math.cos on complex, imaginary part

Percentage Accurate: 65.7% → 99.9%

Time: 11.3s

Alternatives: 19

Speedup: 1.4×

• 49.9% 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: 65.7% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. difference-of-squares N/A

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

   4. lift--.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. cosh-undef N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cosh.f64 N/A

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

4. Applied rewrites 99.9%

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

Alternative 2: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t\_0 \cdot \left(e^{-im} - e^{im}\right)\\ t_2 := \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\left(\sinh \left(-im\right) \cdot 2\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot t\_2, re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, t\_2, 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re)))
        (t_1 (* t_0 (- (exp (- im)) (exp im))))
        (t_2 (fma -0.0001984126984126984 (* re re) 0.008333333333333333)))
   (if (<= t_1 -1e+197)
     (* (* (sinh (- im)) 2.0) (* re 0.5))
     (if (<= t_1 2e-7)
       (*
        t_0
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))
       (*
        (*
         (fma
          (-
           (*
            (fma (* -0.16666666666666666 t_2) (* re re) 0.027777777777777776)
            (* re re))
           0.16666666666666666)
          (* im im)
          (- (* (fma (* (- re) re) t_2 0.16666666666666666) (* re re)) 1.0))
         re)
        im)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = t_0 * (exp(-im) - exp(im));
	double t_2 = fma(-0.0001984126984126984, (re * re), 0.008333333333333333);
	double tmp;
	if (t_1 <= -1e+197) {
		tmp = (sinh(-im) * 2.0) * (re * 0.5);
	} else if (t_1 <= 2e-7) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (fma(((fma((-0.16666666666666666 * t_2), (re * re), 0.027777777777777776) * (re * re)) - 0.16666666666666666), (im * im), ((fma((-re * re), t_2, 0.16666666666666666) * (re * re)) - 1.0)) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) - exp(im)))
	t_2 = fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)
	tmp = 0.0
	if (t_1 <= -1e+197)
		tmp = Float64(Float64(sinh(Float64(-im)) * 2.0) * Float64(re * 0.5));
	elseif (t_1 <= 2e-7)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(fma(Float64(-0.16666666666666666 * t_2), Float64(re * re), 0.027777777777777776) * Float64(re * re)) - 0.16666666666666666), Float64(im * im), Float64(Float64(fma(Float64(Float64(-re) * re), t_2, 0.16666666666666666) * Float64(re * re)) - 1.0)) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+197], N[(N[(N[Sinh[(-im)], $MachinePrecision] * 2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.16666666666666666 * t$95$2), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(N[(N[((-re) * re), $MachinePrecision] * t$95$2 + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $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))) < -9.9999999999999995e196

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 74.0%

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

   if -9.9999999999999995e196 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 33.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if 1.9999999999999999e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) + {re}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)\right)\right) - 1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 8.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot -0.16666666666666666\right), re \cdot re, \left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \left(\left(\left({im}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{36} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right) - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right) - 1\right) \cdot re\right) \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 49.7%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\left(\sinh \left(-im\right) \cdot 2\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot t\_1, re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, t\_1, 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))
        (t_1 (fma -0.0001984126984126984 (* re re) 0.008333333333333333)))
   (if (<= t_0 -1e+197)
     (* (* (sinh (- im)) 2.0) (* re 0.5))
     (if (<= t_0 2e-7)
       (*
        (*
         (sin re)
         (fma
          (* im im)
          (fma -0.008333333333333333 (* im im) -0.16666666666666666)
          -1.0))
        im)
       (*
        (*
         (fma
          (-
           (*
            (fma (* -0.16666666666666666 t_1) (* re re) 0.027777777777777776)
            (* re re))
           0.16666666666666666)
          (* im im)
          (- (* (fma (* (- re) re) t_1 0.16666666666666666) (* re re)) 1.0))
         re)
        im)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double t_1 = fma(-0.0001984126984126984, (re * re), 0.008333333333333333);
	double tmp;
	if (t_0 <= -1e+197) {
		tmp = (sinh(-im) * 2.0) * (re * 0.5);
	} else if (t_0 <= 2e-7) {
		tmp = (sin(re) * fma((im * im), fma(-0.008333333333333333, (im * im), -0.16666666666666666), -1.0)) * im;
	} else {
		tmp = (fma(((fma((-0.16666666666666666 * t_1), (re * re), 0.027777777777777776) * (re * re)) - 0.16666666666666666), (im * im), ((fma((-re * re), t_1, 0.16666666666666666) * (re * re)) - 1.0)) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)
	tmp = 0.0
	if (t_0 <= -1e+197)
		tmp = Float64(Float64(sinh(Float64(-im)) * 2.0) * Float64(re * 0.5));
	elseif (t_0 <= 2e-7)
		tmp = Float64(Float64(sin(re) * fma(Float64(im * im), fma(-0.008333333333333333, Float64(im * im), -0.16666666666666666), -1.0)) * im);
	else
		tmp = Float64(Float64(fma(Float64(Float64(fma(Float64(-0.16666666666666666 * t_1), Float64(re * re), 0.027777777777777776) * Float64(re * re)) - 0.16666666666666666), Float64(im * im), Float64(Float64(fma(Float64(Float64(-re) * re), t_1, 0.16666666666666666) * Float64(re * re)) - 1.0)) * re) * im);
	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]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+197], N[(N[(N[Sinh[(-im)], $MachinePrecision] * 2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-7], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(-0.008333333333333333 * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.16666666666666666 * t$95$1), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(N[(N[((-re) * re), $MachinePrecision] * t$95$1 + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $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))) < -9.9999999999999995e196

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 74.0%

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

   if -9.9999999999999995e196 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 33.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   if 1.9999999999999999e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) + {re}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)\right)\right) - 1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 8.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot -0.16666666666666666\right), re \cdot re, \left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \left(\left(\left({im}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{36} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right) - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right) - 1\right) \cdot re\right) \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 49.7%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\left(\sinh \left(-im\right) \cdot 2\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot t\_1, re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, t\_1, 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))
        (t_1 (fma -0.0001984126984126984 (* re re) 0.008333333333333333)))
   (if (<= t_0 -1e+197)
     (* (* (sinh (- im)) 2.0) (* re 0.5))
     (if (<= t_0 2e-7)
       (* (* (sin re) im) (fma (* -0.16666666666666666 im) im -1.0))
       (*
        (*
         (fma
          (-
           (*
            (fma (* -0.16666666666666666 t_1) (* re re) 0.027777777777777776)
            (* re re))
           0.16666666666666666)
          (* im im)
          (- (* (fma (* (- re) re) t_1 0.16666666666666666) (* re re)) 1.0))
         re)
        im)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double t_1 = fma(-0.0001984126984126984, (re * re), 0.008333333333333333);
	double tmp;
	if (t_0 <= -1e+197) {
		tmp = (sinh(-im) * 2.0) * (re * 0.5);
	} else if (t_0 <= 2e-7) {
		tmp = (sin(re) * im) * fma((-0.16666666666666666 * im), im, -1.0);
	} else {
		tmp = (fma(((fma((-0.16666666666666666 * t_1), (re * re), 0.027777777777777776) * (re * re)) - 0.16666666666666666), (im * im), ((fma((-re * re), t_1, 0.16666666666666666) * (re * re)) - 1.0)) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)
	tmp = 0.0
	if (t_0 <= -1e+197)
		tmp = Float64(Float64(sinh(Float64(-im)) * 2.0) * Float64(re * 0.5));
	elseif (t_0 <= 2e-7)
		tmp = Float64(Float64(sin(re) * im) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(fma(Float64(-0.16666666666666666 * t_1), Float64(re * re), 0.027777777777777776) * Float64(re * re)) - 0.16666666666666666), Float64(im * im), Float64(Float64(fma(Float64(Float64(-re) * re), t_1, 0.16666666666666666) * Float64(re * re)) - 1.0)) * re) * im);
	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]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+197], N[(N[(N[Sinh[(-im)], $MachinePrecision] * 2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-7], N[(N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.16666666666666666 * t$95$1), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(N[(N[((-re) * re), $MachinePrecision] * t$95$1 + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $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))) < -9.9999999999999995e196

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 74.0%

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

   if -9.9999999999999995e196 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 33.4%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

   5. Applied rewrites 99.2%

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

   if 1.9999999999999999e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) + {re}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)\right)\right) - 1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 8.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot -0.16666666666666666\right), re \cdot re, \left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \left(\left(\left({im}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{36} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right) - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right) - 1\right) \cdot re\right) \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 49.7%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-32}:\\ \;\;\;\;\left(\sinh \left(-im\right) \cdot 2\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot t\_0, re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, t\_0, 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma -0.0001984126984126984 (* re re) 0.008333333333333333))
        (t_1 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_1 -1e-32)
     (* (* (sinh (- im)) 2.0) (* re 0.5))
     (if (<= t_1 2e-7)
       (* (- (sin re)) im)
       (*
        (*
         (fma
          (-
           (*
            (fma (* -0.16666666666666666 t_0) (* re re) 0.027777777777777776)
            (* re re))
           0.16666666666666666)
          (* im im)
          (- (* (fma (* (- re) re) t_0 0.16666666666666666) (* re re)) 1.0))
         re)
        im)))))

C

double code(double re, double im) {
	double t_0 = fma(-0.0001984126984126984, (re * re), 0.008333333333333333);
	double t_1 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_1 <= -1e-32) {
		tmp = (sinh(-im) * 2.0) * (re * 0.5);
	} else if (t_1 <= 2e-7) {
		tmp = -sin(re) * im;
	} else {
		tmp = (fma(((fma((-0.16666666666666666 * t_0), (re * re), 0.027777777777777776) * (re * re)) - 0.16666666666666666), (im * im), ((fma((-re * re), t_0, 0.16666666666666666) * (re * re)) - 1.0)) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)
	t_1 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_1 <= -1e-32)
		tmp = Float64(Float64(sinh(Float64(-im)) * 2.0) * Float64(re * 0.5));
	elseif (t_1 <= 2e-7)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(Float64(fma(Float64(-0.16666666666666666 * t_0), Float64(re * re), 0.027777777777777776) * Float64(re * re)) - 0.16666666666666666), Float64(im * im), Float64(Float64(fma(Float64(Float64(-re) * re), t_0, 0.16666666666666666) * Float64(re * re)) - 1.0)) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-32], N[(N[(N[Sinh[(-im)], $MachinePrecision] * 2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.16666666666666666 * t$95$0), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(N[(N[((-re) * re), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $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))) < -1.00000000000000006e-32

   1. Initial program 99.6%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 73.4%

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

   if -1.00000000000000006e-32 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 32.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.9999999999999999e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) + {re}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)\right)\right) - 1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 8.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot -0.16666666666666666\right), re \cdot re, \left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \left(\left(\left({im}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{36} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right) - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right) - 1\right) \cdot re\right) \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 49.7%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot t\_1, re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, t\_1, 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))
        (t_1 (fma -0.0001984126984126984 (* re re) 0.008333333333333333)))
   (if (<= t_0 -1e+197)
     (*
      (*
       (-
        (*
         (*
          (-
           (*
            (*
             (- (* (* im im) -0.0003968253968253968) 0.016666666666666666)
             im)
            im)
           0.3333333333333333)
          im)
         im)
        2.0)
       im)
      (* re 0.5))
     (if (<= t_0 2e-7)
       (* (- (sin re)) im)
       (*
        (*
         (fma
          (-
           (*
            (fma (* -0.16666666666666666 t_1) (* re re) 0.027777777777777776)
            (* re re))
           0.16666666666666666)
          (* im im)
          (- (* (fma (* (- re) re) t_1 0.16666666666666666) (* re re)) 1.0))
         re)
        im)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double t_1 = fma(-0.0001984126984126984, (re * re), 0.008333333333333333);
	double tmp;
	if (t_0 <= -1e+197) {
		tmp = ((((((((((im * im) * -0.0003968253968253968) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * im) * im) - 2.0) * im) * (re * 0.5);
	} else if (t_0 <= 2e-7) {
		tmp = -sin(re) * im;
	} else {
		tmp = (fma(((fma((-0.16666666666666666 * t_1), (re * re), 0.027777777777777776) * (re * re)) - 0.16666666666666666), (im * im), ((fma((-re * re), t_1, 0.16666666666666666) * (re * re)) - 1.0)) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)
	tmp = 0.0
	if (t_0 <= -1e+197)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.0003968253968253968) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * im) * im) - 2.0) * im) * Float64(re * 0.5));
	elseif (t_0 <= 2e-7)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(Float64(fma(Float64(-0.16666666666666666 * t_1), Float64(re * re), 0.027777777777777776) * Float64(re * re)) - 0.16666666666666666), Float64(im * im), Float64(Float64(fma(Float64(Float64(-re) * re), t_1, 0.16666666666666666) * Float64(re * re)) - 1.0)) * re) * im);
	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]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+197], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-7], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.16666666666666666 * t$95$1), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(N[(N[((-re) * re), $MachinePrecision] * t$95$1 + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $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))) < -9.9999999999999995e196

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 74.0%

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

   8. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

   10. Applied rewrites 67.0%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot 0.5\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 67.0%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)} \]

   if -9.9999999999999995e196 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 33.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 1.9999999999999999e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) + {re}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)\right)\right) - 1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 8.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot -0.16666666666666666\right), re \cdot re, \left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \left(\left(\left({im}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{36} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right) - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right) - 1\right) \cdot re\right) \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 49.7%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot t\_0, re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, t\_0, 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma -0.0001984126984126984 (* re re) 0.008333333333333333)))
   (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) 0.0)
     (*
      (*
       (-
        (*
         (*
          (-
           (*
            (*
             (- (* (* im im) -0.0003968253968253968) 0.016666666666666666)
             im)
            im)
           0.3333333333333333)
          im)
         im)
        2.0)
       im)
      (* re 0.5))
     (*
      (*
       (fma
        (-
         (*
          (fma (* -0.16666666666666666 t_0) (* re re) 0.027777777777777776)
          (* re re))
         0.16666666666666666)
        (* im im)
        (- (* (fma (* (- re) re) t_0 0.16666666666666666) (* re re)) 1.0))
       re)
      im))))

C

double code(double re, double im) {
	double t_0 = fma(-0.0001984126984126984, (re * re), 0.008333333333333333);
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = ((((((((((im * im) * -0.0003968253968253968) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * im) * im) - 2.0) * im) * (re * 0.5);
	} else {
		tmp = (fma(((fma((-0.16666666666666666 * t_0), (re * re), 0.027777777777777776) * (re * re)) - 0.16666666666666666), (im * im), ((fma((-re * re), t_0, 0.16666666666666666) * (re * re)) - 1.0)) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.0003968253968253968) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * im) * im) - 2.0) * im) * Float64(re * 0.5));
	else
		tmp = Float64(Float64(fma(Float64(Float64(fma(Float64(-0.16666666666666666 * t_0), Float64(re * re), 0.027777777777777776) * Float64(re * re)) - 0.16666666666666666), Float64(im * im), Float64(Float64(fma(Float64(Float64(-re) * re), t_0, 0.16666666666666666) * Float64(re * re)) - 1.0)) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.16666666666666666 * t$95$0), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(N[(N[((-re) * re), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $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))) < -0.0

   1. Initial program 57.4%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 47.6

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

   5. Applied rewrites 47.6%

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

   7. Applied rewrites 59.3%

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

   8. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

   10. Applied rewrites 56.3%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot 0.5\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 56.3%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)} \]

   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 96.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 58.6%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) + {re}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)\right)\right) - 1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 12.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot -0.16666666666666666\right), re \cdot re, \left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \left(\left(\left({im}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{36} + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right) - \frac{1}{6}\right) + {re}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right)\right)\right) - 1\right) \cdot re\right) \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 51.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, 0.027777777777777776\right) \cdot \left(re \cdot re\right) - 0.16666666666666666, im \cdot im, \mathsf{fma}\left(\left(-re\right) \cdot re, \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 0.0006:\\ \;\;\;\;t\_0 \cdot \left(\left(2 \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \left(\left(0.3333333333333333 \cdot \left(im \cdot im\right) - 1\right) \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 0.0006)
     (*
      t_0
      (*
       (* 2.0 (fma (* im im) 0.5 1.0))
       (* (- (* 0.3333333333333333 (* im im)) 1.0) im)))
     (* t_0 (- (exp (- im)) (exp im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 0.0006) {
		tmp = t_0 * ((2.0 * fma((im * im), 0.5, 1.0)) * (((0.3333333333333333 * (im * im)) - 1.0) * im));
	} else {
		tmp = t_0 * (exp(-im) - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 0.0006)
		tmp = Float64(t_0 * Float64(Float64(2.0 * fma(Float64(im * im), 0.5, 1.0)) * Float64(Float64(Float64(0.3333333333333333 * Float64(im * im)) - 1.0) * im)));
	else
		tmp = Float64(t_0 * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.0006], N[(t$95$0 * N[(N[(2.0 * N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 5.99999999999999947e-4

   1. Initial program 55.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. difference-of-squares N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. cosh-undef N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cosh.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.5, 1\right)\right) \cdot \tanh \left(-im\right)\right) \]

   7. Applied rewrites 83.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, 0.5, 1\right)}\right) \cdot \tanh \left(-im\right)\right) \]

   8. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 66.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \left(\left(0.3333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} - 1\right) \cdot im\right)\right) \]

   10. Applied rewrites 66.1%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot \left(im \cdot im\right) - 1\right) \cdot im\right)}\right) \]

   if 5.99999999999999947e-4 < im

   1. Initial program 100.0%

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

Alternative 9: 74.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 0.94:\\ \;\;\;\;t\_0 \cdot \left(\left(2 \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \left(\left(0.3333333333333333 \cdot \left(im \cdot im\right) - 1\right) \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 0.94)
     (*
      t_0
      (*
       (* 2.0 (fma (* im im) 0.5 1.0))
       (* (- (* 0.3333333333333333 (* im im)) 1.0) im)))
     (* t_0 (- (- 1.0 im) (exp im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 0.94) {
		tmp = t_0 * ((2.0 * fma((im * im), 0.5, 1.0)) * (((0.3333333333333333 * (im * im)) - 1.0) * im));
	} else {
		tmp = t_0 * ((1.0 - im) - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 0.94)
		tmp = Float64(t_0 * Float64(Float64(2.0 * fma(Float64(im * im), 0.5, 1.0)) * Float64(Float64(Float64(0.3333333333333333 * Float64(im * im)) - 1.0) * im)));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.94], N[(t$95$0 * N[(N[(2.0 * N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 - im), $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.93999999999999995

   1. Initial program 55.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. difference-of-squares N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. cosh-undef N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cosh.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.5, 1\right)\right) \cdot \tanh \left(-im\right)\right) \]

   7. Applied rewrites 83.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, 0.5, 1\right)}\right) \cdot \tanh \left(-im\right)\right) \]

   8. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 66.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \left(\left(0.3333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} - 1\right) \cdot im\right)\right) \]

   10. Applied rewrites 66.1%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot \left(im \cdot im\right) - 1\right) \cdot im\right)}\right) \]

   if 0.93999999999999995 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 10: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (*
     (*
      (fma -0.16666666666666666 (* re re) 1.0)
      (-
       (*
        (* (- (* (* im im) -0.008333333333333333) 0.16666666666666666) im)
        im)
       1.0))
     re)
    im)
   (*
    (*
     (-
      (*
       (*
        (-
         (*
          (* (- (* (* im im) -0.0003968253968253968) 0.016666666666666666) im)
          im)
         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 = ((fma(-0.16666666666666666, (re * re), 1.0) * ((((((im * im) * -0.008333333333333333) - 0.16666666666666666) * im) * im) - 1.0)) * re) * im;
	} else {
		tmp = ((((((((((im * im) * -0.0003968253968253968) - 0.016666666666666666) * im) * im) - 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(fma(-0.16666666666666666, Float64(re * re), 1.0) * Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.008333333333333333) - 0.16666666666666666) * im) * im) - 1.0)) * re) * im);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.0003968253968253968) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * im) * im) - 2.0) * im) * Float64(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[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + {im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 26.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)\right) \cdot re\right) \cdot im \]

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

   1. Initial program 72.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   7. Applied rewrites 76.7%

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

   8. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

   10. Applied rewrites 70.9%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot 0.5\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 70.9%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-266}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-266)
   (*
    (*
     (*
      (fma -0.16666666666666666 (* re re) 1.0)
      (-
       (*
        (* (- (* (* im im) -0.008333333333333333) 0.16666666666666666) im)
        im)
       1.0))
     re)
    im)
   (*
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im)
    (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-266) {
		tmp = ((fma(-0.16666666666666666, (re * re), 1.0) * ((((((im * im) * -0.008333333333333333) - 0.16666666666666666) * im) * im) - 1.0)) * re) * im;
	} else {
		tmp = ((((((-0.016666666666666666 * (im * im)) - 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)) <= 5e-266)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.008333333333333333) - 0.16666666666666666) * im) * im) - 1.0)) * re) * im);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-266], N[(N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999992e-266

   1. Initial program 68.1%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + {im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 54.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)\right) \cdot re\right) \cdot im \]

   if 4.99999999999999992e-266 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 67.3%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot {im}^{2}} - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im} - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im} - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right)} \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)} \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\left(\left(\left(\frac{-1}{60} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      12. lower-*.f64 58.8

          \[\leadsto \left(\left(\left(\left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right) \]

   10. Applied rewrites 58.8%

       \[\leadsto \color{blue}{\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)} \cdot \left(re \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (*
     (*
      (fma (* re re) -0.16666666666666666 1.0)
      (- (* (* im im) -0.16666666666666666) 1.0))
     re)
    im)
   (*
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 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 = ((fma((re * re), -0.16666666666666666, 1.0) * (((im * im) * -0.16666666666666666) - 1.0)) * re) * im;
	} else {
		tmp = ((((((-0.016666666666666666 * (im * im)) - 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(fma(Float64(re * re), -0.16666666666666666, 1.0) * Float64(Float64(Float64(im * im) * -0.16666666666666666) - 1.0)) * re) * im);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im) * Float64(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] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

   4. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \sinh \left(-3 \cdot im\right)\right) \cdot \sin re\right) \cdot 0.5}{\mathsf{fma}\left(2, \cosh \left(-2 \cdot im\right), 1\right)}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-3}{2} \cdot \sin re - \frac{-4}{3} \cdot \sin re\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-3}{2} \cdot \sin re - \frac{-4}{3} \cdot \sin re\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-3}{2} \cdot \sin re - \frac{-4}{3} \cdot \sin re\right)\right) \cdot im} \]

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 26.9%

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

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

   1. Initial program 72.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   7. Applied rewrites 76.7%

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

   8. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right)} \cdot \left(re \cdot \frac{1}{2}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot {im}^{2}} - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im} - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im} - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right)} \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)} \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\left(\left(\left(\frac{-1}{60} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot \frac{1}{2}\right) \]

      12. lower-*.f64 69.9

          \[\leadsto \left(\left(\left(\left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right) \]

   10. Applied rewrites 69.9%

       \[\leadsto \color{blue}{\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)} \cdot \left(re \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \left(re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (*
     (*
      (fma (* re re) -0.16666666666666666 1.0)
      (- (* (* im im) -0.16666666666666666) 1.0))
     re)
    im)
   (*
    (-
     (* (* (- (* (* -0.008333333333333333 im) im) 0.16666666666666666) im) im)
     1.0)
    (* re im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = ((fma((re * re), -0.16666666666666666, 1.0) * (((im * im) * -0.16666666666666666) - 1.0)) * re) * im;
	} else {
		tmp = ((((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * (re * im);
	}
	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) * Float64(Float64(Float64(im * im) * -0.16666666666666666) - 1.0)) * re) * im);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * Float64(re * im));
	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] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

   4. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \sinh \left(-3 \cdot im\right)\right) \cdot \sin re\right) \cdot 0.5}{\mathsf{fma}\left(2, \cosh \left(-2 \cdot im\right), 1\right)}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-3}{2} \cdot \sin re - \frac{-4}{3} \cdot \sin re\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-3}{2} \cdot \sin re - \frac{-4}{3} \cdot \sin re\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-3}{2} \cdot \sin re - \frac{-4}{3} \cdot \sin re\right)\right) \cdot im} \]

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 26.9%

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

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

   1. Initial program 72.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 68.8%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 68.8%

       \[\leadsto \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \color{blue}{\left(re \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 55.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \left(re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* (* re im) re) 0.16666666666666666) re)
   (*
    (-
     (* (* (- (* (* -0.008333333333333333 im) im) 0.16666666666666666) im) im)
     1.0)
    (* re im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	} else {
		tmp = ((((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * (re * im);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.01d0)) then
        tmp = (((re * im) * re) * 0.16666666666666666d0) * re
    else
        tmp = (((((((-0.008333333333333333d0) * im) * im) - 0.16666666666666666d0) * im) * im) - 1.0d0) * (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	} else {
		tmp = ((((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.01:
		tmp = (((re * im) * re) * 0.16666666666666666) * re
	else:
		tmp = ((((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * im) * re) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.01)
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	else
		tmp = ((((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 20.1%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 24.4%

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

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

   1. Initial program 72.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 68.8%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 68.8%

       \[\leadsto \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \color{blue}{\left(re \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), im \cdot im, -re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* (* re im) re) 0.16666666666666666) re)
   (*
    (fma
     (* re (fma (* im im) -0.008333333333333333 -0.16666666666666666))
     (* im im)
     (- re))
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	} else {
		tmp = fma((re * fma((im * im), -0.008333333333333333, -0.16666666666666666)), (im * im), -re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * im) * re) * 0.16666666666666666) * re);
	else
		tmp = Float64(fma(Float64(re * fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666)), Float64(im * im), Float64(-re)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(re * N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + (-re)), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 20.1%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 24.4%

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

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

   1. Initial program 72.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 68.8%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 68.8%

       \[\leadsto \left(\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \left(-1 \cdot re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot re + \frac{-1}{120} \cdot \left({im}^{2} \cdot re\right)\right)\right) \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 67.4%

       \[\leadsto \mathsf{fma}\left(re \cdot \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), im \cdot im, -re\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	} else {
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * 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)) <= (-0.01d0)) then
        tmp = (((re * im) * re) * 0.16666666666666666d0) * re
    else
        tmp = ((((im * im) * (-0.16666666666666666d0)) - 1.0d0) * 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)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	} else {
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * im) * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.01:
		tmp = (((re * im) * re) * 0.16666666666666666) * re
	else:
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * im) * re
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.01)
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	else
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * im) * re;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 20.1%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 24.4%

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

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

   1. Initial program 72.9%

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

   4. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \sinh \left(-3 \cdot im\right)\right) \cdot \sin re\right) \cdot 0.5}{\mathsf{fma}\left(2, \cosh \left(-2 \cdot im\right), 1\right)}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-3}{2} \cdot \sin re - \frac{-4}{3} \cdot \sin re\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-3}{2} \cdot \sin re - \frac{-4}{3} \cdot \sin re\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-3}{2} \cdot \sin re - \frac{-4}{3} \cdot \sin re\right)\right) \cdot im} \]

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 66.2%

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

Alternative 17: 49.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 20.1%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 24.4%

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

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

   1. Initial program 72.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 68.8%

      \[\leadsto \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 68.8%

       \[\leadsto \left(\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 63.0%

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

Alternative 18: 34.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.01d0)) then
        tmp = (((re * im) * re) * 0.16666666666666666d0) * re
    else
        tmp = -re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.01:
		tmp = (((re * im) * re) * 0.16666666666666666) * re
	else:
		tmp = -re * im
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.01)
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	else
		tmp = -re * im;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 20.1%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 24.4%

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

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

   1. Initial program 72.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 39.9%

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

Alternative 19: 32.5% accurate, 39.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return -re * 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 = -re * im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 50.4

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

5. Applied rewrites 50.4%

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

6. Taylor expanded in re around 0

   \[\leadsto \left(-1 \cdot re\right) \cdot im \]
7. Step-by-step derivation

8. Applied rewrites 34.2%

   \[\leadsto \left(-re\right) \cdot im \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024346 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

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

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))