math.cos on complex, imaginary part

Percentage Accurate: 64.4% → 99.9%

Time: 5.2s

Alternatives: 12

Speedup: 1.2×

• 48.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-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 * sin(re)) * (exp(-im) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 64.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-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 * sin(re)) * (exp(-im) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (sin(re) * 0.5) * (-2.0 * sinh(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 = (sin(re) * 0.5d0) * ((-2.0d0) * sinh(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sin.f64 64.4

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. negate-sub2 N/A

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

   11. lower-neg.f64 N/A

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

   12. sinh-undef N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sinh.f64 99.9

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

3. Applied rewrites 99.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sinh.f64 N/A

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

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

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lift-sinh.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 90.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1], N[(N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[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 52.8%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.4

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

   4. Applied rewrites 87.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. negate-sub N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. negate-sub N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 87.4

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

   6. Applied rewrites 87.4%

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

   if 2.10000000000000009 < im

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 99.7%

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

Alternative 3: 73.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* (* 0.5 (sin re)) t_0)))
   (if (<= t_1 -4e-72)
     (* (* 0.5 re) t_0)
     (if (<= t_1 20.0)
       (* (* (sin re) (fma -0.16666666666666666 (* im im) -1.0)) im)
       (* (* (* -2.0 im) (* (* re re) -0.08333333333333333)) re)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = (0.5 * sin(re)) * t_0;
	double tmp;
	if (t_1 <= -4e-72) {
		tmp = (0.5 * re) * t_0;
	} else if (t_1 <= 20.0) {
		tmp = (sin(re) * fma(-0.16666666666666666, (im * im), -1.0)) * im;
	} else {
		tmp = ((-2.0 * im) * ((re * re) * -0.08333333333333333)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(Float64(0.5 * sin(re)) * t_0)
	tmp = 0.0
	if (t_1 <= -4e-72)
		tmp = Float64(Float64(0.5 * re) * t_0);
	elseif (t_1 <= 20.0)
		tmp = Float64(Float64(sin(re) * fma(-0.16666666666666666, Float64(im * im), -1.0)) * im);
	else
		tmp = Float64(Float64(Float64(-2.0 * im) * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-72], N[(N[(0.5 * re), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 20.0], N[(N[(N[Sin[re], $MachinePrecision] * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(-2.0 * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -3.9999999999999999e-72

   1. Initial program 98.6%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 73.1%

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

   if -3.9999999999999999e-72 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 20

   1. Initial program 30.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 20 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

   4. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \sinh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 23.1

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   7. Applied rewrites 23.1%

      \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

      3. pow2 N/A

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right) \cdot re \]

      4. lift-*.f64 22.0

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

   10. Applied rewrites 22.0%

       \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* (* 0.5 (sin re)) t_0)))
   (if (<= t_1 -4e-72)
     (* (* 0.5 re) t_0)
     (if (<= t_1 20.0)
       (* (- (sin re)) im)
       (* (* (* -2.0 im) (* (* re re) -0.08333333333333333)) re)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = (0.5 * sin(re)) * t_0;
	double tmp;
	if (t_1 <= -4e-72) {
		tmp = (0.5 * re) * t_0;
	} else if (t_1 <= 20.0) {
		tmp = -sin(re) * im;
	} else {
		tmp = ((-2.0 * im) * ((re * re) * -0.08333333333333333)) * re;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = (0.5d0 * sin(re)) * t_0
    if (t_1 <= (-4d-72)) then
        tmp = (0.5d0 * re) * t_0
    else if (t_1 <= 20.0d0) then
        tmp = -sin(re) * im
    else
        tmp = (((-2.0d0) * im) * ((re * re) * (-0.08333333333333333d0))) * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = (0.5 * Math.sin(re)) * t_0;
	double tmp;
	if (t_1 <= -4e-72) {
		tmp = (0.5 * re) * t_0;
	} else if (t_1 <= 20.0) {
		tmp = -Math.sin(re) * im;
	} else {
		tmp = ((-2.0 * im) * ((re * re) * -0.08333333333333333)) * re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = (0.5 * math.sin(re)) * t_0
	tmp = 0
	if t_1 <= -4e-72:
		tmp = (0.5 * re) * t_0
	elif t_1 <= 20.0:
		tmp = -math.sin(re) * im
	else:
		tmp = ((-2.0 * im) * ((re * re) * -0.08333333333333333)) * re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(Float64(0.5 * sin(re)) * t_0)
	tmp = 0.0
	if (t_1 <= -4e-72)
		tmp = Float64(Float64(0.5 * re) * t_0);
	elseif (t_1 <= 20.0)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(Float64(-2.0 * im) * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = (0.5 * sin(re)) * t_0;
	tmp = 0.0;
	if (t_1 <= -4e-72)
		tmp = (0.5 * re) * t_0;
	elseif (t_1 <= 20.0)
		tmp = -sin(re) * im;
	else
		tmp = ((-2.0 * im) * ((re * re) * -0.08333333333333333)) * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-72], N[(N[(0.5 * re), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 20.0], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(-2.0 * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -3.9999999999999999e-72

   1. Initial program 98.6%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 73.1%

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

   if -3.9999999999999999e-72 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 20

   1. Initial program 30.1%

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

   2. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-sin.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if 20 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

   4. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \sinh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 23.1

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   7. Applied rewrites 23.1%

      \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

      3. pow2 N/A

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right) \cdot re \]

      4. lift-*.f64 22.0

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

   10. Applied rewrites 22.0%

       \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sinh im \cdot 2\right) \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (*
     (* (fma (* re re) -0.16666666666666666 1.0) re)
     (* (* im im) -0.16666666666666666))
    im)
   (* (* (* (sinh im) 2.0) re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = ((fma((re * re), -0.16666666666666666, 1.0) * re) * ((im * im) * -0.16666666666666666)) * im;
	} else {
		tmp = ((sinh(im) * 2.0) * re) * -0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(fma(Float64(re * re), -0.16666666666666666, 1.0) * re) * Float64(Float64(im * im) * -0.16666666666666666)) * im);
	else
		tmp = Float64(Float64(Float64(sinh(im) * 2.0) * re) * -0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[Sinh[im], $MachinePrecision] * 2.0), $MachinePrecision] * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 85.9

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 37.4

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

   7. Applied rewrites 37.4%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 23.8

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

   10. Applied rewrites 23.8%

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

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

   1. Initial program 68.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 68.5

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. negate-sub2 N/A

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

      11. lower-neg.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. rec-exp N/A

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

      6. sinh-undef-rev N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 75.1

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

   6. Applied rewrites 75.1%

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

Alternative 6: 61.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sinh im \cdot 2\right) \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* -2.0 im) (fma (* re re) -0.08333333333333333 0.5)) re)
   (* (* (* (sinh im) 2.0) re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = ((-2.0 * im) * fma((re * re), -0.08333333333333333, 0.5)) * re;
	} else {
		tmp = ((sinh(im) * 2.0) * re) * -0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(-2.0 * im) * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re);
	else
		tmp = Float64(Float64(Float64(sinh(im) * 2.0) * re) * -0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(-2.0 * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[Sinh[im], $MachinePrecision] * 2.0), $MachinePrecision] * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \sinh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 21.6

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   7. Applied rewrites 21.6%

      \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

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

   1. Initial program 68.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 68.5

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. negate-sub2 N/A

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

      11. lower-neg.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. rec-exp N/A

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

      6. sinh-undef-rev N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 75.1

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

   6. Applied rewrites 75.1%

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

Alternative 7: 52.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\left(\left(1 - e^{im}\right) \cdot re\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -4e-72)
   (* (* (- 1.0 (exp im)) re) 0.5)
   (* (* (* -2.0 im) (fma (* re re) -0.08333333333333333 0.5)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -4e-72) {
		tmp = ((1.0 - exp(im)) * re) * 0.5;
	} else {
		tmp = ((-2.0 * im) * fma((re * re), -0.08333333333333333, 0.5)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -4e-72)
		tmp = Float64(Float64(Float64(1.0 - exp(im)) * re) * 0.5);
	else
		tmp = Float64(Float64(Float64(-2.0 * im) * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-72], N[(N[(N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(-2.0 * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -3.9999999999999999e-72

   1. Initial program 98.6%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. negate-sub2 N/A

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

      6. lower-neg.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 73.3

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

   4. Applied rewrites 73.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. sinh-undef-rev N/A

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

      5. negate-sub2 N/A

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

      6. lower--.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-exp.f64 73.1

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 38.0%

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

   if -3.9999999999999999e-72 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 53.2%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \sinh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 41.8

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   7. Applied rewrites 41.8%

      \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 44.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* -2.0 im) (fma (* re re) -0.08333333333333333 0.5)) re)
   (* (* (* (fma -0.3333333333333333 (* im im) -2.0) im) re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = ((-2.0 * im) * fma((re * re), -0.08333333333333333, 0.5)) * re;
	} else {
		tmp = ((fma(-0.3333333333333333, (im * im), -2.0) * im) * re) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(-2.0 * im) * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re);
	else
		tmp = Float64(Float64(Float64(fma(-0.3333333333333333, Float64(im * im), -2.0) * im) * re) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(-2.0 * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \sinh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 21.6

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   7. Applied rewrites 21.6%

      \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

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

   1. Initial program 68.5%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. negate-sub2 N/A

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

      6. lower-neg.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.1

         \[\leadsto \left(\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot re\right) \cdot 0.5 \]

   7. Applied rewrites 63.1%

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

Alternative 9: 44.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -0.005)
     (* (* (* (* im im) im) re) -0.16666666666666666)
     (if (<= t_0 0.0)
       (* (* im re) (fma (* im im) -0.16666666666666666 -1.0))
       (* (* (* -2.0 im) (* (* re re) -0.08333333333333333)) re)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -0.005) {
		tmp = (((im * im) * im) * re) * -0.16666666666666666;
	} else if (t_0 <= 0.0) {
		tmp = (im * re) * fma((im * im), -0.16666666666666666, -1.0);
	} else {
		tmp = ((-2.0 * im) * ((re * re) * -0.08333333333333333)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(Float64(Float64(Float64(im * im) * im) * re) * -0.16666666666666666);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * re) * fma(Float64(im * im), -0.16666666666666666, -1.0));
	else
		tmp = Float64(Float64(Float64(-2.0 * im) * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], N[(N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. negate-sub2 N/A

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

      6. lower-neg.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 74.7

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 49.1

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 55.0

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

   10. Applied rewrites 55.0%

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

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.0

   1. Initial program 29.7%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. negate-sub N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. negate-sub N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in re around 0

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

      1. lower-*.f64 51.4

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

   9. Applied rewrites 51.4%

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

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

   1. Initial program 97.9%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \sinh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 23.8

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   7. Applied rewrites 23.8%

      \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

      3. pow2 N/A

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right) \cdot re \]

      4. lift-*.f64 21.0

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

   10. Applied rewrites 21.0%

       \[\leadsto \left(\left(-2 \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 42.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -4e-72)
   (* (* (* (* im im) im) re) -0.16666666666666666)
   (* (* (* -2.0 im) (fma (* re re) -0.08333333333333333 0.5)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -4e-72) {
		tmp = (((im * im) * im) * re) * -0.16666666666666666;
	} else {
		tmp = ((-2.0 * im) * fma((re * re), -0.08333333333333333, 0.5)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -4e-72)
		tmp = Float64(Float64(Float64(Float64(im * im) * im) * re) * -0.16666666666666666);
	else
		tmp = Float64(Float64(Float64(-2.0 * im) * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-72], N[(N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(N[(N[(-2.0 * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -3.9999999999999999e-72

   1. Initial program 98.6%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. negate-sub2 N/A

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

      6. lower-neg.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 53.6

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

   10. Applied rewrites 53.6%

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

   if -3.9999999999999999e-72 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 53.2%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) + \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \cdot \color{blue}{re} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \sinh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 41.8

         \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   7. Applied rewrites 41.8%

      \[\leadsto \left(\left(-2 \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 40.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -4e-72)
   (* (* (* (* im im) im) re) -0.16666666666666666)
   (* (- im) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -4e-72) {
		tmp = (((im * im) * im) * re) * -0.16666666666666666;
	} else {
		tmp = -im * re;
	}
	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 * sin(re)) * (exp(-im) - exp(im))) <= (-4d-72)) then
        tmp = (((im * im) * im) * re) * (-0.16666666666666666d0)
    else
        tmp = -im * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im))) <= -4e-72) {
		tmp = (((im * im) * im) * re) * -0.16666666666666666;
	} else {
		tmp = -im * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))) <= -4e-72:
		tmp = (((im * im) * im) * re) * -0.16666666666666666
	else:
		tmp = -im * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -4e-72)
		tmp = Float64(Float64(Float64(Float64(im * im) * im) * re) * -0.16666666666666666);
	else
		tmp = Float64(Float64(-im) * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -4e-72)
		tmp = (((im * im) * im) * re) * -0.16666666666666666;
	else
		tmp = -im * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-72], N[(N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[((-im) * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -3.9999999999999999e-72

   1. Initial program 98.6%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. negate-sub2 N/A

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

      6. lower-neg.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 53.6

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

   10. Applied rewrites 53.6%

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

   if -3.9999999999999999e-72 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 53.2%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. negate-sub2 N/A

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

      6. lower-neg.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 39.4

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

   7. Applied rewrites 39.4%

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

Alternative 12: 33.7% accurate, 12.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return -im * re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

2. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. negate-sub2 N/A

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

   6. lower-neg.f64 N/A

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

   7. sinh-undef N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sinh.f64 62.7

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

4. Applied rewrites 62.7%

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

5. Taylor expanded in im around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. lower-*.f64 33.7

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

7. Applied rewrites 33.7%

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

Reproduce

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