math.sin on complex, imaginary part

Percentage Accurate: 53.1% → 99.9%

Time: 5.1s

Alternatives: 12

Speedup: 1.2×

• 48.5% 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: 53.1% 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: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\cos re \cdot \left(-2 \cdot \sinh im\right)\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* (cos re) (* -2.0 (sinh im))) 0.5))

C

double code(double re, double im) {
	return (cos(re) * (-2.0 * sinh(im))) * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (cos(re) * ((-2.0d0) * sinh(im))) * 0.5d0
end function

Java

public static double code(double re, double im) {
	return (Math.cos(re) * (-2.0 * Math.sinh(im))) * 0.5;
}

Python

def code(re, im):
	return (math.cos(re) * (-2.0 * math.sinh(im))) * 0.5

Julia

function code(re, im)
	return Float64(Float64(cos(re) * Float64(-2.0 * sinh(im))) * 0.5)
end

MATLAB

function tmp = code(re, im)
	tmp = (cos(re) * (-2.0 * sinh(im))) * 0.5;
end

Wolfram

code[re_, im_] := N[(N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * N[Sinh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
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. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub0-neg N/A

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

   6. lower-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. negate-sub2 N/A

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

   15. lower-neg.f64 N/A

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

   16. sinh-undef N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sinh.f64 N/A

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

   19. lift-cos.f64 99.9

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

3. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right)} \cdot \cos re\right) \cdot \frac{1}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \sinh im}\right)\right) \cdot \cos re\right) \cdot \frac{1}{2} \]

   4. lift-sinh.f64 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(2 \cdot \color{blue}{\sinh im}\right)\right) \cdot \cos re\right) \cdot \frac{1}{2} \]

   5. lift-cos.f64 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \color{blue}{\cos re}\right) \cdot \frac{1}{2} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\cos re \cdot \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right)\right)} \cdot \frac{1}{2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\cos re \cdot \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right)\right)} \cdot \frac{1}{2} \]

   8. lift-cos.f64 N/A

      \[\leadsto \left(\color{blue}{\cos re} \cdot \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right)\right) \cdot \frac{1}{2} \]

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

      \[\leadsto \left(\cos re \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh im\right)}\right) \cdot \frac{1}{2} \]

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sinh.f64 99.9

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\cos re \cdot \left(-2 \cdot \sinh im\right)\right)} \cdot 0.5 \]
6. Add Preprocessing

Alternative 2: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;\left(-2 \cdot \sinh im\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh im - \mathsf{fma}\left(2, im, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 -100.0)
     (* (* -2.0 (sinh im)) 0.5)
     (if (<= t_0 0.0)
       (* (* (cos re) (fma -0.16666666666666666 (* im im) -1.0)) im)
       (* (- (cosh im) (fma 2.0 im 1.0)) 0.5)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -100.0) {
		tmp = (-2.0 * sinh(im)) * 0.5;
	} else if (t_0 <= 0.0) {
		tmp = (cos(re) * fma(-0.16666666666666666, (im * im), -1.0)) * im;
	} else {
		tmp = (cosh(im) - fma(2.0, im, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(Float64(-2.0 * sinh(im)) * 0.5);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(cos(re) * fma(-0.16666666666666666, Float64(im * im), -1.0)) * im);
	else
		tmp = Float64(Float64(cosh(im) - fma(2.0, im, 1.0)) * 0.5);
	end
	return 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 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], N[(N[(-2.0 * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Cos[re], $MachinePrecision] * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[Cosh[im], $MachinePrecision] - N[(2.0 * im + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -100

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 74.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{0 - im} - e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. sub0-neg N/A

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

      8. negate-sub2 N/A

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

      9. sinh-undef-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{2 \cdot \sinh im}\right)\right) \cdot \frac{1}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2}} \]

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

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh im\right)} \cdot \frac{1}{2} \]

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sinh.f64 74.3

          \[\leadsto \left(-2 \cdot \color{blue}{\sinh im}\right) \cdot 0.5 \]

   6. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\left(-2 \cdot \sinh im\right) \cdot 0.5} \]

   if -100 < (*.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 6.8%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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 97.7%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 73.2

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 73.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      3. lift-sinh.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      4. sinh-undef-rev N/A

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

      5. negate-sub2 N/A

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

      6. negate-sub N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      7. sinh---cosh-rev N/A

         \[\leadsto \left(\left(\cosh im - \sinh im\right) + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      8. associate--r- N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 72.0

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   6. Applied rewrites 72.0%

      \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   7. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 95.1

         \[\leadsto \left(\cosh im - \mathsf{fma}\left(2, im, 1\right)\right) \cdot 0.5 \]

   9. Applied rewrites 95.1%

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

Alternative 3: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;\left(-2 \cdot \sinh im\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\cos re \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh im - \mathsf{fma}\left(2, im, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 -100.0)
     (* (* -2.0 (sinh im)) 0.5)
     (if (<= t_0 0.0)
       (* (* (cos re) im) (fma (* im im) -0.16666666666666666 -1.0))
       (* (- (cosh im) (fma 2.0 im 1.0)) 0.5)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -100.0) {
		tmp = (-2.0 * sinh(im)) * 0.5;
	} else if (t_0 <= 0.0) {
		tmp = (cos(re) * im) * fma((im * im), -0.16666666666666666, -1.0);
	} else {
		tmp = (cosh(im) - fma(2.0, im, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(Float64(-2.0 * sinh(im)) * 0.5);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(cos(re) * im) * fma(Float64(im * im), -0.16666666666666666, -1.0));
	else
		tmp = Float64(Float64(cosh(im) - fma(2.0, im, 1.0)) * 0.5);
	end
	return 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 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], N[(N[(-2.0 * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cosh[im], $MachinePrecision] - N[(2.0 * im + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -100

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 74.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{0 - im} - e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. sub0-neg N/A

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

      8. negate-sub2 N/A

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

      9. sinh-undef-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{2 \cdot \sinh im}\right)\right) \cdot \frac{1}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2}} \]

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

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh im\right)} \cdot \frac{1}{2} \]

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sinh.f64 74.3

          \[\leadsto \left(-2 \cdot \color{blue}{\sinh im}\right) \cdot 0.5 \]

   6. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\left(-2 \cdot \sinh im\right) \cdot 0.5} \]

   if -100 < (*.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 6.8%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right) \cdot im} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. negate-sub N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. negate-sub N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   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 97.7%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 73.2

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 73.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      3. lift-sinh.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      4. sinh-undef-rev N/A

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

      5. negate-sub2 N/A

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

      6. negate-sub N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      7. sinh---cosh-rev N/A

         \[\leadsto \left(\left(\cosh im - \sinh im\right) + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      8. associate--r- N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 72.0

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   6. Applied rewrites 72.0%

      \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   7. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 95.1

         \[\leadsto \left(\cosh im - \mathsf{fma}\left(2, im, 1\right)\right) \cdot 0.5 \]

   9. Applied rewrites 95.1%

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

Alternative 4: 92.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1)
   (* (* (cos re) (fma -0.16666666666666666 (* im im) -1.0)) im)
   (* (* 0.5 (cos re)) (- 1.0 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1) {
		tmp = (cos(re) * fma(-0.16666666666666666, (im * im), -1.0)) * im;
	} else {
		tmp = (0.5 * cos(re)) * (1.0 - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1)
		tmp = Float64(Float64(cos(re) * fma(-0.16666666666666666, Float64(im * im), -1.0)) * im);
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(1.0 - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1], N[(N[(N[Cos[re], $MachinePrecision] * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.10000000000000009

   1. Initial program 37.8%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 89.3

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

   4. Applied rewrites 89.3%

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

   if 2.10000000000000009 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 99.8%

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

Alternative 5: 91.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;\left(-2 \cdot \sinh im\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh im - \mathsf{fma}\left(2, im, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 -100.0)
     (* (* -2.0 (sinh im)) 0.5)
     (if (<= t_0 0.0)
       (* (- (cos re)) im)
       (* (- (cosh im) (fma 2.0 im 1.0)) 0.5)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -100.0) {
		tmp = (-2.0 * sinh(im)) * 0.5;
	} else if (t_0 <= 0.0) {
		tmp = -cos(re) * im;
	} else {
		tmp = (cosh(im) - fma(2.0, im, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(Float64(-2.0 * sinh(im)) * 0.5);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-cos(re)) * im);
	else
		tmp = Float64(Float64(cosh(im) - fma(2.0, im, 1.0)) * 0.5);
	end
	return 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 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], N[(N[(-2.0 * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-N[Cos[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[Cosh[im], $MachinePrecision] - N[(2.0 * im + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -100

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 74.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{0 - im} - e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. sub0-neg N/A

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

      8. negate-sub2 N/A

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

      9. sinh-undef-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{2 \cdot \sinh im}\right)\right) \cdot \frac{1}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2}} \]

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

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh im\right)} \cdot \frac{1}{2} \]

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sinh.f64 74.3

          \[\leadsto \left(-2 \cdot \color{blue}{\sinh im}\right) \cdot 0.5 \]

   6. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\left(-2 \cdot \sinh im\right) \cdot 0.5} \]

   if -100 < (*.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 6.8%

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

   2. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-cos.f64 99.3

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

   4. Applied rewrites 99.3%

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

   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 97.7%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 73.2

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 73.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      3. lift-sinh.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      4. sinh-undef-rev N/A

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

      5. negate-sub2 N/A

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

      6. negate-sub N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      7. sinh---cosh-rev N/A

         \[\leadsto \left(\left(\cosh im - \sinh im\right) + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      8. associate--r- N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 72.0

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   6. Applied rewrites 72.0%

      \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   7. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 95.1

         \[\leadsto \left(\cosh im - \mathsf{fma}\left(2, im, 1\right)\right) \cdot 0.5 \]

   9. Applied rewrites 95.1%

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

Alternative 6: 70.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.01:\\ \;\;\;\;\left(\cosh im - \mathsf{fma}\left(2, im, 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sinh im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.01)
   (* (- (cosh im) (fma 2.0 im 1.0)) 0.5)
   (* (* -2.0 (sinh im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.01) {
		tmp = (cosh(im) - fma(2.0, im, 1.0)) * 0.5;
	} else {
		tmp = (-2.0 * sinh(im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.01)
		tmp = Float64(Float64(cosh(im) - fma(2.0, im, 1.0)) * 0.5);
	else
		tmp = Float64(Float64(-2.0 * sinh(im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[Cosh[im], $MachinePrecision] - N[(2.0 * im + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(-2.0 * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0100000000000000002

   1. Initial program 54.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 2.0

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 2.0%

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

      1. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      3. lift-sinh.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      4. sinh-undef-rev N/A

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

      5. negate-sub2 N/A

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

      6. negate-sub N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      7. sinh---cosh-rev N/A

         \[\leadsto \left(\left(\cosh im - \sinh im\right) + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      8. associate--r- N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 2.7

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   6. Applied rewrites 2.7%

      \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   7. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 26.8

         \[\leadsto \left(\cosh im - \mathsf{fma}\left(2, im, 1\right)\right) \cdot 0.5 \]

   9. Applied rewrites 26.8%

      \[\leadsto \left(\cosh im - \mathsf{fma}\left(2, im, 1\right)\right) \cdot 0.5 \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 52.8%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 52.1%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{0 - im} - e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. sub0-neg N/A

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

      8. negate-sub2 N/A

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

      9. sinh-undef-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{2 \cdot \sinh im}\right)\right) \cdot \frac{1}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2}} \]

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

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh im\right)} \cdot \frac{1}{2} \]

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sinh.f64 85.5

          \[\leadsto \left(-2 \cdot \color{blue}{\sinh im}\right) \cdot 0.5 \]

   6. Applied rewrites 85.5%

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

Alternative 7: 70.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.01:\\ \;\;\;\;\left(\cosh im - 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sinh im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.01)
   (* (- (cosh im) 1.0) 0.5)
   (* (* -2.0 (sinh im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.01) {
		tmp = (cosh(im) - 1.0) * 0.5;
	} else {
		tmp = (-2.0 * sinh(im)) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * cos(re)) <= (-0.01d0)) then
        tmp = (cosh(im) - 1.0d0) * 0.5d0
    else
        tmp = ((-2.0d0) * sinh(im)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.01) {
		tmp = (Math.cosh(im) - 1.0) * 0.5;
	} else {
		tmp = (-2.0 * Math.sinh(im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.01:
		tmp = (math.cosh(im) - 1.0) * 0.5
	else:
		tmp = (-2.0 * math.sinh(im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.01)
		tmp = Float64(Float64(cosh(im) - 1.0) * 0.5);
	else
		tmp = Float64(Float64(-2.0 * sinh(im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.01)
		tmp = (cosh(im) - 1.0) * 0.5;
	else
		tmp = (-2.0 * sinh(im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[Cosh[im], $MachinePrecision] - 1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(-2.0 * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0100000000000000002

   1. Initial program 54.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 2.0

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 2.0%

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

      1. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      3. lift-sinh.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      4. sinh-undef-rev N/A

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

      5. negate-sub2 N/A

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

      6. negate-sub N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      7. sinh---cosh-rev N/A

         \[\leadsto \left(\left(\cosh im - \sinh im\right) + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      8. associate--r- N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 2.7

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   6. Applied rewrites 2.7%

      \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 26.9%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 52.8%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 52.1%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{0 - im} - e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. sub0-neg N/A

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

      8. negate-sub2 N/A

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

      9. sinh-undef-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{2 \cdot \sinh im}\right)\right) \cdot \frac{1}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2}} \]

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

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh im\right)} \cdot \frac{1}{2} \]

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sinh.f64 85.5

          \[\leadsto \left(-2 \cdot \color{blue}{\sinh im}\right) \cdot 0.5 \]

   6. Applied rewrites 85.5%

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

Alternative 8: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh im - 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 -100.0)
     (* 0.5 (- 1.0 (exp im)))
     (if (<= t_0 0.0) (- im) (* (- (cosh im) 1.0) 0.5)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -100.0) {
		tmp = 0.5 * (1.0 - exp(im));
	} else if (t_0 <= 0.0) {
		tmp = -im;
	} else {
		tmp = (cosh(im) - 1.0) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    if (t_0 <= (-100.0d0)) then
        tmp = 0.5d0 * (1.0d0 - exp(im))
    else if (t_0 <= 0.0d0) then
        tmp = -im
    else
        tmp = (cosh(im) - 1.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	double tmp;
	if (t_0 <= -100.0) {
		tmp = 0.5 * (1.0 - Math.exp(im));
	} else if (t_0 <= 0.0) {
		tmp = -im;
	} else {
		tmp = (Math.cosh(im) - 1.0) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	tmp = 0
	if t_0 <= -100.0:
		tmp = 0.5 * (1.0 - math.exp(im))
	elif t_0 <= 0.0:
		tmp = -im
	else:
		tmp = (math.cosh(im) - 1.0) * 0.5
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(0.5 * Float64(1.0 - exp(im)));
	elseif (t_0 <= 0.0)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(cosh(im) - 1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	tmp = 0.0;
	if (t_0 <= -100.0)
		tmp = 0.5 * (1.0 - exp(im));
	elseif (t_0 <= 0.0)
		tmp = -im;
	else
		tmp = (cosh(im) - 1.0) * 0.5;
	end
	tmp_2 = 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 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], N[(0.5 * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-im), N[(N[(N[Cosh[im], $MachinePrecision] - 1.0), $MachinePrecision] * 0.5), $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))) < -100

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 74.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{0 - im} - e^{im}\right) \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 74.4%

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

   if -100 < (*.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 6.8%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 55.6

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\left(-2 \cdot \sinh im\right) \cdot 0.5} \]

   5. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.4

         \[\leadsto -im \]

   7. Applied rewrites 55.4%

      \[\leadsto -im \]

   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 97.7%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 73.2

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 73.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      3. lift-sinh.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      4. sinh-undef-rev N/A

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

      5. negate-sub2 N/A

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

      6. negate-sub N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      7. sinh---cosh-rev N/A

         \[\leadsto \left(\left(\cosh im - \sinh im\right) + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      8. associate--r- N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 72.0

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   6. Applied rewrites 72.0%

      \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 93.9%

      \[\leadsto \left(\cosh im - 1\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 58.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) - 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 -100.0)
     (* 0.5 (- 1.0 (exp im)))
     (if (<= t_0 0.0) (- im) (* (- (fma (* im im) 0.5 1.0) 1.0) 0.5)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -100.0) {
		tmp = 0.5 * (1.0 - exp(im));
	} else if (t_0 <= 0.0) {
		tmp = -im;
	} else {
		tmp = (fma((im * im), 0.5, 1.0) - 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(0.5 * Float64(1.0 - exp(im)));
	elseif (t_0 <= 0.0)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(fma(Float64(im * im), 0.5, 1.0) - 1.0) * 0.5);
	end
	return 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 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], N[(0.5 * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-im), N[(N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] - 1.0), $MachinePrecision] * 0.5), $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))) < -100

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 74.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{0 - im} - e^{im}\right) \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 74.4%

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

   if -100 < (*.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 6.8%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 55.6

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\left(-2 \cdot \sinh im\right) \cdot 0.5} \]

   5. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.4

         \[\leadsto -im \]

   7. Applied rewrites 55.4%

      \[\leadsto -im \]

   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 97.7%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 73.2

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 73.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      3. lift-sinh.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      4. sinh-undef-rev N/A

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

      5. negate-sub2 N/A

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

      6. negate-sub N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      7. sinh---cosh-rev N/A

         \[\leadsto \left(\left(\cosh im - \sinh im\right) + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      8. associate--r- N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 72.0

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   6. Applied rewrites 72.0%

      \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 47.4

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

   12. Applied rewrites 47.4%

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

Alternative 10: 56.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq 5.8 \cdot 10^{-254}:\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) - 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) 5.8e-254)
   (* (- (fma (* im im) 0.5 1.0) 1.0) 0.5)
   (* (fma (* im im) -0.16666666666666666 -1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= 5.8e-254) {
		tmp = (fma((im * im), 0.5, 1.0) - 1.0) * 0.5;
	} else {
		tmp = fma((im * im), -0.16666666666666666, -1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= 5.8e-254)
		tmp = Float64(Float64(fma(Float64(im * im), 0.5, 1.0) - 1.0) * 0.5);
	else
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], 5.8e-254], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] - 1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < 5.7999999999999999e-254

   1. Initial program 54.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub2 N/A

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

      4. lower-neg.f64 N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sinh.f64 2.0

         \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

   4. Applied rewrites 2.0%

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

      1. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      3. lift-sinh.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \sinh im\right)\right) \cdot \frac{1}{2} \]

      4. sinh-undef-rev N/A

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

      5. negate-sub2 N/A

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

      6. negate-sub N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      7. sinh---cosh-rev N/A

         \[\leadsto \left(\left(\cosh im - \sinh im\right) + \left(\mathsf{neg}\left(e^{im}\right)\right)\right) \cdot \frac{1}{2} \]

      8. associate--r- N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(\mathsf{neg}\left(e^{im}\right)\right)\right)\right) \cdot \frac{1}{2} \]

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 2.7

          \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   6. Applied rewrites 2.7%

      \[\leadsto \left(\cosh im - \left(\sinh im - \left(-e^{im}\right)\right)\right) \cdot 0.5 \]

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 26.9%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 15.1

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

   12. Applied rewrites 15.1%

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

   if 5.7999999999999999e-254 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 52.8%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in re around 0

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

      1. negate-sub N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 70.1

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

   7. Applied rewrites 70.1%

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

Alternative 11: 53.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (fma (* im im) -0.16666666666666666 -1.0) im))

C

double code(double re, double im) {
	return fma((im * im), -0.16666666666666666, -1.0) * im;
}

Julia

function code(re, im)
	return Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * im)
end

Wolfram

code[re_, im_] := N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

1. Initial program 53.1%

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

2. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 84.2

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

4. Applied rewrites 84.2%

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

5. Taylor expanded in re around 0

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

   1. negate-sub N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 53.0

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

7. Applied rewrites 53.0%

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

Alternative 12: 30.2% accurate, 32.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

2. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. negate-sub2 N/A

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

   4. lower-neg.f64 N/A

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

   5. sinh-undef N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sinh.f64 64.7

      \[\leadsto \left(-2 \cdot \sinh im\right) \cdot 0.5 \]

4. Applied rewrites 64.7%

   \[\leadsto \color{blue}{\left(-2 \cdot \sinh im\right) \cdot 0.5} \]

5. Taylor expanded in im around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 30.2

      \[\leadsto -im \]

7. Applied rewrites 30.2%

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

Reproduce

herbie shell --seed 2025110 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))