math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 21

Speedup: 1.0×

• 24.7% 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, 1.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (/ (sin im) (exp (- re))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh---cosh-rev N/A

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

   6. associate-*l/ N/A

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

   7. sinh-cosh N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (* (* re re) 0.5) (fma (pow im 3.0) -0.16666666666666666 im))
     (if (<= t_0 -0.05)
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (sin im))
       (if (<= t_0 2e-23)
         (/ im (exp (- re)))
         (if (<= t_0 1.0)
           (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
           (* (exp re) im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((re * re) * 0.5) * fma(pow(im, 3.0), -0.16666666666666666, im);
	} else if (t_0 <= -0.05) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else if (t_0 <= 2e-23) {
		tmp = im / exp(-re);
	} else if (t_0 <= 1.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma((im ^ 3.0), -0.16666666666666666, im));
	elseif (t_0 <= -0.05)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	elseif (t_0 <= 2e-23)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-23], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 49.6%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 48.0%

       \[\leadsto \left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot \sin im \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999992e-23

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 99.3%

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

   if 1.99999999999999992e-23 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 99.9%

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

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.9%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (* (* re re) 0.5) (fma (pow im 3.0) -0.16666666666666666 im))
     (if (<= t_0 -0.05)
       t_1
       (if (<= t_0 2e-23)
         (/ im (exp (- re)))
         (if (<= t_0 1.0) t_1 (* (exp re) im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((re * re) * 0.5) * fma(pow(im, 3.0), -0.16666666666666666, im);
	} else if (t_0 <= -0.05) {
		tmp = t_1;
	} else if (t_0 <= 2e-23) {
		tmp = im / exp(-re);
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma((im ^ 3.0), -0.16666666666666666, im));
	elseif (t_0 <= -0.05)
		tmp = t_1;
	elseif (t_0 <= 2e-23)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], t$95$1, If[LessEqual[t$95$0, 2e-23], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], t$95$1, N[(N[Exp[re], $MachinePrecision] * im), $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. Add Preprocessing

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 49.6%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 48.0%

       \[\leadsto \left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1.99999999999999992e-23 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 98.7%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999992e-23

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 99.3%

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

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.9%

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

Alternative 4: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* -0.16666666666666666 (* im im)) im im))
     (if (<= t_0 -0.05)
       t_1
       (if (<= t_0 2e-23)
         (/ im (exp (- re)))
         (if (<= t_0 1.0) t_1 (* (exp re) im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma((-0.16666666666666666 * (im * im)), im, im);
	} else if (t_0 <= -0.05) {
		tmp = t_1;
	} else if (t_0 <= 2e-23) {
		tmp = im / exp(-re);
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im));
	elseif (t_0 <= -0.05)
		tmp = t_1;
	elseif (t_0 <= 2e-23)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], t$95$1, If[LessEqual[t$95$0, 2e-23], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], t$95$1, N[(N[Exp[re], $MachinePrecision] * im), $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. Add Preprocessing

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1.99999999999999992e-23 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 98.7%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999992e-23

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 99.3%

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

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.9%

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

Alternative 5: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* -0.16666666666666666 (* im im)) im im))
     (if (<= t_0 -0.05)
       (* (+ 1.0 re) (sin im))
       (if (<= t_0 2e-23)
         (/ im (exp (- re)))
         (if (<= t_0 1.0) (/ (sin im) (- 1.0 re)) (* (exp re) im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma((-0.16666666666666666 * (im * im)), im, im);
	} else if (t_0 <= -0.05) {
		tmp = (1.0 + re) * sin(im);
	} else if (t_0 <= 2e-23) {
		tmp = im / exp(-re);
	} else if (t_0 <= 1.0) {
		tmp = sin(im) / (1.0 - re);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im));
	elseif (t_0 <= -0.05)
		tmp = Float64(Float64(1.0 + re) * sin(im));
	elseif (t_0 <= 2e-23)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im) / Float64(1.0 - re));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-23], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 97.6%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999992e-23

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 99.3%

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

   if 1.99999999999999992e-23 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 99.3%

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

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.9%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05 \lor \neg \left(t\_0 \leq 2 \cdot 10^{-23} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* -0.16666666666666666 (* im im)) im im))
     (if (or (<= t_0 -0.05) (not (or (<= t_0 2e-23) (not (<= t_0 1.0)))))
       (* (+ 1.0 re) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma((-0.16666666666666666 * (im * im)), im, im);
	} else if ((t_0 <= -0.05) || !((t_0 <= 2e-23) || !(t_0 <= 1.0))) {
		tmp = (1.0 + re) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im));
	elseif ((t_0 <= -0.05) || !((t_0 <= 2e-23) || !(t_0 <= 1.0)))
		tmp = Float64(Float64(1.0 + re) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.05], N[Not[Or[LessEqual[t$95$0, 2e-23], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $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. Add Preprocessing

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1.99999999999999992e-23 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 98.3%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999992e-23 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 93.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05 \lor \neg \left(e^{re} \cdot \sin im \leq 2 \cdot 10^{-23} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \left(1 + re\right) \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (+ 1.0 re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* -0.16666666666666666 (* im im)) im im))
     (if (<= t_0 -0.05)
       t_1
       (if (<= t_0 2e-23)
         (/ im (exp (- re)))
         (if (<= t_0 1.0) t_1 (* (exp re) im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = (1.0 + re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma((-0.16666666666666666 * (im * im)), im, im);
	} else if (t_0 <= -0.05) {
		tmp = t_1;
	} else if (t_0 <= 2e-23) {
		tmp = im / exp(-re);
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(Float64(1.0 + re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im));
	elseif (t_0 <= -0.05)
		tmp = t_1;
	elseif (t_0 <= 2e-23)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], t$95$1, If[LessEqual[t$95$0, 2e-23], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], t$95$1, N[(N[Exp[re], $MachinePrecision] * im), $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. Add Preprocessing

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1.99999999999999992e-23 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 98.3%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999992e-23

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 99.3%

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

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.9%

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

Alternative 8: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05 \lor \neg \left(t\_0 \leq 2 \cdot 10^{-23} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* -0.16666666666666666 (* im im)) im im))
     (if (or (<= t_0 -0.05) (not (or (<= t_0 2e-23) (not (<= t_0 1.0)))))
       (sin im)
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma((-0.16666666666666666 * (im * im)), im, im);
	} else if ((t_0 <= -0.05) || !((t_0 <= 2e-23) || !(t_0 <= 1.0))) {
		tmp = sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im));
	elseif ((t_0 <= -0.05) || !((t_0 <= 2e-23) || !(t_0 <= 1.0)))
		tmp = sin(im);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.05], N[Not[Or[LessEqual[t$95$0, 2e-23], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $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. Add Preprocessing

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1.99999999999999992e-23 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 96.1%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999992e-23 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 93.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05 \lor \neg \left(e^{re} \cdot \sin im \leq 2 \cdot 10^{-23} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{im}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(0.16666666666666666, re, 0.5\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* -0.16666666666666666 (* im im)) im im))
     (if (<= t_0 -0.05)
       (sin im)
       (if (<= t_0 2e-23)
         (/ im (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
         (if (<= t_0 1.0)
           (sin im)
           (* (* (* re re) (fma 0.16666666666666666 re 0.5)) im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma((-0.16666666666666666 * (im * im)), im, im);
	} else if (t_0 <= -0.05) {
		tmp = sin(im);
	} else if (t_0 <= 2e-23) {
		tmp = im / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = ((re * re) * fma(0.16666666666666666, re, 0.5)) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im));
	elseif (t_0 <= -0.05)
		tmp = sin(im);
	elseif (t_0 <= 2e-23)
		tmp = Float64(im / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(Float64(re * re) * fma(0.16666666666666666, re, 0.5)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 2e-23], N[(im / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision]), $MachinePrecision] * im), $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. Add Preprocessing

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1.99999999999999992e-23 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 96.1%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999992e-23

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 86.3%

       \[\leadsto \frac{im}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}} \]

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 31.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 31.5%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, re, 0.5\right)}\right) \cdot im \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0.55:\\ \;\;\;\;\frac{im}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(0.16666666666666666, re, 0.5\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.05)
     (* (+ 1.0 re) (fma (* -0.16666666666666666 (* im im)) im im))
     (if (<= t_0 0.55)
       (/ im (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
       (* (* (* re re) (fma 0.16666666666666666 re 0.5)) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = (1.0 + re) * fma((-0.16666666666666666 * (im * im)), im, im);
	} else if (t_0 <= 0.55) {
		tmp = im / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0);
	} else {
		tmp = ((re * re) * fma(0.16666666666666666, re, 0.5)) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im));
	elseif (t_0 <= 0.55)
		tmp = Float64(im / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0));
	else
		tmp = Float64(Float64(Float64(re * re) * fma(0.16666666666666666, re, 0.5)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.55], N[(im / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 14.4%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 14.4%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.55000000000000004

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 92.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 80.1%

       \[\leadsto \frac{im}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}} \]

   if 0.55000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 38.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 20.8%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, re, 0.5\right)}\right) \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.05)
     (* (+ 1.0 re) (fma (* -0.16666666666666666 (* im im)) im im))
     (if (<= t_0 0.0)
       (/ im (fma (fma 0.5 re -1.0) re 1.0))
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = (1.0 + re) * fma((-0.16666666666666666 * (im * im)), im, im);
	} else if (t_0 <= 0.0) {
		tmp = im / fma(fma(0.5, re, -1.0), re, 1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im));
	elseif (t_0 <= 0.0)
		tmp = Float64(im / fma(fma(0.5, re, -1.0), re, 1.0));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im / N[(N[(0.5 * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 14.4%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 14.4%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 71.6%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 49.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (/ im (fma (fma 0.5 re -1.0) re 1.0))
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im / fma(fma(0.5, re, -1.0), re, 1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im / N[(N[(0.5 * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 41.6%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 39.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (/ im (- 1.0 re))
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im / (1.0 - re);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 31.6%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 39.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.55:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(0.16666666666666666, re, 0.5\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.55)
   (/ im (- 1.0 re))
   (* (* (* re re) (fma 0.16666666666666666 re 0.5)) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.55) {
		tmp = im / (1.0 - re);
	} else {
		tmp = ((re * re) * fma(0.16666666666666666, re, 0.5)) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.55)
		tmp = Float64(im / Float64(1.0 - re));
	else
		tmp = Float64(Float64(Float64(re * re) * fma(0.16666666666666666, re, 0.5)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.55], N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.55000000000000004

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 72.2%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 41.5%

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

   if 0.55000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 38.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 20.8%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, re, 0.5\right)}\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 38.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (/ im (- 1.0 re))
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im / (1.0 - re);
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(im / Float64(1.0 - re));
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 31.6%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 50.9%

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

Alternative 16: 38.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.55:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.55)
   (/ im (- 1.0 re))
   (* (* (* re re) 0.5) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.55) {
		tmp = im / (1.0 - re);
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= 0.55d0) then
        tmp = im / (1.0d0 - re)
    else
        tmp = ((re * re) * 0.5d0) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 0.55) {
		tmp = im / (1.0 - re);
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.55:
		tmp = im / (1.0 - re)
	else:
		tmp = ((re * re) * 0.5) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.55)
		tmp = Float64(im / Float64(1.0 - re));
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 0.55)
		tmp = im / (1.0 - re);
	else
		tmp = ((re * re) * 0.5) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.55], N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.55000000000000004

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 72.2%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 41.5%

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

   if 0.55000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 24.5%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 18.1%

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

Alternative 17: 34.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0) (/ im (- 1.0 re)) (* (- re -1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im / (1.0 - re);
	} else {
		tmp = (re - -1.0) * im;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= 0.0d0) then
        tmp = im / (1.0d0 - re)
    else
        tmp = (re - (-1.0d0)) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 0.0) {
		tmp = im / (1.0 - re);
	} else {
		tmp = (re - -1.0) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.0:
		tmp = im / (1.0 - re)
	else:
		tmp = (re - -1.0) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(im / Float64(1.0 - re));
	else
		tmp = Float64(Float64(re - -1.0) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 0.0)
		tmp = im / (1.0 - re);
	else
		tmp = (re - -1.0) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(N[(re - -1.0), $MachinePrecision] * im), $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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 31.6%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.1%

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

Alternative 18: 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]

Derivation

1. Initial program 100.0%

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

Alternative 19: 28.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+21}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 6.8e+21) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+21) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (im <= 6.8d+21) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+21) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.8e+21:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.8e+21)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.8e+21)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.8e+21], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.8e21

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 38.2%

      \[\leadsto im \]

   if 6.8e21 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in re around inf

      \[\leadsto re \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 5.9%

       \[\leadsto re \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 30.1% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \left(re - -1\right) \cdot im \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Applied rewrites 67.2%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.2%

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

Alternative 21: 27.0% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

5. Applied rewrites 54.1%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 29.2%

   \[\leadsto im \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))