math.sin on complex, imaginary part

Percentage Accurate: 54.3% → 99.8%

Time: 11.0s

Alternatives: 12

Speedup: 5.6×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(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 = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

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

Initial Program: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(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 = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;\left(-im\_m\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.6e-6)
    (* (- im_m) (cos re))
    (* (- (exp (- im_m)) (exp im_m)) (* (cos re) 0.5)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.6e-6) {
		tmp = -im_m * cos(re);
	} else {
		tmp = (exp(-im_m) - exp(im_m)) * (cos(re) * 0.5);
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.6d-6) then
        tmp = -im_m * cos(re)
    else
        tmp = (exp(-im_m) - exp(im_m)) * (cos(re) * 0.5d0)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.6e-6) {
		tmp = -im_m * Math.cos(re);
	} else {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (Math.cos(re) * 0.5);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.6e-6:
		tmp = -im_m * math.cos(re)
	else:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (math.cos(re) * 0.5)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.6e-6)
		tmp = Float64(Float64(-im_m) * cos(re));
	else
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(cos(re) * 0.5));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.6e-6)
		tmp = -im_m * cos(re);
	else
		tmp = (exp(-im_m) - exp(im_m)) * (cos(re) * 0.5);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.6e-6], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.6e-6

   1. Initial program 34.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 72.1%

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

   if 4.6e-6 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-literal N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2}} \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. lift-var N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos \color{blue}{re}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\cos re}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

      5. lift-literal N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{\color{blue}{0} - im} - e^{im}\right) \]

      6. lift-var N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - \color{blue}{im}} - e^{im}\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{\color{blue}{0 - im}} - e^{im}\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{e^{0 - im}} - e^{im}\right) \]

      9. lift-var N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{\color{blue}{im}}\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

   4. Applied rewrites 100.0%

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

Alternative 2: 86.6% accurate, 0.4× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (/ (/ (/ (* (* im_m im_m) (* im_m im_m)) (- im_m)) im_m) im_m)
      (if (<= t_0 0.0)
        (* (- im_m) (cos re))
        (/ (* (* (* im_m im_m) im_m) (/ (cos re) im_m)) (- im_m)))))))

C

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

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m;
	} else if (t_0 <= 0.0) {
		tmp = -im_m * Math.cos(re);
	} else {
		tmp = (((im_m * im_m) * im_m) * (Math.cos(re) / im_m)) / -im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m
	elif t_0 <= 0.0:
		tmp = -im_m * math.cos(re)
	else:
		tmp = (((im_m * im_m) * im_m) * (math.cos(re) / im_m)) / -im_m
	return im_s * tmp

Julia

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

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m;
	elseif (t_0 <= 0.0)
		tmp = -im_m * cos(re);
	else
		tmp = (((im_m * im_m) * im_m) * (cos(re) / im_m)) / -im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 5.7%

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{-1 \cdot {im}^{2}}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

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

   12. Applied rewrites 68.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 6.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.7%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 7.1%

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 57.4%

      \[\leadsto \frac{\left(\left(\left(-im\right) \cdot im\right) \cdot im\right) \cdot \frac{\cos re}{im}}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{-im}}{im}}{im}\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{\cos re}{im}}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{\frac{\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)}{-im\_m}}{im\_m}}{im\_m}\\ \mathbf{elif}\;t\_0 \leq 10^{-12}:\\ \;\;\;\;\left(-im\_m\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re}{im\_m} \cdot \left(\left(-im\_m\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (/ (/ (/ (* (* im_m im_m) (* im_m im_m)) (- im_m)) im_m) im_m)
      (if (<= t_0 1e-12)
        (* (- im_m) (cos re))
        (* (/ (cos re) im_m) (* (- im_m) im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m;
	} else if (t_0 <= 1e-12) {
		tmp = -im_m * cos(re);
	} else {
		tmp = (cos(re) / im_m) * (-im_m * im_m);
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m;
	} else if (t_0 <= 1e-12) {
		tmp = -im_m * Math.cos(re);
	} else {
		tmp = (Math.cos(re) / im_m) * (-im_m * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m
	elif t_0 <= 1e-12:
		tmp = -im_m * math.cos(re)
	else:
		tmp = (math.cos(re) / im_m) * (-im_m * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)) / Float64(-im_m)) / im_m) / im_m);
	elseif (t_0 <= 1e-12)
		tmp = Float64(Float64(-im_m) * cos(re));
	else
		tmp = Float64(Float64(cos(re) / im_m) * Float64(Float64(-im_m) * im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m;
	elseif (t_0 <= 1e-12)
		tmp = -im_m * cos(re);
	else
		tmp = (cos(re) / im_m) * (-im_m * im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision] / im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-12], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] / im$95$m), $MachinePrecision] * N[((-im$95$m) * im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 5.7%

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{-1 \cdot {im}^{2}}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

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

   12. Applied rewrites 68.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 9.9999999999999998e-13

   1. Initial program 6.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.7%

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

   if 9.9999999999999998e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 7.1%

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 45.4%

      \[\leadsto \frac{\cos re}{im} \cdot \color{blue}{\left(\left(-im\right) \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{-im}}{im}}{im}\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 10^{-12}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re}{im} \cdot \left(\left(-im\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 0.7× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) (- INFINITY))
    (/ (/ (/ (* (* im_m im_m) (* im_m im_m)) (- im_m)) im_m) im_m)
    (* (- im_m) (cos re)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 5.7%

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{-1 \cdot {im}^{2}}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

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

   12. Applied rewrites 68.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 34.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 72.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{-im}}{im}}{im}\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -0.05:\\ \;\;\;\;\frac{\frac{\frac{\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)}{-im\_m}}{im\_m}}{im\_m}\\ \mathbf{else}:\\ \;\;\;\;-im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) -0.05)
    (/ (/ (/ (* (* im_m im_m) (* im_m im_m)) (- im_m)) im_m) im_m)
    (- im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= -0.05) {
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m;
	} else {
		tmp = -im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im_m) - exp(im_m))) <= (-0.05d0)) then
        tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m
    else
        tmp = -im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= -0.05) {
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m;
	} else {
		tmp = -im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))) <= -0.05:
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m
	else:
		tmp = -im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= -0.05)
		tmp = Float64(Float64(Float64(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)) / Float64(-im_m)) / im_m) / im_m);
	else
		tmp = Float64(-im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= -0.05)
		tmp = ((((im_m * im_m) * (im_m * im_m)) / -im_m) / im_m) / im_m;
	else
		tmp = -im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision] / im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision], (-im$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 5.7%

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{-1 \cdot {im}^{2}}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

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

   12. Applied rewrites 68.0%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 34.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.0%

      \[\leadsto -im \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.05:\\ \;\;\;\;\frac{\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{-im}}{im}}{im}\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.4% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) (- INFINITY))
    (/ (/ (* (* (- im_m) im_m) im_m) im_m) im_m)
    (- im_m))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 5.7%

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{-1 \cdot {im}^{2}}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

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

   12. Applied rewrites 62.4%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 34.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.0%

      \[\leadsto -im \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{\left(\left(-im\right) \cdot im\right) \cdot im}{im}}{im}\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.6% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) (- INFINITY))
    (/ (* (- im_m) im_m) im_m)
    (- im_m))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 5.7%

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{-1 \cdot {im}^{2}}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 34.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.0%

      \[\leadsto -im \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\frac{\left(-im\right) \cdot im}{im}\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(-im\_m\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\left(-im\_m\right) \cdot \cos re\\ \mathbf{elif}\;im\_m \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot t\_0}{im\_m \cdot im\_m} \cdot \cos re}{im\_m \cdot im\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re}{im\_m} \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- im_m) im_m)))
   (*
    im_s
    (if (<= im_m 5e+41)
      (* (- im_m) (cos re))
      (if (<= im_m 1.4e+154)
        (/
         (* (/ (* (* (* im_m im_m) im_m) t_0) (* im_m im_m)) (cos re))
         (* im_m im_m))
        (* (/ (cos re) im_m) t_0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -im_m * im_m;
	double tmp;
	if (im_m <= 5e+41) {
		tmp = -im_m * cos(re);
	} else if (im_m <= 1.4e+154) {
		tmp = (((((im_m * im_m) * im_m) * t_0) / (im_m * im_m)) * cos(re)) / (im_m * im_m);
	} else {
		tmp = (cos(re) / im_m) * t_0;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -im_m * im_m
    if (im_m <= 5d+41) then
        tmp = -im_m * cos(re)
    else if (im_m <= 1.4d+154) then
        tmp = (((((im_m * im_m) * im_m) * t_0) / (im_m * im_m)) * cos(re)) / (im_m * im_m)
    else
        tmp = (cos(re) / im_m) * t_0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = -im_m * im_m;
	double tmp;
	if (im_m <= 5e+41) {
		tmp = -im_m * Math.cos(re);
	} else if (im_m <= 1.4e+154) {
		tmp = (((((im_m * im_m) * im_m) * t_0) / (im_m * im_m)) * Math.cos(re)) / (im_m * im_m);
	} else {
		tmp = (Math.cos(re) / im_m) * t_0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = -im_m * im_m
	tmp = 0
	if im_m <= 5e+41:
		tmp = -im_m * math.cos(re)
	elif im_m <= 1.4e+154:
		tmp = (((((im_m * im_m) * im_m) * t_0) / (im_m * im_m)) * math.cos(re)) / (im_m * im_m)
	else:
		tmp = (math.cos(re) / im_m) * t_0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(-im_m) * im_m)
	tmp = 0.0
	if (im_m <= 5e+41)
		tmp = Float64(Float64(-im_m) * cos(re));
	elseif (im_m <= 1.4e+154)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(im_m * im_m) * im_m) * t_0) / Float64(im_m * im_m)) * cos(re)) / Float64(im_m * im_m));
	else
		tmp = Float64(Float64(cos(re) / im_m) * t_0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = -im_m * im_m;
	tmp = 0.0;
	if (im_m <= 5e+41)
		tmp = -im_m * cos(re);
	elseif (im_m <= 1.4e+154)
		tmp = (((((im_m * im_m) * im_m) * t_0) / (im_m * im_m)) * cos(re)) / (im_m * im_m);
	else
		tmp = (cos(re) / im_m) * t_0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[((-im$95$m) * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 5e+41], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.4e+154], N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision] / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] / im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 5.00000000000000022e41

   1. Initial program 36.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 69.3%

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

   if 5.00000000000000022e41 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 4.1%

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

   7. Applied rewrites 56.7%

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

   9. Applied rewrites 85.5%

      \[\leadsto \frac{\frac{\left(\left(\left(-im\right) \cdot im\right) \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im} \cdot \cos re}{im \cdot im} \]

   if 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 7.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

      \[\leadsto \frac{\cos re}{im} \cdot \color{blue}{\left(\left(-im\right) \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\left(\left(im \cdot im\right) \cdot im\right) \cdot \left(\left(-im\right) \cdot im\right)}{im \cdot im} \cdot \cos re}{im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re}{im} \cdot \left(\left(-im\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;\left(-im\_m\right) \cdot \cos re\\ \mathbf{elif}\;im\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right) \cdot \cos re}{\left(\left(-im\_m\right) \cdot im\_m\right) \cdot im\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot \frac{\cos re}{im\_m}}{-im\_m}\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 6.5e-35)
    (* (- im_m) (cos re))
    (if (<= im_m 5.6e+102)
      (/
       (* (* (* im_m im_m) (* im_m im_m)) (cos re))
       (* (* (- im_m) im_m) im_m))
      (/ (* (* (* im_m im_m) im_m) (/ (cos re) im_m)) (- im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.5e-35) {
		tmp = -im_m * cos(re);
	} else if (im_m <= 5.6e+102) {
		tmp = (((im_m * im_m) * (im_m * im_m)) * cos(re)) / ((-im_m * im_m) * im_m);
	} else {
		tmp = (((im_m * im_m) * im_m) * (cos(re) / im_m)) / -im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 6.5d-35) then
        tmp = -im_m * cos(re)
    else if (im_m <= 5.6d+102) then
        tmp = (((im_m * im_m) * (im_m * im_m)) * cos(re)) / ((-im_m * im_m) * im_m)
    else
        tmp = (((im_m * im_m) * im_m) * (cos(re) / im_m)) / -im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.5e-35) {
		tmp = -im_m * Math.cos(re);
	} else if (im_m <= 5.6e+102) {
		tmp = (((im_m * im_m) * (im_m * im_m)) * Math.cos(re)) / ((-im_m * im_m) * im_m);
	} else {
		tmp = (((im_m * im_m) * im_m) * (Math.cos(re) / im_m)) / -im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 6.5e-35:
		tmp = -im_m * math.cos(re)
	elif im_m <= 5.6e+102:
		tmp = (((im_m * im_m) * (im_m * im_m)) * math.cos(re)) / ((-im_m * im_m) * im_m)
	else:
		tmp = (((im_m * im_m) * im_m) * (math.cos(re) / im_m)) / -im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 6.5e-35)
		tmp = Float64(Float64(-im_m) * cos(re));
	elseif (im_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)) * cos(re)) / Float64(Float64(Float64(-im_m) * im_m) * im_m));
	else
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * im_m) * Float64(cos(re) / im_m)) / Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 6.5e-35)
		tmp = -im_m * cos(re);
	elseif (im_m <= 5.6e+102)
		tmp = (((im_m * im_m) * (im_m * im_m)) * cos(re)) / ((-im_m * im_m) * im_m);
	else
		tmp = (((im_m * im_m) * im_m) * (cos(re) / im_m)) / -im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 6.5e-35], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.6e+102], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision] / N[(N[((-im$95$m) * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] / im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6.4999999999999999e-35

   1. Initial program 34.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 71.3%

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

   if 6.4999999999999999e-35 < im < 5.60000000000000037e102

   1. Initial program 79.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 27.8%

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

   7. Applied rewrites 27.8%

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

   9. Applied rewrites 27.8%

      \[\leadsto \frac{\left(\left(\left(-im\right) \cdot im\right) \cdot im\right) \cdot \frac{\cos re}{im}}{im} \]

   11. Applied rewrites 44.6%

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

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 6.4%

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

   7. Applied rewrites 79.4%

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

   9. Applied rewrites 100.0%

      \[\leadsto \frac{\left(\left(\left(-im\right) \cdot im\right) \cdot im\right) \cdot \frac{\cos re}{im}}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \cos re}{\left(\left(-im\right) \cdot im\right) \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{\cos re}{im}}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(-im\_m\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 10^{-34}:\\ \;\;\;\;-im\_m\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot im\_m}{t\_0}}{im\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot im\_m}{im\_m}}{im\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- im_m) im_m)))
   (*
    im_s
    (if (<= im_m 1e-34)
      (- im_m)
      (if (<= im_m 2e+102)
        (/ (/ (* (* (* im_m im_m) im_m) im_m) t_0) im_m)
        (/ (/ (* t_0 im_m) im_m) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -im_m * im_m;
	double tmp;
	if (im_m <= 1e-34) {
		tmp = -im_m;
	} else if (im_m <= 2e+102) {
		tmp = ((((im_m * im_m) * im_m) * im_m) / t_0) / im_m;
	} else {
		tmp = ((t_0 * im_m) / im_m) / im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -im_m * im_m
    if (im_m <= 1d-34) then
        tmp = -im_m
    else if (im_m <= 2d+102) then
        tmp = ((((im_m * im_m) * im_m) * im_m) / t_0) / im_m
    else
        tmp = ((t_0 * im_m) / im_m) / im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = -im_m * im_m;
	double tmp;
	if (im_m <= 1e-34) {
		tmp = -im_m;
	} else if (im_m <= 2e+102) {
		tmp = ((((im_m * im_m) * im_m) * im_m) / t_0) / im_m;
	} else {
		tmp = ((t_0 * im_m) / im_m) / im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = -im_m * im_m
	tmp = 0
	if im_m <= 1e-34:
		tmp = -im_m
	elif im_m <= 2e+102:
		tmp = ((((im_m * im_m) * im_m) * im_m) / t_0) / im_m
	else:
		tmp = ((t_0 * im_m) / im_m) / im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(-im_m) * im_m)
	tmp = 0.0
	if (im_m <= 1e-34)
		tmp = Float64(-im_m);
	elseif (im_m <= 2e+102)
		tmp = Float64(Float64(Float64(Float64(Float64(im_m * im_m) * im_m) * im_m) / t_0) / im_m);
	else
		tmp = Float64(Float64(Float64(t_0 * im_m) / im_m) / im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = -im_m * im_m;
	tmp = 0.0;
	if (im_m <= 1e-34)
		tmp = -im_m;
	elseif (im_m <= 2e+102)
		tmp = ((((im_m * im_m) * im_m) * im_m) / t_0) / im_m;
	else
		tmp = ((t_0 * im_m) / im_m) / im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[((-im$95$m) * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 1e-34], (-im$95$m), If[LessEqual[im$95$m, 2e+102], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] / t$95$0), $MachinePrecision] / im$95$m), $MachinePrecision], N[(N[(N[(t$95$0 * im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 9.99999999999999928e-35

   1. Initial program 34.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.3%

      \[\leadsto -im \]

   if 9.99999999999999928e-35 < im < 1.99999999999999995e102

   1. Initial program 87.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 20.9%

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

   7. Applied rewrites 20.9%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{-1 \cdot {im}^{2}}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 8.2%

       \[\leadsto \frac{\left(-im\right) \cdot im}{im} \]
   11. Step-by-step derivation

   12. Applied rewrites 26.5%

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

   if 1.99999999999999995e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 6.4%

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{-1 \cdot {im}^{2}}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.7%

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

   12. Applied rewrites 82.4%

       \[\leadsto \frac{\frac{\left(\left(-im\right) \cdot im\right) \cdot im}{im}}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10^{-34}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\left(\left(im \cdot im\right) \cdot im\right) \cdot im}{\left(-im\right) \cdot im}}{im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(-im\right) \cdot im\right) \cdot im}{im}}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(-im\_m\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 10^{+45}:\\ \;\;\;\;-im\_m\\ \mathbf{elif}\;im\_m \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{t\_0 \cdot im\_m}{im\_m \cdot im\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{im\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- im_m) im_m)))
   (*
    im_s
    (if (<= im_m 1e+45)
      (- im_m)
      (if (<= im_m 1.4e+154) (/ (* t_0 im_m) (* im_m im_m)) (/ t_0 im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -im_m * im_m;
	double tmp;
	if (im_m <= 1e+45) {
		tmp = -im_m;
	} else if (im_m <= 1.4e+154) {
		tmp = (t_0 * im_m) / (im_m * im_m);
	} else {
		tmp = t_0 / im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -im_m * im_m
    if (im_m <= 1d+45) then
        tmp = -im_m
    else if (im_m <= 1.4d+154) then
        tmp = (t_0 * im_m) / (im_m * im_m)
    else
        tmp = t_0 / im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = -im_m * im_m;
	double tmp;
	if (im_m <= 1e+45) {
		tmp = -im_m;
	} else if (im_m <= 1.4e+154) {
		tmp = (t_0 * im_m) / (im_m * im_m);
	} else {
		tmp = t_0 / im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = -im_m * im_m
	tmp = 0
	if im_m <= 1e+45:
		tmp = -im_m
	elif im_m <= 1.4e+154:
		tmp = (t_0 * im_m) / (im_m * im_m)
	else:
		tmp = t_0 / im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(-im_m) * im_m)
	tmp = 0.0
	if (im_m <= 1e+45)
		tmp = Float64(-im_m);
	elseif (im_m <= 1.4e+154)
		tmp = Float64(Float64(t_0 * im_m) / Float64(im_m * im_m));
	else
		tmp = Float64(t_0 / im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = -im_m * im_m;
	tmp = 0.0;
	if (im_m <= 1e+45)
		tmp = -im_m;
	elseif (im_m <= 1.4e+154)
		tmp = (t_0 * im_m) / (im_m * im_m);
	else
		tmp = t_0 / im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[((-im$95$m) * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 1e+45], (-im$95$m), If[LessEqual[im$95$m, 1.4e+154], N[(N[(t$95$0 * im$95$m), $MachinePrecision] / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 9.9999999999999993e44

   1. Initial program 37.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.8%

      \[\leadsto -im \]

   if 9.9999999999999993e44 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 4.1%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 62.4%

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

   if 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 7.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{-1 \cdot {im}^{2}}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.5%

       \[\leadsto \frac{\left(-im\right) \cdot im}{im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 29.6% accurate, 105.7× speedup

Math

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

FPCore

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

C

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

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * -im_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 35.0%

   \[\leadsto -im \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(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 (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025021 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (cos re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (cos re)) (- (exp (- 0 im)) (exp im)))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))