math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 20

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% 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]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \cosh im\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 2.0 (cosh im)))
        (t_1 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (* t_0 (* (* re re) -0.08333333333333333)) re)
     (if (<= t_1 1.0)
       (fma
        (* (sin re) (fma 0.041666666666666664 (* im im) 0.5))
        (* im im)
        (sin re))
       (* (* re 0.5) t_0)))))

C

double code(double re, double im) {
	double t_0 = 2.0 * cosh(im);
	double t_1 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 * ((re * re) * -0.08333333333333333)) * re;
	} else if (t_1 <= 1.0) {
		tmp = fma((sin(re) * fma(0.041666666666666664, (im * im), 0.5)), (im * im), sin(re));
	} else {
		tmp = (re * 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(2.0 * cosh(im))
	t_1 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	elseif (t_1 <= 1.0)
		tmp = fma(Float64(sin(re) * fma(0.041666666666666664, Float64(im * im), 0.5)), Float64(im * im), sin(re));
	else
		tmp = Float64(Float64(re * 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 * N[Cosh[im], $MachinePrecision]), $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, (-Infinity)], N[(N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(N[Sin[re], $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 28.4

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

   8. Applied rewrites 28.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \color{blue}{\sin re} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin \color{blue}{re} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, \color{blue}{{im}^{2}}, \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      13. lower-sin.f64 99.4

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 2.0 (cosh im)))
        (t_1 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (* t_0 (* (* re re) -0.08333333333333333)) re)
     (if (<= t_1 1.0)
       (fma (* (* im im) 0.5) (sin re) (sin re))
       (* (* re 0.5) t_0)))))

C

double code(double re, double im) {
	double t_0 = 2.0 * cosh(im);
	double t_1 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 * ((re * re) * -0.08333333333333333)) * re;
	} else if (t_1 <= 1.0) {
		tmp = fma(((im * im) * 0.5), sin(re), sin(re));
	} else {
		tmp = (re * 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(2.0 * cosh(im))
	t_1 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	elseif (t_1 <= 1.0)
		tmp = fma(Float64(Float64(im * im) * 0.5), sin(re), sin(re));
	else
		tmp = Float64(Float64(re * 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 * N[Cosh[im], $MachinePrecision]), $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, (-Infinity)], N[(N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision] * N[Sin[re], $MachinePrecision] + N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 28.4

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

   8. Applied rewrites 28.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \color{blue}{\sin re} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin \color{blue}{re} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, \color{blue}{{im}^{2}}, \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      13. lower-sin.f64 99.4

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right)} \]

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.5, \sin re, \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \cosh im\\ t_1 := 0.5 \cdot \sin re\\ t_2 := t\_1 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 2.0 (cosh im)))
        (t_1 (* 0.5 (sin re)))
        (t_2 (* t_1 (+ (exp (- im)) (exp im)))))
   (if (<= t_2 (- INFINITY))
     (* (* t_0 (* (* re re) -0.08333333333333333)) re)
     (if (<= t_2 1.0) (* t_1 (fma im im 2.0)) (* (* re 0.5) t_0)))))

C

double code(double re, double im) {
	double t_0 = 2.0 * cosh(im);
	double t_1 = 0.5 * sin(re);
	double t_2 = t_1 * (exp(-im) + exp(im));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 * ((re * re) * -0.08333333333333333)) * re;
	} else if (t_2 <= 1.0) {
		tmp = t_1 * fma(im, im, 2.0);
	} else {
		tmp = (re * 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(2.0 * cosh(im))
	t_1 = Float64(0.5 * sin(re))
	t_2 = Float64(t_1 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	elseif (t_2 <= 1.0)
		tmp = Float64(t_1 * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(re * 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(t$95$1 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 28.4

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

   8. Applied rewrites 28.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma 0.002777777777777778 (* im im) 0.08333333333333333)
         (* im im)
         1.0)
        (* im im)
        2.0)
       (* (* re re) -0.08333333333333333))
      re)
     (if (<= t_1 1.0)
       (* t_0 (fma im im 2.0))
       (* (* re 0.5) (* 2.0 (cosh 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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * ((re * re) * -0.08333333333333333)) * re;
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = (re * 0.5) * (2.0 * cosh(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)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(2.0 * cosh(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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      14. lower-*.f64 71.8

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 71.8%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot re \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

       3. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right) \cdot re \]

       4. lower-*.f64 28.4

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

   11. Applied rewrites 28.4%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma 0.002777777777777778 (* im im) 0.08333333333333333)
         (* im im)
         1.0)
        (* im im)
        2.0)
       (* (* re re) -0.08333333333333333))
      re)
     (if (<= t_0 1.0) (sin re) (* (* re 0.5) (* 2.0 (cosh im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * ((re * re) * -0.08333333333333333)) * re;
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = (re * 0.5) * (2.0 * cosh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(re * 0.5) * Float64(2.0 * cosh(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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      14. lower-*.f64 71.8

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 71.8%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot re \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

       3. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right) \cdot re \]

       4. lower-*.f64 28.4

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

   11. Applied rewrites 28.4%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 98.3

         \[\leadsto \sin re \]

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\sin re} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma 0.002777777777777778 (* im im) 0.08333333333333333)
         (* im im)
         1.0)
        (* im im)
        2.0)
       (* (* re re) -0.08333333333333333))
      re)
     (if (<= t_0 1.0)
       (sin re)
       (*
        (fma
         (fma
          (fma (* im im) 0.001388888888888889 0.041666666666666664)
          (* im im)
          0.5)
         (* im im)
         1.0)
        re)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * ((re * re) * -0.08333333333333333)) * re;
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = fma(fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(fma(fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      14. lower-*.f64 71.8

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 71.8%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot re \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

       3. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right) \cdot re \]

       4. lower-*.f64 28.4

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

   11. Applied rewrites 28.4%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 98.3

         \[\leadsto \sin re \]

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\sin re} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right), {im}^{\color{blue}{2}}, re\right) \]

   8. Applied rewrites 61.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.001388888888888889, 0.041666666666666664 \cdot re\right), im \cdot im, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot re \]

   11. Applied rewrites 64.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.0005)
   (*
    (*
     (fma (fma 0.08333333333333333 (* im im) 1.0) (* im im) 2.0)
     (fma (* re re) -0.08333333333333333 0.5))
    re)
   (*
    (fma
     (fma
      (fma (* im im) 0.001388888888888889 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.0005) {
		tmp = (fma(fma(0.08333333333333333, (im * im), 1.0), (im * im), 2.0) * fma((re * re), -0.08333333333333333, 0.5)) * re;
	} else {
		tmp = fma(fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0005)
		tmp = Float64(Float64(fma(fma(0.08333333333333333, Float64(im * im), 1.0), Float64(im * im), 2.0) * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re);
	else
		tmp = Float64(fma(fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0005], N[(N[(N[(N[(0.08333333333333333 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 67.8

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 67.8%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   if 5.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right), {im}^{\color{blue}{2}}, re\right) \]

   8. Applied rewrites 39.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.001388888888888889, 0.041666666666666664 \cdot re\right), im \cdot im, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot re \]

   11. Applied rewrites 41.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.0005)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (fma im im 2.0))
   (* (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.0005) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0005)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 60.3

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

   8. Applied rewrites 60.3%

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

   if 5.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \color{blue}{\sin re} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin \color{blue}{re} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, \color{blue}{{im}^{2}}, \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      13. lower-sin.f64 78.7

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2} + 1\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2}, 1\right) \cdot re \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}, {im}^{2}, 1\right) \cdot re \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{24} + \frac{1}{2}, {im}^{2}, 1\right) \cdot re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot re \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot re \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), im \cdot im, 1\right) \cdot re \]

      12. lower-*.f64 39.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re \]

   8. Applied rewrites 39.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.05)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.05) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.05)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 27.8

         \[\leadsto \sin re \]

   5. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 12.9

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

   8. Applied rewrites 12.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \color{blue}{\sin re} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin \color{blue}{re} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, \color{blue}{{im}^{2}}, \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      13. lower-sin.f64 86.1

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right) \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2} + 1\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2}, 1\right) \cdot re \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}, {im}^{2}, 1\right) \cdot re \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \frac{1}{24} + \frac{1}{2}, {im}^{2}, 1\right) \cdot re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot re \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot re \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), im \cdot im, 1\right) \cdot re \]

      12. lower-*.f64 61.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re \]

   8. Applied rewrites 61.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664, im \cdot im, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.0005)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (fma (* (* (* im im) re) 0.041666666666666664) (* im im) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.0005) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma((((im * im) * re) * 0.041666666666666664), (im * im), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0005)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = fma(Float64(Float64(Float64(im * im) * re) * 0.041666666666666664), Float64(im * im), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0005], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 53.1

         \[\leadsto \sin re \]

   5. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 43.7

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

   8. Applied rewrites 43.7%

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

   if 5.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re, {im}^{\color{blue}{2}}, re\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + \frac{1}{24} \cdot \left({im}^{2} \cdot re\right), {im}^{2}, re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re, {im}^{2}, re\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right), {im}^{2}, re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot re, {im}^{2}, re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot re, {im}^{2}, re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot \frac{1}{24} + \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right) \cdot re, im \cdot im, re\right) \]

      15. lower-*.f64 31.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re, im \cdot im, re\right) \]

   8. Applied rewrites 31.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re, \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right), im \cdot im, re\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{24}, im \cdot im, re\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{24}, im \cdot im, re\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{24}, im \cdot im, re\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot \frac{1}{24}, im \cdot im, re\right) \]

       5. lower-*.f64 31.5

          \[\leadsto \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664, im \cdot im, re\right) \]

   11. Applied rewrites 31.5%

       \[\leadsto \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664, im \cdot im, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664, im \cdot im, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.001388888888888889, im \cdot im, re \cdot 0.5\right), im \cdot im, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.0002)
   (*
    (*
     (fma
      (fma
       (fma 0.002777777777777778 (* im im) 0.08333333333333333)
       (* im im)
       1.0)
      (* im im)
      2.0)
     (fma (* re re) -0.08333333333333333 0.5))
    re)
   (fma
    (fma (* (* (* im im) re) 0.001388888888888889) (* im im) (* re 0.5))
    (* im im)
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.0002) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * fma((re * re), -0.08333333333333333, 0.5)) * re;
	} else {
		tmp = fma(fma((((im * im) * re) * 0.001388888888888889), (im * im), (re * 0.5)), (im * im), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.0002)
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re);
	else
		tmp = fma(fma(Float64(Float64(Float64(im * im) * re) * 0.001388888888888889), Float64(im * im), Float64(re * 0.5)), Float64(im * im), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(re * 0.5), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 2.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      14. lower-*.f64 73.4

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 73.4%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   if 2.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 25.1

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

   5. Applied rewrites 25.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right), {im}^{\color{blue}{2}}, re\right) \]

   8. Applied rewrites 22.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.001388888888888889, 0.041666666666666664 \cdot re\right), im \cdot im, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right), im \cdot im, re \cdot \frac{1}{2}\right), im \cdot im, re\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{720}, im \cdot im, re \cdot \frac{1}{2}\right), im \cdot im, re\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{720}, im \cdot im, re \cdot \frac{1}{2}\right), im \cdot im, re\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{720}, im \cdot im, re \cdot \frac{1}{2}\right), im \cdot im, re\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot \frac{1}{720}, im \cdot im, re \cdot \frac{1}{2}\right), im \cdot im, re\right) \]

       5. lower-*.f64 22.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.001388888888888889, im \cdot im, re \cdot 0.5\right), im \cdot im, re\right) \]

   11. Applied rewrites 22.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.001388888888888889, im \cdot im, re \cdot 0.5\right), im \cdot im, re\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.001388888888888889 \cdot \left(im \cdot im\right) - 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (*
     (fma
      (fma
       (fma 0.002777777777777778 (* im im) 0.08333333333333333)
       (* im im)
       1.0)
      (* im im)
      2.0)
     (* (* re re) -0.08333333333333333))
    re)
   (*
    (-
     (*
      (-
       (*
        (- (* -0.001388888888888889 (* im im)) 0.041666666666666664)
        (* im im))
       0.5)
      (* im im))
     1.0)
    (- re))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * ((re * re) * -0.08333333333333333)) * re;
	} else {
		tmp = ((((((-0.001388888888888889 * (im * im)) - 0.041666666666666664) * (im * im)) - 0.5) * (im * im)) - 1.0) * -re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * Float64(Float64(re * re) * -0.08333333333333333)) * re);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.001388888888888889 * Float64(im * im)) - 0.041666666666666664) * Float64(im * im)) - 0.5) * Float64(im * im)) - 1.0) * Float64(-re));
	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[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $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 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      14. lower-*.f64 33.6

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 33.6%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot re \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

       3. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right) \cdot re \]

       4. lower-*.f64 33.6

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

   11. Applied rewrites 33.6%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right), {im}^{\color{blue}{2}}, re\right) \]

   8. Applied rewrites 65.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.001388888888888889, 0.041666666666666664 \cdot re\right), im \cdot im, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in re around -inf

      \[\leadsto -1 \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right)\right) \]

       2. lower-neg.f64 N/A

          \[\leadsto -re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right) \]

       3. *-commutative N/A

          \[\leadsto -\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right) \cdot re \]

       4. lower-*.f64 N/A

          \[\leadsto -\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right) \cdot re \]

   11. Applied rewrites 67.4%

       \[\leadsto -\left(\left(\left(-0.001388888888888889 \cdot \left(im \cdot im\right) - 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.001388888888888889 \cdot \left(im \cdot im\right) - 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.001388888888888889 \cdot \left(im \cdot im\right) - 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (*
     (fma
      (-
       (* (fma -9.92063492063492e-5 (* re re) 0.004166666666666667) (* re re))
       0.08333333333333333)
      (* re re)
      0.5)
     re)
    (fma im im 2.0))
   (*
    (-
     (*
      (-
       (*
        (- (* -0.001388888888888889 (* im im)) 0.041666666666666664)
        (* im im))
       0.5)
      (* im im))
     1.0)
    (- re))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = ((((((-0.001388888888888889 * (im * im)) - 0.041666666666666664) * (im * im)) - 0.5) * (im * im)) - 1.0) * -re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.001388888888888889 * Float64(im * im)) - 0.041666666666666664) * Float64(im * im)) - 0.5) * Float64(im * im)) - 1.0) * Float64(-re));
	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[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $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 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right), {im}^{\color{blue}{2}}, re\right) \]

   8. Applied rewrites 65.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.001388888888888889, 0.041666666666666664 \cdot re\right), im \cdot im, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in re around -inf

      \[\leadsto -1 \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right)\right) \]

       2. lower-neg.f64 N/A

          \[\leadsto -re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right) \]

       3. *-commutative N/A

          \[\leadsto -\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right) \cdot re \]

       4. lower-*.f64 N/A

          \[\leadsto -\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right) \cdot re \]

   11. Applied rewrites 67.4%

       \[\leadsto -\left(\left(\left(-0.001388888888888889 \cdot \left(im \cdot im\right) - 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.001388888888888889 \cdot \left(im \cdot im\right) - 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.001388888888888889 \cdot \left(im \cdot im\right) - 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (*
     (fma
      (-
       (* (fma -9.92063492063492e-5 (* re re) 0.004166666666666667) (* re re))
       0.08333333333333333)
      (* re re)
      0.5)
     re)
    (* im im))
   (*
    (-
     (*
      (-
       (*
        (- (* -0.001388888888888889 (* im im)) 0.041666666666666664)
        (* im im))
       0.5)
      (* im im))
     1.0)
    (- re))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * (im * im);
	} else {
		tmp = ((((((-0.001388888888888889 * (im * im)) - 0.041666666666666664) * (im * im)) - 0.5) * (im * im)) - 1.0) * -re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(im * im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.001388888888888889 * Float64(im * im)) - 0.041666666666666664) * Float64(im * im)) - 0.5) * Float64(im * im)) - 1.0) * Float64(-re));
	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[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $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 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(im \cdot im\right) \]

       2. lower-*.f64 31.6

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(im \cdot im\right) \]

   11. Applied rewrites 31.6%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(im \cdot \color{blue}{im}\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right), {im}^{\color{blue}{2}}, re\right) \]

   8. Applied rewrites 65.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.001388888888888889, 0.041666666666666664 \cdot re\right), im \cdot im, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in re around -inf

      \[\leadsto -1 \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right)\right) \]

       2. lower-neg.f64 N/A

          \[\leadsto -re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right) \]

       3. *-commutative N/A

          \[\leadsto -\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right) \cdot re \]

       4. lower-*.f64 N/A

          \[\leadsto -\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2} - \frac{1}{24}\right) - \frac{1}{2}\right) - 1\right) \cdot re \]

   11. Applied rewrites 67.4%

       \[\leadsto -\left(\left(\left(-0.001388888888888889 \cdot \left(im \cdot im\right) - 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.001388888888888889 \cdot \left(im \cdot im\right) - 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (fma im im 2.0))
   (*
    (fma
     (fma
      (fma (* im im) 0.001388888888888889 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = fma(fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * re);
	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 * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $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 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 27.8

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

   8. Applied rewrites 27.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right)\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{24} \cdot re\right), {im}^{\color{blue}{2}}, re\right) \]

   8. Applied rewrites 65.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.001388888888888889, 0.041666666666666664 \cdot re\right), im \cdot im, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot re \]

   11. Applied rewrites 67.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right) \cdot re, im \cdot im, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (fma (* (fma im (* im 0.041666666666666664) 0.5) re) (* im im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma((fma(im, (im * 0.041666666666666664), 0.5) * re), (im * im), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = fma(Float64(fma(im, Float64(im * 0.041666666666666664), 0.5) * re), Float64(im * im), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * 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 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 42.1

         \[\leadsto \sin re \]

   5. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 18.9

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

   8. Applied rewrites 18.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re, {im}^{\color{blue}{2}}, re\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + \frac{1}{24} \cdot \left({im}^{2} \cdot re\right), {im}^{2}, re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re, {im}^{2}, re\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right), {im}^{2}, re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot re, {im}^{2}, re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot re, {im}^{2}, re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot \frac{1}{24} + \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right) \cdot re, {im}^{2}, re\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right) \cdot re, im \cdot im, re\right) \]

      15. lower-*.f64 59.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re, im \cdot im, re\right) \]

   8. Applied rewrites 59.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re, \color{blue}{im \cdot im}, re\right) \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot \left(im \cdot \frac{1}{24}\right) + \frac{1}{2}\right) \cdot re, im \cdot im, re\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right) \cdot re, im \cdot im, re\right) \]

      3. lower-*.f64 59.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right) \cdot re, im \cdot im, re\right) \]

   10. Applied rewrites 59.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right) \cdot re, im \cdot im, re\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 47.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * 0.5 + 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 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 42.1

         \[\leadsto \sin re \]

   5. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 18.9

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

   8. Applied rewrites 18.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 54.6

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

   8. Applied rewrites 54.6%

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

Alternative 19: 33.8% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. lower-sin.f64 47.9

      \[\leadsto \sin re \]

5. Applied rewrites 47.9%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lower-*.f64 32.0

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

8. Applied rewrites 32.0%

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

Alternative 20: 26.3% accurate, 317.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. cosh-undef N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cosh.f64 60.8

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

5. Applied rewrites 60.8%

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

6. Taylor expanded in im around 0

   \[\leadsto re \]
7. Step-by-step derivation

8. Applied rewrites 23.9%

   \[\leadsto re \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025044 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))