math.sin on complex, imaginary part

Percentage Accurate: 54.0% → 99.9%

Time: 5.7s

Alternatives: 9

Speedup: 1.5×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 54.0% accurate, 1.0× speedup

Math

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

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: 99.9% accurate, 1.5× speedup

Math

\[\sinh \left(-im\right) \cdot \cos re \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (sinh (- im)) (cos re)))

C

double code(double re, double im) {
	return sinh(-im) * cos(re);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{e^{0 - im} - e^{im}}{2}} \cdot \cos re \]

   7. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{0 - im} - e^{im}}}{2} \cdot \cos re \]

   8. lift-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{0 - im}} - e^{im}}{2} \cdot \cos re \]

   9. lift-exp.f64 N/A

      \[\leadsto \frac{e^{0 - im} - \color{blue}{e^{im}}}{2} \cdot \cos re \]

   10. --rgt-identity N/A

       \[\leadsto \frac{e^{0 - im} - e^{\color{blue}{im - 0}}}{2} \cdot \cos re \]

   11. sub-negate-rev N/A

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

   12. lift--.f64 N/A

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

   13. sinh-def N/A

       \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sinh \left(0 - im\right) \cdot \cos re} \]

   15. lower-sinh.f64 99.9%

       \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

   16. lift--.f64 N/A

       \[\leadsto \sinh \color{blue}{\left(0 - im\right)} \cdot \cos re \]

   17. sub0-neg N/A

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

   18. lower-neg.f64 99.9%

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sinh \left(-im\right) \cdot \cos re} \]
4. Add Preprocessing

Alternative 2: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - \left|im\right|} - e^{\left|im\right|}\right)\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\sinh \left(-\left|im\right|\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\left(-\cos re\right) \cdot \left|im\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot {re}^{2}\right) - \left|im\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0
        (*
         (* 0.5 (cos re))
         (- (exp (- 0.0 (fabs im))) (exp (fabs im))))))
  (*
   (copysign 1.0 im)
   (if (<= t_0 -0.005)
     (* (sinh (- (fabs im))) 1.0)
     (if (<= t_0 1.0)
       (* (- (cos re)) (fabs im))
       (- (* 0.5 (* (fabs im) (pow re 2.0))) (fabs im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - fabs(im))) - exp(fabs(im)));
	double tmp;
	if (t_0 <= -0.005) {
		tmp = sinh(-fabs(im)) * 1.0;
	} else if (t_0 <= 1.0) {
		tmp = -cos(re) * fabs(im);
	} else {
		tmp = (0.5 * (fabs(im) * pow(re, 2.0))) - fabs(im);
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp((0.0 - Math.abs(im))) - Math.exp(Math.abs(im)));
	double tmp;
	if (t_0 <= -0.005) {
		tmp = Math.sinh(-Math.abs(im)) * 1.0;
	} else if (t_0 <= 1.0) {
		tmp = -Math.cos(re) * Math.abs(im);
	} else {
		tmp = (0.5 * (Math.abs(im) * Math.pow(re, 2.0))) - Math.abs(im);
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp((0.0 - math.fabs(im))) - math.exp(math.fabs(im)))
	tmp = 0
	if t_0 <= -0.005:
		tmp = math.sinh(-math.fabs(im)) * 1.0
	elif t_0 <= 1.0:
		tmp = -math.cos(re) * math.fabs(im)
	else:
		tmp = (0.5 * (math.fabs(im) * math.pow(re, 2.0))) - math.fabs(im)
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - abs(im))) - exp(abs(im))))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(sinh(Float64(-abs(im))) * 1.0);
	elseif (t_0 <= 1.0)
		tmp = Float64(Float64(-cos(re)) * abs(im));
	else
		tmp = Float64(Float64(0.5 * Float64(abs(im) * (re ^ 2.0))) - abs(im));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (exp((0.0 - abs(im))) - exp(abs(im)));
	tmp = 0.0;
	if (t_0 <= -0.005)
		tmp = sinh(-abs(im)) * 1.0;
	elseif (t_0 <= 1.0)
		tmp = -cos(re) * abs(im);
	else
		tmp = (0.5 * (abs(im) * (re ^ 2.0))) - abs(im);
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, -0.005], N[(N[Sinh[(-N[Abs[im], $MachinePrecision])], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[((-N[Cos[re], $MachinePrecision]) * N[Abs[im], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[Abs[im], $MachinePrecision] * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Abs[im], $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))) < -0.0050000000000000001

   1. Initial program 54.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{0 - im} - e^{im}}{2}} \cdot \cos re \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{0 - im} - e^{im}}}{2} \cdot \cos re \]

      8. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{0 - im}} - e^{im}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{e^{0 - im} - \color{blue}{e^{im}}}{2} \cdot \cos re \]

      10. --rgt-identity N/A

          \[\leadsto \frac{e^{0 - im} - e^{\color{blue}{im - 0}}}{2} \cdot \cos re \]

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. sinh-def N/A

          \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sinh \left(0 - im\right) \cdot \cos re} \]

      15. lower-sinh.f64 99.9%

          \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

      16. lift--.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(0 - im\right)} \cdot \cos re \]

      17. sub0-neg N/A

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

      18. lower-neg.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.1%

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

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 54.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.5%

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

   4. Applied rewrites 52.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\cos re \cdot im\right) \]

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 52.5%

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

   6. Applied rewrites 52.5%

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

   if 1 < (*.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 54.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.5%

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto -1 \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{im} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      3. lower-neg.f64 29.5%

         \[\leadsto -im \]

   9. Applied rewrites 29.5%

      \[\leadsto -im \]

   10. Taylor expanded in re around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 35.7%

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

   12. Applied rewrites 35.7%

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

Alternative 3: 96.1% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - \left|im\right|} - e^{\left|im\right|}\right) \leq -0.005:\\ \;\;\;\;\sinh \left(-\left|im\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(\left(\left(\left|im\right| \cdot \left|im\right|\right) \cdot -0.3333333333333333 - 2\right) \cdot \left|im\right|\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<=
      (* (* 0.5 (cos re)) (- (exp (- 0.0 (fabs im))) (exp (fabs im))))
      -0.005)
   (* (sinh (- (fabs im))) 1.0)
   (*
    (* (cos re) 0.5)
    (*
     (- (* (* (fabs im) (fabs im)) -0.3333333333333333) 2.0)
     (fabs im))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - fabs(im))) - exp(fabs(im)))) <= -0.005) {
		tmp = sinh(-fabs(im)) * 1.0;
	} else {
		tmp = (cos(re) * 0.5) * ((((fabs(im) * fabs(im)) * -0.3333333333333333) - 2.0) * fabs(im));
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp((0.0 - Math.abs(im))) - Math.exp(Math.abs(im)))) <= -0.005) {
		tmp = Math.sinh(-Math.abs(im)) * 1.0;
	} else {
		tmp = (Math.cos(re) * 0.5) * ((((Math.abs(im) * Math.abs(im)) * -0.3333333333333333) - 2.0) * Math.abs(im));
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp((0.0 - math.fabs(im))) - math.exp(math.fabs(im)))) <= -0.005:
		tmp = math.sinh(-math.fabs(im)) * 1.0
	else:
		tmp = (math.cos(re) * 0.5) * ((((math.fabs(im) * math.fabs(im)) * -0.3333333333333333) - 2.0) * math.fabs(im))
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - abs(im))) - exp(abs(im)))) <= -0.005)
		tmp = Float64(sinh(Float64(-abs(im))) * 1.0);
	else
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(Float64(Float64(Float64(abs(im) * abs(im)) * -0.3333333333333333) - 2.0) * abs(im)));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp((0.0 - abs(im))) - exp(abs(im)))) <= -0.005)
		tmp = sinh(-abs(im)) * 1.0;
	else
		tmp = (cos(re) * 0.5) * ((((abs(im) * abs(im)) * -0.3333333333333333) - 2.0) * abs(im));
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[Sinh[(-N[Abs[im], $MachinePrecision])], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $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))) < -0.0050000000000000001

   1. Initial program 54.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{0 - im} - e^{im}}{2}} \cdot \cos re \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{0 - im} - e^{im}}}{2} \cdot \cos re \]

      8. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{0 - im}} - e^{im}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{e^{0 - im} - \color{blue}{e^{im}}}{2} \cdot \cos re \]

      10. --rgt-identity N/A

          \[\leadsto \frac{e^{0 - im} - e^{\color{blue}{im - 0}}}{2} \cdot \cos re \]

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. sinh-def N/A

          \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sinh \left(0 - im\right) \cdot \cos re} \]

      15. lower-sinh.f64 99.9%

          \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

      16. lift--.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(0 - im\right)} \cdot \cos re \]

      17. sub0-neg N/A

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

      18. lower-neg.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.1%

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

   if -0.0050000000000000001 < (*.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 54.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \color{blue}{2}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      4. lower-pow.f64 84.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

   4. Applied rewrites 84.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      3. lower-*.f64 84.5%

         \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      6. lower-*.f64 84.5%

         \[\leadsto \left(\cos re \cdot 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{3} - 2\right) \cdot im\right) \]

      9. lower-*.f64 84.5%

         \[\leadsto \left(\cos re \cdot 0.5\right) \cdot \left(\left({im}^{2} \cdot -0.3333333333333333 - 2\right) \cdot im\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{3} - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\cos re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 2\right) \cdot im\right) \]

      12. lower-*.f64 84.5%

          \[\leadsto \left(\cos re \cdot 0.5\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right) \]

   6. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.0% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - \left|im\right|} - e^{\left|im\right|}\right) \leq -0.005:\\ \;\;\;\;\sinh \left(-\left|im\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(-\cos re\right) \cdot \left|im\right|\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<=
      (* (* 0.5 (cos re)) (- (exp (- 0.0 (fabs im))) (exp (fabs im))))
      -0.005)
   (* (sinh (- (fabs im))) 1.0)
   (* (- (cos re)) (fabs im)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - fabs(im))) - exp(fabs(im)))) <= -0.005) {
		tmp = sinh(-fabs(im)) * 1.0;
	} else {
		tmp = -cos(re) * fabs(im);
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp((0.0 - Math.abs(im))) - Math.exp(Math.abs(im)))) <= -0.005) {
		tmp = Math.sinh(-Math.abs(im)) * 1.0;
	} else {
		tmp = -Math.cos(re) * Math.abs(im);
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp((0.0 - math.fabs(im))) - math.exp(math.fabs(im)))) <= -0.005:
		tmp = math.sinh(-math.fabs(im)) * 1.0
	else:
		tmp = -math.cos(re) * math.fabs(im)
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - abs(im))) - exp(abs(im)))) <= -0.005)
		tmp = Float64(sinh(Float64(-abs(im))) * 1.0);
	else
		tmp = Float64(Float64(-cos(re)) * abs(im));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp((0.0 - abs(im))) - exp(abs(im)))) <= -0.005)
		tmp = sinh(-abs(im)) * 1.0;
	else
		tmp = -cos(re) * abs(im);
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[Sinh[(-N[Abs[im], $MachinePrecision])], $MachinePrecision] * 1.0), $MachinePrecision], N[((-N[Cos[re], $MachinePrecision]) * N[Abs[im], $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))) < -0.0050000000000000001

   1. Initial program 54.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{0 - im} - e^{im}}{2}} \cdot \cos re \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{0 - im} - e^{im}}}{2} \cdot \cos re \]

      8. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{0 - im}} - e^{im}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{e^{0 - im} - \color{blue}{e^{im}}}{2} \cdot \cos re \]

      10. --rgt-identity N/A

          \[\leadsto \frac{e^{0 - im} - e^{\color{blue}{im - 0}}}{2} \cdot \cos re \]

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. sinh-def N/A

          \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sinh \left(0 - im\right) \cdot \cos re} \]

      15. lower-sinh.f64 99.9%

          \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

      16. lift--.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(0 - im\right)} \cdot \cos re \]

      17. sub0-neg N/A

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

      18. lower-neg.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.1%

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

   if -0.0050000000000000001 < (*.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 54.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.5%

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

   4. Applied rewrites 52.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\cos re \cdot im\right) \]

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 52.5%

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

   6. Applied rewrites 52.5%

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

Alternative 5: 60.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := -0.3333333333333333 \cdot \left(\left|im\right| \cdot \left|im\right|\right)\\ t_1 := 1 - t\_0\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - \left|im\right|} - e^{\left|im\right|}\right) \leq -0.005:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \left(\frac{t\_1 \cdot t\_1}{t\_0} - \frac{1}{t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\cos re\right) \cdot \left|im\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* -0.3333333333333333 (* (fabs im) (fabs im))))
       (t_1 (- 1.0 t_0)))
  (*
   (copysign 1.0 im)
   (if (<=
        (*
         (* 0.5 (cos re))
         (- (exp (- 0.0 (fabs im))) (exp (fabs im))))
        -0.005)
     (* 0.5 (* (fabs im) (- (/ (* t_1 t_1) t_0) (/ 1.0 t_0))))
     (* (- (cos re)) (fabs im))))))

C

double code(double re, double im) {
	double t_0 = -0.3333333333333333 * (fabs(im) * fabs(im));
	double t_1 = 1.0 - t_0;
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - fabs(im))) - exp(fabs(im)))) <= -0.005) {
		tmp = 0.5 * (fabs(im) * (((t_1 * t_1) / t_0) - (1.0 / t_0)));
	} else {
		tmp = -cos(re) * fabs(im);
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.3333333333333333 * (Math.abs(im) * Math.abs(im));
	double t_1 = 1.0 - t_0;
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp((0.0 - Math.abs(im))) - Math.exp(Math.abs(im)))) <= -0.005) {
		tmp = 0.5 * (Math.abs(im) * (((t_1 * t_1) / t_0) - (1.0 / t_0)));
	} else {
		tmp = -Math.cos(re) * Math.abs(im);
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = -0.3333333333333333 * (math.fabs(im) * math.fabs(im))
	t_1 = 1.0 - t_0
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp((0.0 - math.fabs(im))) - math.exp(math.fabs(im)))) <= -0.005:
		tmp = 0.5 * (math.fabs(im) * (((t_1 * t_1) / t_0) - (1.0 / t_0)))
	else:
		tmp = -math.cos(re) * math.fabs(im)
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(-0.3333333333333333 * Float64(abs(im) * abs(im)))
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - abs(im))) - exp(abs(im)))) <= -0.005)
		tmp = Float64(0.5 * Float64(abs(im) * Float64(Float64(Float64(t_1 * t_1) / t_0) - Float64(1.0 / t_0))));
	else
		tmp = Float64(Float64(-cos(re)) * abs(im));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.3333333333333333 * (abs(im) * abs(im));
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp((0.0 - abs(im))) - exp(abs(im)))) <= -0.005)
		tmp = 0.5 * (abs(im) * (((t_1 * t_1) / t_0) - (1.0 / t_0)));
	else
		tmp = -cos(re) * abs(im);
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(0.5 * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Cos[re], $MachinePrecision]) * N[Abs[im], $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))) < -0.0050000000000000001

   1. Initial program 54.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \color{blue}{2}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      4. lower-pow.f64 84.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

   4. Applied rewrites 84.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \color{blue}{2}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \left(1 + \color{blue}{1}\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - \color{blue}{1}\right)\right) \]

      4. flip-- N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) + 1}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) + 1}}\right) \]

   6. Applied rewrites 13.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) - 1 \cdot 1}{\color{blue}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) + 1}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right)} + 1}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - \color{blue}{1}\right) + 1}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - \color{blue}{1}\right) + 1}\right) \]

      5. div-sub N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right)}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1} - \color{blue}{\frac{1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1}}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right)}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1} - \color{blue}{\frac{1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1}}\right)\right) \]

   8. Applied rewrites 14.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{\left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right)}{-0.3333333333333333 \cdot \left(im \cdot im\right)} - \color{blue}{\frac{1}{-0.3333333333333333 \cdot \left(im \cdot im\right)}}\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(im \cdot \left(\frac{\left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right)}{-0.3333333333333333 \cdot \left(im \cdot im\right)} - \frac{1}{-0.3333333333333333 \cdot \left(im \cdot im\right)}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 11.0%

       \[\leadsto \color{blue}{0.5} \cdot \left(im \cdot \left(\frac{\left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right)}{-0.3333333333333333 \cdot \left(im \cdot im\right)} - \frac{1}{-0.3333333333333333 \cdot \left(im \cdot im\right)}\right)\right) \]

   if -0.0050000000000000001 < (*.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 54.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.5%

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

   4. Applied rewrites 52.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\cos re \cdot im\right) \]

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 52.5%

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

   6. Applied rewrites 52.5%

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

Alternative 6: 38.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := -0.3333333333333333 \cdot \left(\left|im\right| \cdot \left|im\right|\right)\\ t_1 := 1 - t\_0\\ t_2 := -\left|im\right|\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|im\right| \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\left|im\right| \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \left(\frac{t\_1 \cdot t\_1}{t\_0} - \frac{1}{t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* -0.3333333333333333 (* (fabs im) (fabs im))))
       (t_1 (- 1.0 t_0))
       (t_2 (- (fabs im))))
  (*
   (copysign 1.0 im)
   (if (<= (fabs im) 8.2e-10)
     t_2
     (if (<= (fabs im) 1.35e+154)
       (* 0.5 (* (fabs im) (- (/ (* t_1 t_1) t_0) (/ 1.0 t_0))))
       t_2)))))

C

double code(double re, double im) {
	double t_0 = -0.3333333333333333 * (fabs(im) * fabs(im));
	double t_1 = 1.0 - t_0;
	double t_2 = -fabs(im);
	double tmp;
	if (fabs(im) <= 8.2e-10) {
		tmp = t_2;
	} else if (fabs(im) <= 1.35e+154) {
		tmp = 0.5 * (fabs(im) * (((t_1 * t_1) / t_0) - (1.0 / t_0)));
	} else {
		tmp = t_2;
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.3333333333333333 * (Math.abs(im) * Math.abs(im));
	double t_1 = 1.0 - t_0;
	double t_2 = -Math.abs(im);
	double tmp;
	if (Math.abs(im) <= 8.2e-10) {
		tmp = t_2;
	} else if (Math.abs(im) <= 1.35e+154) {
		tmp = 0.5 * (Math.abs(im) * (((t_1 * t_1) / t_0) - (1.0 / t_0)));
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = -0.3333333333333333 * (math.fabs(im) * math.fabs(im))
	t_1 = 1.0 - t_0
	t_2 = -math.fabs(im)
	tmp = 0
	if math.fabs(im) <= 8.2e-10:
		tmp = t_2
	elif math.fabs(im) <= 1.35e+154:
		tmp = 0.5 * (math.fabs(im) * (((t_1 * t_1) / t_0) - (1.0 / t_0)))
	else:
		tmp = t_2
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(-0.3333333333333333 * Float64(abs(im) * abs(im)))
	t_1 = Float64(1.0 - t_0)
	t_2 = Float64(-abs(im))
	tmp = 0.0
	if (abs(im) <= 8.2e-10)
		tmp = t_2;
	elseif (abs(im) <= 1.35e+154)
		tmp = Float64(0.5 * Float64(abs(im) * Float64(Float64(Float64(t_1 * t_1) / t_0) - Float64(1.0 / t_0))));
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.3333333333333333 * (abs(im) * abs(im));
	t_1 = 1.0 - t_0;
	t_2 = -abs(im);
	tmp = 0.0;
	if (abs(im) <= 8.2e-10)
		tmp = t_2;
	elseif (abs(im) <= 1.35e+154)
		tmp = 0.5 * (abs(im) * (((t_1 * t_1) / t_0) - (1.0 / t_0)));
	else
		tmp = t_2;
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-N[Abs[im], $MachinePrecision])}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[im], $MachinePrecision], 8.2e-10], t$95$2, If[LessEqual[N[Abs[im], $MachinePrecision], 1.35e+154], N[(0.5 * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if im < 8.1999999999999996e-10 or 1.35e154 < im

   1. Initial program 54.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.5%

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto -1 \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{im} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      3. lower-neg.f64 29.5%

         \[\leadsto -im \]

   9. Applied rewrites 29.5%

      \[\leadsto -im \]

   if 8.1999999999999996e-10 < im < 1.35e154

   1. Initial program 54.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \color{blue}{2}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      4. lower-pow.f64 84.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

   4. Applied rewrites 84.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \color{blue}{2}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \left(1 + \color{blue}{1}\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - \color{blue}{1}\right)\right) \]

      4. flip-- N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) + 1}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) + 1}}\right) \]

   6. Applied rewrites 13.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) - 1 \cdot 1}{\color{blue}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) + 1}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right)} + 1}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - \color{blue}{1}\right) + 1}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - \color{blue}{1}\right) + 1}\right) \]

      5. div-sub N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right)}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1} - \color{blue}{\frac{1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1}}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right)}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1} - \color{blue}{\frac{1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1}}\right)\right) \]

   8. Applied rewrites 14.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{\left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right)}{-0.3333333333333333 \cdot \left(im \cdot im\right)} - \color{blue}{\frac{1}{-0.3333333333333333 \cdot \left(im \cdot im\right)}}\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(im \cdot \left(\frac{\left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right)}{-0.3333333333333333 \cdot \left(im \cdot im\right)} - \frac{1}{-0.3333333333333333 \cdot \left(im \cdot im\right)}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 11.0%

       \[\leadsto \color{blue}{0.5} \cdot \left(im \cdot \left(\frac{\left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(1 - -0.3333333333333333 \cdot \left(im \cdot im\right)\right)}{-0.3333333333333333 \cdot \left(im \cdot im\right)} - \frac{1}{-0.3333333333333333 \cdot \left(im \cdot im\right)}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 38.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|im\right| \cdot \left|im\right|\right) \cdot -0.3333333333333333 - 1\\ t_1 := -\left|im\right|\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|im\right| \leq 4 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|im\right| \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \frac{t\_0 \cdot t\_0 - 1 \cdot 1}{t\_0 + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (- (* (* (fabs im) (fabs im)) -0.3333333333333333) 1.0))
       (t_1 (- (fabs im))))
  (*
   (copysign 1.0 im)
   (if (<= (fabs im) 4e-7)
     t_1
     (if (<= (fabs im) 1.35e+154)
       (*
        0.5
        (* (fabs im) (/ (- (* t_0 t_0) (* 1.0 1.0)) (+ t_0 1.0))))
       t_1)))))

C

double code(double re, double im) {
	double t_0 = ((fabs(im) * fabs(im)) * -0.3333333333333333) - 1.0;
	double t_1 = -fabs(im);
	double tmp;
	if (fabs(im) <= 4e-7) {
		tmp = t_1;
	} else if (fabs(im) <= 1.35e+154) {
		tmp = 0.5 * (fabs(im) * (((t_0 * t_0) - (1.0 * 1.0)) / (t_0 + 1.0)));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = ((Math.abs(im) * Math.abs(im)) * -0.3333333333333333) - 1.0;
	double t_1 = -Math.abs(im);
	double tmp;
	if (Math.abs(im) <= 4e-7) {
		tmp = t_1;
	} else if (Math.abs(im) <= 1.35e+154) {
		tmp = 0.5 * (Math.abs(im) * (((t_0 * t_0) - (1.0 * 1.0)) / (t_0 + 1.0)));
	} else {
		tmp = t_1;
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = ((math.fabs(im) * math.fabs(im)) * -0.3333333333333333) - 1.0
	t_1 = -math.fabs(im)
	tmp = 0
	if math.fabs(im) <= 4e-7:
		tmp = t_1
	elif math.fabs(im) <= 1.35e+154:
		tmp = 0.5 * (math.fabs(im) * (((t_0 * t_0) - (1.0 * 1.0)) / (t_0 + 1.0)))
	else:
		tmp = t_1
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(abs(im) * abs(im)) * -0.3333333333333333) - 1.0)
	t_1 = Float64(-abs(im))
	tmp = 0.0
	if (abs(im) <= 4e-7)
		tmp = t_1;
	elseif (abs(im) <= 1.35e+154)
		tmp = Float64(0.5 * Float64(abs(im) * Float64(Float64(Float64(t_0 * t_0) - Float64(1.0 * 1.0)) / Float64(t_0 + 1.0))));
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((abs(im) * abs(im)) * -0.3333333333333333) - 1.0;
	t_1 = -abs(im);
	tmp = 0.0;
	if (abs(im) <= 4e-7)
		tmp = t_1;
	elseif (abs(im) <= 1.35e+154)
		tmp = 0.5 * (abs(im) * (((t_0 * t_0) - (1.0 * 1.0)) / (t_0 + 1.0)));
	else
		tmp = t_1;
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[im], $MachinePrecision])}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[im], $MachinePrecision], 4e-7], t$95$1, If[LessEqual[N[Abs[im], $MachinePrecision], 1.35e+154], N[(0.5 * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 3.9999999999999998e-7 or 1.35e154 < im

   1. Initial program 54.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.5%

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto -1 \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{im} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      3. lower-neg.f64 29.5%

         \[\leadsto -im \]

   9. Applied rewrites 29.5%

      \[\leadsto -im \]

   if 3.9999999999999998e-7 < im < 1.35e154

   1. Initial program 54.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \color{blue}{2}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      4. lower-pow.f64 84.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

   4. Applied rewrites 84.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \color{blue}{2}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \left(1 + \color{blue}{1}\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - \color{blue}{1}\right)\right) \]

      4. flip-- N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) + 1}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) + 1}}\right) \]

   6. Applied rewrites 13.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) - 1 \cdot 1}{\color{blue}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) + 1}}\right) \]

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) - 1 \cdot 1}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) + 1}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 10.2%

      \[\leadsto \color{blue}{0.5} \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) - 1 \cdot 1}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) + 1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 38.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := -0.3333333333333333 \cdot \left(\left|im\right| \cdot \left|im\right|\right)\\ t_1 := -\left|im\right|\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|im\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|im\right| \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(t\_0 - 2\right) \cdot t\_0\right) \cdot \left(\frac{1}{t\_0} \cdot \left|im\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* -0.3333333333333333 (* (fabs im) (fabs im))))
       (t_1 (- (fabs im))))
  (*
   (copysign 1.0 im)
   (if (<= (fabs im) 5e-12)
     t_1
     (if (<= (fabs im) 1.35e+154)
       (* 0.5 (* (* (- t_0 2.0) t_0) (* (/ 1.0 t_0) (fabs im))))
       t_1)))))

C

double code(double re, double im) {
	double t_0 = -0.3333333333333333 * (fabs(im) * fabs(im));
	double t_1 = -fabs(im);
	double tmp;
	if (fabs(im) <= 5e-12) {
		tmp = t_1;
	} else if (fabs(im) <= 1.35e+154) {
		tmp = 0.5 * (((t_0 - 2.0) * t_0) * ((1.0 / t_0) * fabs(im)));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.3333333333333333 * (Math.abs(im) * Math.abs(im));
	double t_1 = -Math.abs(im);
	double tmp;
	if (Math.abs(im) <= 5e-12) {
		tmp = t_1;
	} else if (Math.abs(im) <= 1.35e+154) {
		tmp = 0.5 * (((t_0 - 2.0) * t_0) * ((1.0 / t_0) * Math.abs(im)));
	} else {
		tmp = t_1;
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = -0.3333333333333333 * (math.fabs(im) * math.fabs(im))
	t_1 = -math.fabs(im)
	tmp = 0
	if math.fabs(im) <= 5e-12:
		tmp = t_1
	elif math.fabs(im) <= 1.35e+154:
		tmp = 0.5 * (((t_0 - 2.0) * t_0) * ((1.0 / t_0) * math.fabs(im)))
	else:
		tmp = t_1
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(-0.3333333333333333 * Float64(abs(im) * abs(im)))
	t_1 = Float64(-abs(im))
	tmp = 0.0
	if (abs(im) <= 5e-12)
		tmp = t_1;
	elseif (abs(im) <= 1.35e+154)
		tmp = Float64(0.5 * Float64(Float64(Float64(t_0 - 2.0) * t_0) * Float64(Float64(1.0 / t_0) * abs(im))));
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.3333333333333333 * (abs(im) * abs(im));
	t_1 = -abs(im);
	tmp = 0.0;
	if (abs(im) <= 5e-12)
		tmp = t_1;
	elseif (abs(im) <= 1.35e+154)
		tmp = 0.5 * (((t_0 - 2.0) * t_0) * ((1.0 / t_0) * abs(im)));
	else
		tmp = t_1;
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[im], $MachinePrecision])}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[im], $MachinePrecision], 5e-12], t$95$1, If[LessEqual[N[Abs[im], $MachinePrecision], 1.35e+154], N[(0.5 * N[(N[(N[(t$95$0 - 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 4.9999999999999997e-12 or 1.35e154 < im

   1. Initial program 54.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.5%

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto -1 \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{im} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      3. lower-neg.f64 29.5%

         \[\leadsto -im \]

   9. Applied rewrites 29.5%

      \[\leadsto -im \]

   if 4.9999999999999997e-12 < im < 1.35e154

   1. Initial program 54.0%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \color{blue}{2}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      4. lower-pow.f64 84.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right) \]

   4. Applied rewrites 84.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \color{blue}{2}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - \left(1 + \color{blue}{1}\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - \color{blue}{1}\right)\right) \]

      4. flip-- N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) + 1}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 1\right) - 1 \cdot 1}{\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 1\right) + 1}}\right) \]

   6. Applied rewrites 13.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \frac{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) - 1 \cdot 1}{\color{blue}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 1\right) + 1}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(im \cdot \color{blue}{\frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1} \cdot \color{blue}{im}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\frac{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1} \cdot im\right) \]

      4. mult-flip N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1\right) \cdot \frac{1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1}\right) \cdot im\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1\right) \cdot \color{blue}{\left(\frac{1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1} \cdot im\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) - 1 \cdot 1\right) \cdot \color{blue}{\left(\frac{1}{\left(\left(im \cdot im\right) \cdot \frac{-1}{3} - 1\right) + 1} \cdot im\right)}\right) \]

   8. Applied rewrites 37.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{-0.3333333333333333 \cdot \left(im \cdot im\right)} \cdot im\right)}\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\frac{1}{-0.3333333333333333 \cdot \left(im \cdot im\right)} \cdot im\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 23.3%

       \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\frac{1}{-0.3333333333333333 \cdot \left(im \cdot im\right)} \cdot im\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 29.5% accurate, 105.7× speedup

Math

\[-im \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 52.5%

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

4. Applied rewrites 52.5%

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

5. Taylor expanded in re around 0

   \[\leadsto -1 \cdot im \]
6. Step-by-step derivation

7. Applied rewrites 29.5%

   \[\leadsto -1 \cdot im \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{im} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   3. lower-neg.f64 29.5%

      \[\leadsto -im \]

9. Applied rewrites 29.5%

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

Reproduce

herbie shell --seed 2025258 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))