math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 15

Speedup: 1.0×

• 24.3% 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

Alternative 2: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re - -1\right) \cdot \sin im\\ t_1 := e^{re} \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq -0.004:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- re -1.0) (sin im)))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* (* (* im im) im) -0.16666666666666666))
     (if (<= t_1 -0.004)
       t_0
       (if (<= t_1 2e-157) t_2 (if (<= t_1 1.0) t_0 t_2))))))

C

double code(double re, double im) {
	double t_0 = (re - -1.0) * sin(im);
	double t_1 = exp(re) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else if (t_1 <= -0.004) {
		tmp = t_0;
	} else if (t_1 <= 2e-157) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (re - -1.0) * Math.sin(im);
	double t_1 = Math.exp(re) * Math.sin(im);
	double t_2 = Math.exp(re) * im;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else if (t_1 <= -0.004) {
		tmp = t_0;
	} else if (t_1 <= 2e-157) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re - -1.0) * math.sin(im)
	t_1 = math.exp(re) * math.sin(im)
	t_2 = math.exp(re) * im
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.exp(re) * (((im * im) * im) * -0.16666666666666666)
	elif t_1 <= -0.004:
		tmp = t_0
	elif t_1 <= 2e-157:
		tmp = t_2
	elif t_1 <= 1.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re - -1.0) * sin(im))
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	elseif (t_1 <= -0.004)
		tmp = t_0;
	elseif (t_1 <= 2e-157)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re - -1.0) * sin(im);
	t_1 = exp(re) * sin(im);
	t_2 = exp(re) * im;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	elseif (t_1 <= -0.004)
		tmp = t_0;
	elseif (t_1 <= 2e-157)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re - -1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.004], t$95$0, If[LessEqual[t$95$1, 2e-157], t$95$2, If[LessEqual[t$95$1, 1.0], t$95$0, 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. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 23.9

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

   7. Applied rewrites 23.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0040000000000000001 or 1.99999999999999989e-157 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 98.9

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

   4. Applied rewrites 98.9%

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

   if -0.0040000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999989e-157 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 94.4%

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

Alternative 3: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.004:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* (* im im) im) -0.16666666666666666))
     (if (<= t_0 -0.004)
       (sin im)
       (if (<= t_0 1.2e-135) 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 = exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else if (t_0 <= -0.004) {
		tmp = sin(im);
	} else if (t_0 <= 1.2e-135) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.sin(im);
	double t_1 = Math.exp(re) * im;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else if (t_0 <= -0.004) {
		tmp = Math.sin(im);
	} else if (t_0 <= 1.2e-135) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.sin(im)
	t_1 = math.exp(re) * im
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * (((im * im) * im) * -0.16666666666666666)
	elif t_0 <= -0.004:
		tmp = math.sin(im)
	elif t_0 <= 1.2e-135:
		tmp = t_1
	elif t_0 <= 1.0:
		tmp = math.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(exp(re) * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	elseif (t_0 <= -0.004)
		tmp = sin(im);
	elseif (t_0 <= 1.2e-135)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * sin(im);
	t_1 = exp(re) * im;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	elseif (t_0 <= -0.004)
		tmp = sin(im);
	elseif (t_0 <= 1.2e-135)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	tmp_2 = 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[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.004], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1.2e-135], 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. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 23.9

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

   7. Applied rewrites 23.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0040000000000000001 or 1.1999999999999999e-135 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lift-sin.f64 98.0

         \[\leadsto \sin im \]

   4. Applied rewrites 98.0%

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

   if -0.0040000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.1999999999999999e-135 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 94.5%

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

Alternative 4: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -0.004:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))))
   (if (<= t_1 -0.004)
     t_2
     (if (<= t_1 2e-157) t_0 (if (<= t_1 1.0) t_2 t_0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double t_2 = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	double tmp;
	if (t_1 <= -0.004) {
		tmp = t_2;
	} else if (t_1 <= 2e-157) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im))
	tmp = 0.0
	if (t_1 <= -0.004)
		tmp = t_2;
	elseif (t_1 <= 2e-157)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.004], t$95$2, If[LessEqual[t$95$1, 2e-157], t$95$0, If[LessEqual[t$95$1, 1.0], t$95$2, t$95$0]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0040000000000000001 or 1.99999999999999989e-157 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 86.5

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

   4. Applied rewrites 86.5%

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

   if -0.0040000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999989e-157 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 94.4%

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

Alternative 5: 69.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.004:\\ \;\;\;\;e^{re} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.004)
   (* (exp re) (* (fma (* im im) -0.16666666666666666 1.0) im))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.004) {
		tmp = exp(re) * (fma((im * im), -0.16666666666666666, 1.0) * im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.004)
		tmp = Float64(exp(re) * Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.5

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

   4. Applied rewrites 37.5%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 80.6%

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

Alternative 6: 63.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.004:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.004)
   (* (exp re) (* (* (* im im) im) -0.16666666666666666))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.004) {
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else {
		tmp = exp(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 ((exp(re) * sin(im)) <= (-0.004d0)) then
        tmp = exp(re) * (((im * im) * im) * (-0.16666666666666666d0))
    else
        tmp = exp(re) * 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)) <= -0.004) {
		tmp = Math.exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= -0.004:
		tmp = math.exp(re) * (((im * im) * im) * -0.16666666666666666)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.004)
		tmp = Float64(exp(re) * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= -0.004)
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 12.7

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

   7. Applied rewrites 12.7%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 80.6%

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

Alternative 7: 63.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.004)
   (* (fma (fma 0.5 re 1.0) re 1.0) (* (* (* im im) -0.16666666666666666) im))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.004) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * (((im * im) * -0.16666666666666666) * im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.004)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 12.7

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

   7. Applied rewrites 12.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 12.3

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

   10. Applied rewrites 12.3%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 80.6%

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

Alternative 8: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.004:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.004)
   (* (* (* re re) 0.5) (* (* (* im im) -0.16666666666666666) im))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.004) {
		tmp = ((re * re) * 0.5) * (((im * im) * -0.16666666666666666) * im);
	} else {
		tmp = exp(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 ((exp(re) * sin(im)) <= (-0.004d0)) then
        tmp = ((re * re) * 0.5d0) * (((im * im) * (-0.16666666666666666d0)) * im)
    else
        tmp = exp(re) * 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)) <= -0.004) {
		tmp = ((re * re) * 0.5) * (((im * im) * -0.16666666666666666) * im);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= -0.004:
		tmp = ((re * re) * 0.5) * (((im * im) * -0.16666666666666666) * im)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.004)
		tmp = Float64(Float64(Float64(re * re) * 0.5) * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= -0.004)
		tmp = ((re * re) * 0.5) * (((im * im) * -0.16666666666666666) * im);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 12.7

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

   7. Applied rewrites 12.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 12.3

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

   10. Applied rewrites 12.3%

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

   11. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 12.0

          \[\leadsto \left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right) \]

   13. Applied rewrites 12.0%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 80.6%

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

Alternative 9: 63.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.004:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.004)
   (* re (* (fma (* im im) -0.16666666666666666 1.0) im))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.004) {
		tmp = re * (fma((im * im), -0.16666666666666666, 1.0) * im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.004)
		tmp = Float64(re * Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(re * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in re around inf

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

   7. Applied rewrites 4.1%

      \[\leadsto re \cdot \sin im \]

   8. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 11.9

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

   10. Applied rewrites 11.9%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 80.6%

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

Alternative 10: 62.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.004)
   (* (fma -0.16666666666666666 (* im im) 1.0) im)
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.004) {
		tmp = fma(-0.16666666666666666, (im * im), 1.0) * im;
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.004)
		tmp = Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * im);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lift-sin.f64 51.7

         \[\leadsto \sin im \]

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

      3. +-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right) \cdot im \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} + 1\right) \cdot im \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, im \cdot im, 1\right) \cdot im \]

      12. lift-*.f64 11.5

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot im \]

   7. Applied rewrites 11.5%

      \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot \color{blue}{im} \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 9.5%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 80.6%

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

Alternative 11: 30.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* 1.0 (* (* (* im im) im) -0.16666666666666666))
   (* (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 = 1.0 * (((im * im) * im) * -0.16666666666666666);
	} 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(1.0 * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	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[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $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. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 32.5

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 18.1%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.0

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

   4. Applied rewrites 61.0%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 10.1

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

   10. Applied rewrites 10.1%

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

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot im \]
   12. Step-by-step derivation

   13. Applied rewrites 49.7%

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

Alternative 12: 29.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;1 \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 0.945:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 0.0)
     (* 1.0 (* (* (* im im) im) -0.16666666666666666))
     (if (<= t_0 0.945) im (* re im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 * (((im * im) * im) * -0.16666666666666666);
	} else if (t_0 <= 0.945) {
		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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * sin(im)
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 * (((im * im) * im) * (-0.16666666666666666d0))
    else if (t_0 <= 0.945d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.sin(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 * (((im * im) * im) * -0.16666666666666666);
	} else if (t_0 <= 0.945) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.sin(im)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 * (((im * im) * im) * -0.16666666666666666)
	elif t_0 <= 0.945:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	elseif (t_0 <= 0.945)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * sin(im);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 1.0 * (((im * im) * im) * -0.16666666666666666);
	elseif (t_0 <= 0.945)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = 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, 0.0], N[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.945], im, N[(re * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 32.5

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 18.1%

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

   if -0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.944999999999999951

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lift-sin.f64 97.8

         \[\leadsto \sin im \]

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

      3. +-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right) \cdot im \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} + 1\right) \cdot im \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, im \cdot im, 1\right) \cdot im \]

      12. lift-*.f64 57.3

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot im \]

   7. Applied rewrites 57.3%

      \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot \color{blue}{im} \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 57.3%

       \[\leadsto im \]

   if 0.944999999999999951 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 18.9

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

   4. Applied rewrites 18.9%

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

   5. Taylor expanded in re around inf

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

   7. Applied rewrites 4.3%

      \[\leadsto re \cdot \sin im \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 12.0%

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

Alternative 13: 28.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 2e-10)
   (* (fma -0.16666666666666666 (* im im) 1.0) im)
   (* re im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 2e-10) {
		tmp = fma(-0.16666666666666666, (im * im), 1.0) * im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 2e-10)
		tmp = Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * im);
	else
		tmp = Float64(re * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 2e-10], N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 2.00000000000000007e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lift-sin.f64 51.4

         \[\leadsto \sin im \]

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

      3. +-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right) \cdot im \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} + 1\right) \cdot im \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, im \cdot im, 1\right) \cdot im \]

      12. lift-*.f64 37.7

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot im \]

   7. Applied rewrites 37.7%

      \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot \color{blue}{im} \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 37.1%

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

   if 2.00000000000000007e-10 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 51.2

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in re around inf

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

   7. Applied rewrites 4.1%

      \[\leadsto re \cdot \sin im \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 8.9%

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

Alternative 14: 27.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.15:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 3.15) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.15) {
		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 <= 3.15d0) 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 <= 3.15) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.15:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.15)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.15)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.15], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.14999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lift-sin.f64 50.9

         \[\leadsto \sin im \]

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

      3. +-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right) \cdot im \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} + 1\right) \cdot im \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, im \cdot im, 1\right) \cdot im \]

      12. lift-*.f64 38.6

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot im \]

   7. Applied rewrites 38.6%

      \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot \color{blue}{im} \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 35.5%

       \[\leadsto im \]

   if 3.14999999999999991 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 52.3

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in re around inf

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

   7. Applied rewrites 3.8%

      \[\leadsto re \cdot \sin im \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 8.3%

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

Alternative 15: 25.9% accurate, 45.8× 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. Taylor expanded in re around 0

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

   1. lift-sin.f64 51.1

      \[\leadsto \sin im \]

4. Applied rewrites 51.1%

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

5. Taylor expanded in im around 0

   \[\leadsto im \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot im \]

   3. +-commutative N/A

      \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right) \cdot im \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} + 1\right) \cdot im \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

   6. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

   9. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, {im}^{2}, 1\right) \cdot im \]

   11. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{120} - \frac{1}{6}, im \cdot im, 1\right) \cdot im \]

   12. lift-*.f64 31.5

       \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot im \]

7. Applied rewrites 31.5%

   \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.008333333333333333 - 0.16666666666666666, im \cdot im, 1\right) \cdot \color{blue}{im} \]

8. Taylor expanded in im around 0

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

10. Applied rewrites 27.4%

    \[\leadsto im \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025122 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))