math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 12

Speedup: 1.0×

• 24.9% 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

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

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 2: 90.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (fma re (fma re 0.5 1.0) 1.0) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
      (* im (fma im (* im -0.16666666666666666) 1.0)))
     (if (<= t_0 -0.1) t_1 (if (<= t_0 0.0) t_2 (if (<= t_0 1.0) t_1 t_2))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(re, fma(re, 0.5, 1.0), 1.0) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * (im * fma(im, (im * -0.16666666666666666), 1.0));
	} else if (t_0 <= -0.1) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * Float64(im * fma(im, Float64(im * -0.16666666666666666), 1.0)));
	elseif (t_0 <= -0.1)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_2;
	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[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], t$95$1, If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$2]]]]]]]

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

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 or 0.0 < (*.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

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

   5. Applied rewrites 99.2%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 90.8%

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

Alternative 3: 90.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \left(re + 1\right) \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-49}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (+ re 1.0) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
      (* im (fma im (* im -0.16666666666666666) 1.0)))
     (if (<= t_0 -0.1)
       t_1
       (if (<= t_0 2e-49) t_2 (if (<= t_0 1.0) t_1 t_2))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = (re + 1.0) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * (im * fma(im, (im * -0.16666666666666666), 1.0));
	} else if (t_0 <= -0.1) {
		tmp = t_1;
	} else if (t_0 <= 2e-49) {
		tmp = t_2;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(Float64(re + 1.0) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * Float64(im * fma(im, Float64(im * -0.16666666666666666), 1.0)));
	elseif (t_0 <= -0.1)
		tmp = t_1;
	elseif (t_0 <= 2e-49)
		tmp = t_2;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_2;
	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[(re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], t$95$1, If[LessEqual[t$95$0, 2e-49], t$95$2, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$2]]]]]]]

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

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 or 1.99999999999999987e-49 < (*.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

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

   5. Applied rewrites 98.4%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999987e-49 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

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

   5. Applied rewrites 92.3%

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

Alternative 4: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
      (* im (fma im (* im -0.16666666666666666) 1.0)))
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 2e-49) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * (im * fma(im, (im * -0.16666666666666666), 1.0));
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 2e-49) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * Float64(im * fma(im, Float64(im * -0.16666666666666666), 1.0)));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 2e-49)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	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[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 2e-49], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

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

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 or 1.99999999999999987e-49 < (*.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

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

   5. Applied rewrites 97.7%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999987e-49 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

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

   5. Applied rewrites 92.3%

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

Alternative 5: 58.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))
   (if (<= t_0 (- INFINITY))
     (* t_1 (* im (fma im (* im -0.16666666666666666) 1.0)))
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 0.0)
         (* im (* im (* im -0.16666666666666666)))
         (if (<= t_0 1.0) (sin im) (* t_1 im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * (im * fma(im, (im * -0.16666666666666666), 1.0));
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 0.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1 * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * Float64(im * fma(im, Float64(im * -0.16666666666666666), 1.0)));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(im * Float64(im * Float64(im * -0.16666666666666666)));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(t_1 * 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[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(t$95$1 * 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

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 or 0.0 < (*.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

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

   5. Applied rewrites 98.3%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 32.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.9%

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

   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

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 46.2%

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

Alternative 6: 30.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 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(im * Float64(im * -0.16666666666666666)));
	else
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 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[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 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. Taylor expanded in re around 0

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 25.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.6%

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

   if 0.0 < (*.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

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 60.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 52.9%

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

Alternative 7: 29.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else {
		tmp = fma(re, fma(re, 0.5, 1.0), 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(im * Float64(im * -0.16666666666666666)));
	else
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 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[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 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. Taylor expanded in re around 0

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 25.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.6%

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

   if 0.0 < (*.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

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 60.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 47.3%

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

Alternative 8: 25.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} 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 * (im * (im * (-0.16666666666666666d0)))
    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 * (im * (im * -0.16666666666666666));
	} 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 * (im * (im * -0.16666666666666666))
	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(im * Float64(im * -0.16666666666666666)));
	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 * (im * (im * -0.16666666666666666));
	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[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $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. Taylor expanded in re around 0

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 25.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.6%

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

   if 0.0 < (*.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

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 40.6%

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

Alternative 9: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \sin im\\ \mathbf{if}\;re \leq -0.0225:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.11:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, \left(-0.16666666666666666 \cdot im\right) \cdot im, im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
          (sin im))))
   (if (<= re -0.0225)
     (* (exp re) im)
     (if (<= re 0.11)
       t_0
       (if (<= re 1e+103)
         (* (exp re) (fma im (* (* -0.16666666666666666 im) im) im))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * sin(im);
	double tmp;
	if (re <= -0.0225) {
		tmp = exp(re) * im;
	} else if (re <= 0.11) {
		tmp = t_0;
	} else if (re <= 1e+103) {
		tmp = exp(re) * fma(im, ((-0.16666666666666666 * im) * im), im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * sin(im))
	tmp = 0.0
	if (re <= -0.0225)
		tmp = Float64(exp(re) * im);
	elseif (re <= 0.11)
		tmp = t_0;
	elseif (re <= 1e+103)
		tmp = Float64(exp(re) * fma(im, Float64(Float64(-0.16666666666666666 * im) * im), im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0225], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 0.11], t$95$0, If[LessEqual[re, 1e+103], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.022499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   if -0.022499999999999999 < re < 0.110000000000000001 or 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 99.4%

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

   if 0.110000000000000001 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 85.7%

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

Alternative 10: 27.8% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.8 \cdot 10^{+28}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 5.8e+28) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.8e+28) {
		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 <= 5.8d+28) 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 <= 5.8e+28) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.8e+28:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.8e+28)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.8e+28)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.8e+28], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.8000000000000002e28

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto im \]

   8. Applied rewrites 33.4%

      \[\leadsto im \]

   if 5.8000000000000002e28 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 22.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 31.2%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 6.9%

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

   12. Taylor expanded in re around inf

       \[\leadsto re \cdot im \]

   14. Applied rewrites 7.8%

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

Alternative 11: 29.5% 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 re around 0

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 29.3%

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

Alternative 12: 26.2% 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

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

5. Applied rewrites 49.6%

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

6. Taylor expanded in im around 0

   \[\leadsto im \]

8. Applied rewrites 26.1%

   \[\leadsto im \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025036 -o generate:simplify -o generate:proofs
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))