.004:
		tmp = t_1
	elif t_0 <= 1.0:
		tmp = math.sin(im_m)
	else:
		tmp = t_1
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(re) * sin(im_m))
	t_1 = Float64(exp(re) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(Float64(im_m * im_m) * im_m) * -0.16666666666666666));
	elseif (t_0 <= -0.05)
		tmp = sin(im_m);
	elseif (t_0 <= 0.004)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im_m);
	else
		tmp = t_1;
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(re) * sin(im_m);
	t_1 = exp(re) * im_m;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	elseif (t_0 <= -0.05)
		tmp = sin(im_m);
	elseif (t_0 <= 0.004)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im_m);
	else
		tmp = t_1;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Sin[im$95$m], $MachinePrecision], If[LessEqual[t$95$0, 0.004], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im$95$m], $MachinePrecision], t$95$1]]]]), $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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 0.0040000000000000001 < (*.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. lift-sin.f64 97.2

         \[\leadsto \sin im \]

   4. Applied rewrites 97.2%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0040000000000000001 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

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

   4. Applied rewrites 99.0%

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

Alternative 5: 75.6% accurate, 0.6× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) -0.05)
    (* (exp re) (* (* (* im_m im_m) im_m) -0.16666666666666666))
    (* (exp re) im_m))))

C

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

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(re) * sin(im_m)) <= (-0.05d0)) then
        tmp = exp(re) * (((im_m * im_m) * im_m) * (-0.16666666666666666d0))
    else
        tmp = exp(re) * im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im_m)) <= -0.05) {
		tmp = Math.exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	} else {
		tmp = Math.exp(re) * im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(re) * math.sin(im_m)) <= -0.05:
		tmp = math.exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666)
	else:
		tmp = math.exp(re) * im_m
	return im_s * tmp

Julia

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

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(re) * sin(im_m)) <= -0.05)
		tmp = exp(re) * (((im_m * im_m) * im_m) * -0.16666666666666666);
	else
		tmp = exp(re) * im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], -0.05], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im$95$m), $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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.0

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

   4. Applied rewrites 36.0%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 36.0

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

   7. Applied rewrites 36.0%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 84.4%

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

Alternative 6: 73.6% accurate, 0.7× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) -0.05)
    (* 1.0 (* (* im_m im_m) (* -0.16666666666666666 im_m)))
    (* (exp re) im_m))))

C

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

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(re) * sin(im_m)) <= (-0.05d0)) then
        tmp = 1.0d0 * ((im_m * im_m) * ((-0.16666666666666666d0) * im_m))
    else
        tmp = exp(re) * im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im_m)) <= -0.05) {
		tmp = 1.0 * ((im_m * im_m) * (-0.16666666666666666 * im_m));
	} else {
		tmp = Math.exp(re) * im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(re) * math.sin(im_m)) <= -0.05:
		tmp = 1.0 * ((im_m * im_m) * (-0.16666666666666666 * im_m))
	else:
		tmp = math.exp(re) * im_m
	return im_s * tmp

Julia

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

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(re) * sin(im_m)) <= -0.05)
		tmp = 1.0 * ((im_m * im_m) * (-0.16666666666666666 * im_m));
	else
		tmp = exp(re) * im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], -0.05], N[(1.0 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im$95$m), $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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.0

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

   4. Applied rewrites 36.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 24.8

         \[\leadsto 1 \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right) \]

   10. Applied rewrites 24.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 24.8

           \[\leadsto 1 \cdot \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right) \]

   12. Applied rewrites 24.8%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 84.4%

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

Alternative 7: 49.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;1 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 \cdot im\_m\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;\left(re - -1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_0 0.0)
      (* 1.0 (* (* im_m im_m) (* -0.16666666666666666 im_m)))
      (if (<= t_0 0.995)
        (* (- re -1.0) im_m)
        (* (fma (* re re) 0.5 re) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 * ((im_m * im_m) * (-0.16666666666666666 * im_m));
	} else if (t_0 <= 0.995) {
		tmp = (re - -1.0) * im_m;
	} else {
		tmp = fma((re * re), 0.5, re) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 * im_m)));
	elseif (t_0 <= 0.995)
		tmp = Float64(Float64(re - -1.0) * im_m);
	else
		tmp = Float64(fma(Float64(re * re), 0.5, re) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, 0.0], N[(1.0 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.995], N[(N[(re - -1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5 + re), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.2

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 30.3

         \[\leadsto 1 \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right) \]

   10. Applied rewrites 30.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 30.3

           \[\leadsto 1 \cdot \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right) \]

   12. Applied rewrites 30.3%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 67.7

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \]

   7. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot im \]

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 67.4

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

   10. Applied rewrites 67.4%

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

   if 0.994999999999999996 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 59.1

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \]

   7. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot im \]

   8. Taylor expanded in re around inf

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

      1. distribute-lft-in N/A

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

      2. inv-pow N/A

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

      3. pow-prod-up N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 59.0

         \[\leadsto \mathsf{fma}\left(re \cdot re, 0.5, re\right) \cdot im \]

   10. Applied rewrites 59.0%

       \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{0.5}, re\right) \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 49.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0:\\ \;\;\;\;1 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 \cdot im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* 1.0 (* (* im_m im_m) (* -0.16666666666666666 im_m)))
    (* (fma (fma 0.5 re 1.0) re 1.0) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = 1.0 * ((im_m * im_m) * (-0.16666666666666666 * im_m));
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

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

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.2

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 30.3

         \[\leadsto 1 \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right) \]

   10. Applied rewrites 30.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 30.3

           \[\leadsto 1 \cdot \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right) \]

   12. Applied rewrites 30.3%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 64.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \]

   7. Applied rewrites 64.6%

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

Alternative 9: 37.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.995:\\ \;\;\;\;1 \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, re\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.995)
    (* 1.0 im_m)
    (* (fma (* re re) 0.5 re) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.995) {
		tmp = 1.0 * im_m;
	} else {
		tmp = fma((re * re), 0.5, re) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.995)
		tmp = Float64(1.0 * im_m);
	else
		tmp = Float64(fma(Float64(re * re), 0.5, re) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.995], N[(1.0 * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5 + re), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 31.9%

      \[\leadsto \color{blue}{1} \cdot im \]

   if 0.994999999999999996 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 59.1

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \]

   7. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot im \]

   8. Taylor expanded in re around inf

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

      1. distribute-lft-in N/A

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

      2. inv-pow N/A

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

      3. pow-prod-up N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 59.0

         \[\leadsto \mathsf{fma}\left(re \cdot re, 0.5, re\right) \cdot im \]

   10. Applied rewrites 59.0%

       \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{0.5}, re\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 37.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.98:\\ \;\;\;\;1 \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.98)
    (* 1.0 im_m)
    (* (* (* re re) 0.5) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.98) {
		tmp = 1.0 * im_m;
	} else {
		tmp = ((re * re) * 0.5) * im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(re) * sin(im_m)) <= 0.98d0) then
        tmp = 1.0d0 * im_m
    else
        tmp = ((re * re) * 0.5d0) * im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im_m)) <= 0.98) {
		tmp = 1.0 * im_m;
	} else {
		tmp = ((re * re) * 0.5) * im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(re) * math.sin(im_m)) <= 0.98:
		tmp = 1.0 * im_m
	else:
		tmp = ((re * re) * 0.5) * im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.98)
		tmp = Float64(1.0 * im_m);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(re) * sin(im_m)) <= 0.98)
		tmp = 1.0 * im_m;
	else
		tmp = ((re * re) * 0.5) * im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.98], N[(1.0 * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.97999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 63.1%

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

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 32.3%

      \[\leadsto \color{blue}{1} \cdot im \]

   if 0.97999999999999998 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 56.7

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \]

   7. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot im \]

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 56.7

         \[\leadsto \left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im \]

   10. Applied rewrites 56.7%

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

Alternative 11: 29.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(re - -1\right) \cdot im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* (- re -1.0) im_m)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * ((re - -1.0) * im_m);
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * ((re - (-1.0d0)) * im_m)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * ((re - -1.0) * im_m);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((re - -1.0) * im_m)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(re - -1.0) * im_m))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * ((re - -1.0) * im_m);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(N[(re - -1.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 69.2%

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

5. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-fma.f64 37.3

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \]

7. Applied rewrites 37.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot im \]

8. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. add-flip N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 29.5

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

10. Applied rewrites 29.5%

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

Alternative 12: 26.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(1 \cdot im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* 1.0 im_m)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (1.0 * im_m);
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (1.0d0 * im_m)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (1.0 * im_m);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (1.0 * im_m)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(1.0 * im_m))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (1.0 * im_m);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(1.0 * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 69.2%

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

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \cdot im \]
6. Step-by-step derivation

7. Applied rewrites 26.2%

   \[\leadsto \color{blue}{1} \cdot im \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025128 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))