math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 2.8s

Alternatives: 15

Speedup: 1.0×

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

double code(double re, double im) {
	return exp(re) * cos(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) * cos(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

double code(double re, double im) {
	return exp(re) * cos(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) * cos(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{e^{-re}}} \cdot \cos im \]

4. Taylor expanded in re around inf

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

6. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

double code(double re, double im) {
	return exp(re) * cos(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) * cos(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 3: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(1 + \left(-0.5 + -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999997:\\ \;\;\;\;\frac{\cos im}{1 + -1 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) -0.5)
     (if (<= t_0 -0.02)
       (* (+ 1.0 re) (cos im))
       (if (<= t_0 0.0)
         (* (+ 1.0 re) (+ 1.0 (+ -0.5 -0.5)))
         (if (<= t_0 0.9999999999999997)
           (/ (cos im) (+ 1.0 (* -1.0 re)))
           (* (exp re) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * -0.5;
	} else if (t_0 <= -0.02) {
		tmp = (1.0 + re) * cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	} else if (t_0 <= 0.9999999999999997) {
		tmp = cos(im) / (1.0 + (-1.0 * re));
	} else {
		tmp = exp(re) * 1.0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * -0.5;
	} else if (t_0 <= -0.02) {
		tmp = (1.0 + re) * Math.cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	} else if (t_0 <= 0.9999999999999997) {
		tmp = Math.cos(im) / (1.0 + (-1.0 * re));
	} else {
		tmp = Math.exp(re) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * -0.5
	elif t_0 <= -0.02:
		tmp = (1.0 + re) * math.cos(im)
	elif t_0 <= 0.0:
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5))
	elif t_0 <= 0.9999999999999997:
		tmp = math.cos(im) / (1.0 + (-1.0 * re))
	else:
		tmp = math.exp(re) * 1.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * -0.5);
	elseif (t_0 <= -0.02)
		tmp = Float64(Float64(1.0 + re) * cos(im));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(1.0 + Float64(-0.5 + -0.5)));
	elseif (t_0 <= 0.9999999999999997)
		tmp = Float64(cos(im) / Float64(1.0 + Float64(-1.0 * re)));
	else
		tmp = Float64(exp(re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * -0.5;
	elseif (t_0 <= -0.02)
		tmp = (1.0 + re) * cos(im);
	elseif (t_0 <= 0.0)
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	elseif (t_0 <= 0.9999999999999997)
		tmp = cos(im) / (1.0 + (-1.0 * re));
	else
		tmp = exp(re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(1.0 + N[(-0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999997], N[(N[Cos[im], $MachinePrecision] / N[(1.0 + N[(-1.0 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 49.7%

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

   8. Applied rewrites 33.7%

      \[\leadsto \color{blue}{e^{re} \cdot -0.5} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 26.0%

      \[\leadsto e^{re} \cdot \left(1 + \left(-0.5 + \color{blue}{-0.5}\right)\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + \left(-0.5 + -0.5\right)\right) \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999667

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-re}}} \cdot \cos im \]

   4. Taylor expanded in re around inf

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 51.6%

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

   if 0.999999999999999667 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto e^{re} \cdot \color{blue}{1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + re\right) \cdot \cos im\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(1 + \left(-0.5 + -0.5\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999997:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 re) (cos im))) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) -0.5)
     (if (<= t_1 -0.02)
       t_0
       (if (<= t_1 0.0)
         (* (+ 1.0 re) (+ 1.0 (+ -0.5 -0.5)))
         (if (<= t_1 0.9999999999999997) t_0 (* (exp re) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = (1.0 + re) * cos(im);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * -0.5;
	} else if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	} else if (t_1 <= 0.9999999999999997) {
		tmp = t_0;
	} else {
		tmp = exp(re) * 1.0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (1.0 + re) * Math.cos(im);
	double t_1 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * -0.5;
	} else if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	} else if (t_1 <= 0.9999999999999997) {
		tmp = t_0;
	} else {
		tmp = Math.exp(re) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (1.0 + re) * math.cos(im)
	t_1 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.exp(re) * -0.5
	elif t_1 <= -0.02:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5))
	elif t_1 <= 0.9999999999999997:
		tmp = t_0
	else:
		tmp = math.exp(re) * 1.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(1.0 + re) * cos(im))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * -0.5);
	elseif (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(1.0 + Float64(-0.5 + -0.5)));
	elseif (t_1 <= 0.9999999999999997)
		tmp = t_0;
	else
		tmp = Float64(exp(re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (1.0 + re) * cos(im);
	t_1 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = exp(re) * -0.5;
	elseif (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	elseif (t_1 <= 0.9999999999999997)
		tmp = t_0;
	else
		tmp = exp(re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(1.0 + N[(-0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999997], t$95$0, N[(N[Exp[re], $MachinePrecision] * 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 49.7%

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

   8. Applied rewrites 33.7%

      \[\leadsto \color{blue}{e^{re} \cdot -0.5} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999667

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 26.0%

      \[\leadsto e^{re} \cdot \left(1 + \left(-0.5 + \color{blue}{-0.5}\right)\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + \left(-0.5 + -0.5\right)\right) \]

   if 0.999999999999999667 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto e^{re} \cdot \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(1 + \left(-0.5 + -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999997:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) -0.5)
     (if (<= t_0 -0.02)
       (cos im)
       (if (<= t_0 0.0)
         (* (+ 1.0 re) (+ 1.0 (+ -0.5 -0.5)))
         (if (<= t_0 0.9999999999999997) (cos im) (* (exp re) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * -0.5;
	} else if (t_0 <= -0.02) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	} else if (t_0 <= 0.9999999999999997) {
		tmp = cos(im);
	} else {
		tmp = exp(re) * 1.0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * -0.5;
	} else if (t_0 <= -0.02) {
		tmp = Math.cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	} else if (t_0 <= 0.9999999999999997) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * -0.5
	elif t_0 <= -0.02:
		tmp = math.cos(im)
	elif t_0 <= 0.0:
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5))
	elif t_0 <= 0.9999999999999997:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re) * 1.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * -0.5);
	elseif (t_0 <= -0.02)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(1.0 + Float64(-0.5 + -0.5)));
	elseif (t_0 <= 0.9999999999999997)
		tmp = cos(im);
	else
		tmp = Float64(exp(re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * -0.5;
	elseif (t_0 <= -0.02)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	elseif (t_0 <= 0.9999999999999997)
		tmp = cos(im);
	else
		tmp = exp(re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(1.0 + N[(-0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999997], N[Cos[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 49.7%

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

   8. Applied rewrites 33.7%

      \[\leadsto \color{blue}{e^{re} \cdot -0.5} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999667

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.1%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 26.0%

      \[\leadsto e^{re} \cdot \left(1 + \left(-0.5 + \color{blue}{-0.5}\right)\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + \left(-0.5 + -0.5\right)\right) \]

   if 0.999999999999999667 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto e^{re} \cdot \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;e^{re} \cdot \left(\left(1 - -0.5\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re + re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.02)
   (* (exp re) (* (- 1.0 -0.5) -0.5))
   (* (exp (+ re re)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.02) {
		tmp = exp(re) * ((1.0 - -0.5) * -0.5);
	} else {
		tmp = exp((re + re)) * 1.0;
	}
	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) * cos(im)) <= (-0.02d0)) then
        tmp = exp(re) * ((1.0d0 - (-0.5d0)) * (-0.5d0))
    else
        tmp = exp((re + re)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= -0.02) {
		tmp = Math.exp(re) * ((1.0 - -0.5) * -0.5);
	} else {
		tmp = Math.exp((re + re)) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= -0.02:
		tmp = math.exp(re) * ((1.0 - -0.5) * -0.5)
	else:
		tmp = math.exp((re + re)) * 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.02)
		tmp = Float64(exp(re) * Float64(Float64(1.0 - -0.5) * -0.5));
	else
		tmp = Float64(exp(Float64(re + re)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= -0.02)
		tmp = exp(re) * ((1.0 - -0.5) * -0.5);
	else
		tmp = exp((re + re)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[Exp[re], $MachinePrecision] * N[(N[(1.0 - -0.5), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(re + re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 33.8%

      \[\leadsto e^{re} \cdot \left(\left(1 - -0.5\right) \cdot \color{blue}{-0.5}\right) \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto e^{re} \cdot \color{blue}{1} \]

   10. Applied rewrites 69.9%

       \[\leadsto e^{\color{blue}{re + re}} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;e^{re} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;e^{re + re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.02)
   (* (exp re) -0.5)
   (* (exp (+ re re)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.02) {
		tmp = exp(re) * -0.5;
	} else {
		tmp = exp((re + re)) * 1.0;
	}
	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) * cos(im)) <= (-0.02d0)) then
        tmp = exp(re) * (-0.5d0)
    else
        tmp = exp((re + re)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= -0.02) {
		tmp = Math.exp(re) * -0.5;
	} else {
		tmp = Math.exp((re + re)) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= -0.02:
		tmp = math.exp(re) * -0.5
	else:
		tmp = math.exp((re + re)) * 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.02)
		tmp = Float64(exp(re) * -0.5);
	else
		tmp = Float64(exp(Float64(re + re)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= -0.02)
		tmp = exp(re) * -0.5;
	else
		tmp = exp((re + re)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[Exp[re], $MachinePrecision] * -0.5), $MachinePrecision], N[(N[Exp[N[(re + re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 49.7%

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

   8. Applied rewrites 33.7%

      \[\leadsto \color{blue}{e^{re} \cdot -0.5} \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto e^{re} \cdot \color{blue}{1} \]

   10. Applied rewrites 69.9%

       \[\leadsto e^{\color{blue}{re + re}} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;e^{re} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.02) (* (exp re) -0.5) (* (exp re) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.02) {
		tmp = exp(re) * -0.5;
	} else {
		tmp = exp(re) * 1.0;
	}
	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) * cos(im)) <= (-0.02d0)) then
        tmp = exp(re) * (-0.5d0)
    else
        tmp = exp(re) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= -0.02) {
		tmp = Math.exp(re) * -0.5;
	} else {
		tmp = Math.exp(re) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= -0.02:
		tmp = math.exp(re) * -0.5
	else:
		tmp = math.exp(re) * 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.02)
		tmp = Float64(exp(re) * -0.5);
	else
		tmp = Float64(exp(re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= -0.02)
		tmp = exp(re) * -0.5;
	else
		tmp = exp(re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[Exp[re], $MachinePrecision] * -0.5), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 49.7%

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

   8. Applied rewrites 33.7%

      \[\leadsto \color{blue}{e^{re} \cdot -0.5} \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto e^{re} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;e^{re} \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\left(1 + re\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(im + re\right) \cdot \left(im + -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* (exp re) -0.5)
     (if (<= t_0 2.0) (* (+ 1.0 re) 1.0) (* (+ im re) (+ im -0.5))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = exp(re) * -0.5;
	} else if (t_0 <= 2.0) {
		tmp = (1.0 + re) * 1.0;
	} else {
		tmp = (im + re) * (im + -0.5);
	}
	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) * cos(im)
    if (t_0 <= 0.0d0) then
        tmp = exp(re) * (-0.5d0)
    else if (t_0 <= 2.0d0) then
        tmp = (1.0d0 + re) * 1.0d0
    else
        tmp = (im + re) * (im + (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.exp(re) * -0.5;
	} else if (t_0 <= 2.0) {
		tmp = (1.0 + re) * 1.0;
	} else {
		tmp = (im + re) * (im + -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.exp(re) * -0.5
	elif t_0 <= 2.0:
		tmp = (1.0 + re) * 1.0
	else:
		tmp = (im + re) * (im + -0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(exp(re) * -0.5);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(1.0 + re) * 1.0);
	else
		tmp = Float64(Float64(im + re) * Float64(im + -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = exp(re) * -0.5;
	elseif (t_0 <= 2.0)
		tmp = (1.0 + re) * 1.0;
	else
		tmp = (im + re) * (im + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Exp[re], $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(1.0 + re), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(im + re), $MachinePrecision] * N[(im + -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 49.7%

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

   8. Applied rewrites 33.7%

      \[\leadsto \color{blue}{e^{re} \cdot -0.5} \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto e^{re} \cdot \color{blue}{1} \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 29.0%

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

   if 2 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + -0.5\right) \]

   11. Applied rewrites 3.5%

       \[\leadsto \left(im + re\right) \cdot \left(1 + -0.5\right) \]

   13. Applied rewrites 6.5%

       \[\leadsto \left(im + re\right) \cdot \left(im + -0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(1 + \left(-0.5 + -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\left(1 + re\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(im + re\right) \cdot \left(im + -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.02)
     (* (+ 1.0 re) -0.5)
     (if (<= t_0 0.0)
       (* (+ 1.0 re) (+ 1.0 (+ -0.5 -0.5)))
       (if (<= t_0 2.0) (* (+ 1.0 re) 1.0) (* (+ im re) (+ im -0.5)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (1.0 + re) * -0.5;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	} else if (t_0 <= 2.0) {
		tmp = (1.0 + re) * 1.0;
	} else {
		tmp = (im + re) * (im + -0.5);
	}
	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) * cos(im)
    if (t_0 <= (-0.02d0)) then
        tmp = (1.0d0 + re) * (-0.5d0)
    else if (t_0 <= 0.0d0) then
        tmp = (1.0d0 + re) * (1.0d0 + ((-0.5d0) + (-0.5d0)))
    else if (t_0 <= 2.0d0) then
        tmp = (1.0d0 + re) * 1.0d0
    else
        tmp = (im + re) * (im + (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (1.0 + re) * -0.5;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	} else if (t_0 <= 2.0) {
		tmp = (1.0 + re) * 1.0;
	} else {
		tmp = (im + re) * (im + -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -0.02:
		tmp = (1.0 + re) * -0.5
	elif t_0 <= 0.0:
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5))
	elif t_0 <= 2.0:
		tmp = (1.0 + re) * 1.0
	else:
		tmp = (im + re) * (im + -0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(Float64(1.0 + re) * -0.5);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(1.0 + Float64(-0.5 + -0.5)));
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(1.0 + re) * 1.0);
	else
		tmp = Float64(Float64(im + re) * Float64(im + -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = (1.0 + re) * -0.5;
	elseif (t_0 <= 0.0)
		tmp = (1.0 + re) * (1.0 + (-0.5 + -0.5));
	elseif (t_0 <= 2.0)
		tmp = (1.0 + re) * 1.0;
	else
		tmp = (im + re) * (im + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(1.0 + re), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(1.0 + N[(-0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(1.0 + re), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(im + re), $MachinePrecision] * N[(im + -0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + -0.5\right) \]

   11. Applied rewrites 4.1%

       \[\leadsto \left(1 + re\right) \cdot -0.5 \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 26.0%

      \[\leadsto e^{re} \cdot \left(1 + \left(-0.5 + \color{blue}{-0.5}\right)\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + \left(-0.5 + -0.5\right)\right) \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto e^{re} \cdot \color{blue}{1} \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 29.0%

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

   if 2 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + -0.5\right) \]

   11. Applied rewrites 3.5%

       \[\leadsto \left(im + re\right) \cdot \left(1 + -0.5\right) \]

   13. Applied rewrites 6.5%

       \[\leadsto \left(im + re\right) \cdot \left(im + -0.5\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 34.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im + re\right) \cdot \left(im + -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.02)
     (* (+ 1.0 re) -0.5)
     (if (<= t_0 2.0) 1.0 (* (+ im re) (+ im -0.5))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (1.0 + re) * -0.5;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (im + re) * (im + -0.5);
	}
	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) * cos(im)
    if (t_0 <= (-0.02d0)) then
        tmp = (1.0d0 + re) * (-0.5d0)
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (im + re) * (im + (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (1.0 + re) * -0.5;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (im + re) * (im + -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -0.02:
		tmp = (1.0 + re) * -0.5
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = (im + re) * (im + -0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(Float64(1.0 + re) * -0.5);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(im + re) * Float64(im + -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = (1.0 + re) * -0.5;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = (im + re) * (im + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(1.0 + re), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(N[(im + re), $MachinePrecision] * N[(im + -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + -0.5\right) \]

   11. Applied rewrites 4.1%

       \[\leadsto \left(1 + re\right) \cdot -0.5 \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 29.4%

      \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]

   9. Applied rewrites 3.3%

      \[\leadsto im \]

   11. Applied rewrites 28.6%

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

   if 2 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + -0.5\right) \]

   11. Applied rewrites 3.5%

       \[\leadsto \left(im + re\right) \cdot \left(1 + -0.5\right) \]

   13. Applied rewrites 6.5%

       \[\leadsto \left(im + re\right) \cdot \left(im + -0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 31.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0) -0.5 (* (+ 1.0 re) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = -0.5;
	} else {
		tmp = (1.0 + re) * 1.0;
	}
	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) * cos(im)) <= 0.0d0) then
        tmp = -0.5d0
    else
        tmp = (1.0d0 + re) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 0.0) {
		tmp = -0.5;
	} else {
		tmp = (1.0 + re) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 0.0:
		tmp = -0.5
	else:
		tmp = (1.0 + re) * 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], -0.5, N[(N[(1.0 + re), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 29.4%

      \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]

   9. Applied rewrites 8.8%

      \[\leadsto 1 + -0.5 \cdot {\left(im \cdot \frac{1}{im}\right)}^{2} \]

   11. Applied rewrites 4.2%

       \[\leadsto \color{blue}{-0.5} \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto e^{re} \cdot \color{blue}{1} \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 29.0%

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

Alternative 13: 31.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.02) (* (+ 1.0 re) -0.5) 1.0))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.02) {
		tmp = (1.0 + re) * -0.5;
	} else {
		tmp = 1.0;
	}
	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) * cos(im)) <= (-0.02d0)) then
        tmp = (1.0d0 + re) * (-0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= -0.02) {
		tmp = (1.0 + re) * -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= -0.02:
		tmp = (1.0 + re) * -0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.02)
		tmp = Float64(Float64(1.0 + re) * -0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= -0.02)
		tmp = (1.0 + re) * -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(1.0 + re), $MachinePrecision] * -0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 62.6%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto e^{re} \cdot \left(1 + -0.5\right) \]

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \left(1 + -0.5\right) \]

   11. Applied rewrites 4.1%

       \[\leadsto \left(1 + re\right) \cdot -0.5 \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 29.4%

      \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]

   9. Applied rewrites 3.3%

      \[\leadsto im \]

   11. Applied rewrites 28.6%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 31.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	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) * cos(im)) <= 0.0d0) then
        tmp = -0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 0.0) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 0.0:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 29.4%

      \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]

   9. Applied rewrites 8.8%

      \[\leadsto 1 + -0.5 \cdot {\left(im \cdot \frac{1}{im}\right)}^{2} \]

   11. Applied rewrites 4.2%

       \[\leadsto \color{blue}{-0.5} \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 29.4%

      \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]

   9. Applied rewrites 3.3%

      \[\leadsto im \]

   11. Applied rewrites 28.6%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 4.2% accurate, 46.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -0.5)

C

double code(double re, double im) {
	return -0.5;
}

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 = -0.5d0
end function

Java

public static double code(double re, double im) {
	return -0.5;
}

Python

def code(re, im):
	return -0.5

Julia

function code(re, im)
	return -0.5
end

MATLAB

function tmp = code(re, im)
	tmp = -0.5;
end

Wolfram

code[re_, im_] := -0.5

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0

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

4. Applied rewrites 51.1%

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

5. Taylor expanded in im around 0

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

7. Applied rewrites 29.4%

   \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]

9. Applied rewrites 8.8%

   \[\leadsto 1 + -0.5 \cdot {\left(im \cdot \frac{1}{im}\right)}^{2} \]

11. Applied rewrites 4.2%

    \[\leadsto \color{blue}{-0.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))