math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 18

Speedup: 1.5×

• 49.8% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[Exp[(-im)], $MachinePrecision] + N[(N[(N[Exp[im], $MachinePrecision] * 0.5), $MachinePrecision] * N[Sin[re], $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub0-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot 0.5, e^{-im}, \left(e^{im} \cdot 0.5\right) \cdot \sin re\right)} \]
5. Add Preprocessing

Alternative 2: 73.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (* (* re re) -0.08333333333333333) re) (+ 1.0 (exp im)))
     (if (<= t_0 1.0)
       (*
        (sin re)
        (fma
         (fma
          (fma 0.001388888888888889 (* im im) 0.041666666666666664)
          (* im im)
          0.5)
         (* im im)
         1.0))
       (* re (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 = (((re * re) * -0.08333333333333333) * re) * (1.0 + exp(im));
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0);
	} else {
		tmp = re * 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(Float64(Float64(re * re) * -0.08333333333333333) * re) * Float64(1.0 + exp(im)));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0));
	else
		tmp = Float64(re * 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[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[Cosh[im], $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 im around 0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 47.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(1 + e^{im}\right) \]

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 23.1%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(1 + e^{im}\right) \]

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sub0-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. lift-exp.f64 N/A

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

      16. sub0-neg N/A

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

      17. cosh-undef-rev N/A

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

      18. lift-cosh.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      5. lower-*.f64 100.0

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 99.7%

       \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \]

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sub0-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. lift-exp.f64 N/A

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

      16. sub0-neg N/A

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

      17. cosh-undef-rev N/A

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

      18. lift-cosh.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      5. lower-*.f64 100.0

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   10. Step-by-step derivation

   11. Applied rewrites 72.5%

       \[\leadsto \color{blue}{re} \cdot \cosh im \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.8%

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

Alternative 3: 73.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (* (* re re) -0.08333333333333333) re) (+ 1.0 (exp im)))
     (if (<= t_0 1.0)
       (*
        (sin re)
        (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0))
       (* re (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 = (((re * re) * -0.08333333333333333) * re) * (1.0 + exp(im));
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0);
	} else {
		tmp = re * 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(Float64(Float64(re * re) * -0.08333333333333333) * re) * Float64(1.0 + exp(im)));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0));
	else
		tmp = Float64(re * 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[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[Cosh[im], $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 im around 0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 47.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(1 + e^{im}\right) \]

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 23.1%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(1 + e^{im}\right) \]

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sub0-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. lift-exp.f64 N/A

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

      16. sub0-neg N/A

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

      17. cosh-undef-rev N/A

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

      18. lift-cosh.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      5. lower-*.f64 100.0

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 99.6%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sub0-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. lift-exp.f64 N/A

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

      16. sub0-neg N/A

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

      17. cosh-undef-rev N/A

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

      18. lift-cosh.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      5. lower-*.f64 100.0

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   10. Step-by-step derivation

   11. Applied rewrites 72.5%

       \[\leadsto \color{blue}{re} \cdot \cosh im \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.8%

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

Alternative 4: 73.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (* (* re re) -0.08333333333333333) re) (+ 1.0 (exp im)))
     (if (<= t_0 1.0) (* (fma (* 0.5 im) im 1.0) (sin re)) (* re (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 = (((re * re) * -0.08333333333333333) * re) * (1.0 + exp(im));
	} else if (t_0 <= 1.0) {
		tmp = fma((0.5 * im), im, 1.0) * sin(re);
	} else {
		tmp = re * 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(Float64(Float64(re * re) * -0.08333333333333333) * re) * Float64(1.0 + exp(im)));
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re));
	else
		tmp = Float64(re * 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[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(re * N[Cosh[im], $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 im around 0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 47.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(1 + e^{im}\right) \]

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 23.1%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(1 + e^{im}\right) \]

   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

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

   5. Applied rewrites 99.5%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sub0-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. lift-exp.f64 N/A

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

      16. sub0-neg N/A

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

      17. cosh-undef-rev N/A

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

      18. lift-cosh.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      5. lower-*.f64 100.0

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   10. Step-by-step derivation

   11. Applied rewrites 72.5%

       \[\leadsto \color{blue}{re} \cdot \cosh im \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.7%

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

Alternative 5: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (-
         (*
          (* 0.5 (fma -0.0001984126984126984 (* re re) 0.008333333333333333))
          (* re re))
         0.08333333333333333)
        (* re re)
        0.5)
       (* im im))
      re)
     (if (<= t_0 1.0) (* (fma (* 0.5 im) im 1.0) (sin re)) (* re (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((((0.5 * fma(-0.0001984126984126984, (re * re), 0.008333333333333333)) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * (im * im)) * re;
	} else if (t_0 <= 1.0) {
		tmp = fma((0.5 * im), im, 1.0) * sin(re);
	} else {
		tmp = re * 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(Float64(Float64(Float64(0.5 * fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * Float64(im * im)) * re);
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re));
	else
		tmp = Float64(re * 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[(N[(0.5 * N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(re * N[Cosh[im], $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 im around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 15.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right), re \cdot re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right) - \frac{1}{12}\right)\right)\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 49.4%

       \[\leadsto \left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re \]

   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

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

   5. Applied rewrites 99.5%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sub0-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. lift-exp.f64 N/A

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

      16. sub0-neg N/A

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

      17. cosh-undef-rev N/A

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

      18. lift-cosh.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      5. lower-*.f64 100.0

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   10. Step-by-step derivation

   11. Applied rewrites 72.5%

       \[\leadsto \color{blue}{re} \cdot \cosh im \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (-
         (*
          (* 0.5 (fma -0.0001984126984126984 (* re re) 0.008333333333333333))
          (* re re))
         0.08333333333333333)
        (* re re)
        0.5)
       (* im im))
      re)
     (if (<= t_0 1.0) (sin re) (* re (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((((0.5 * fma(-0.0001984126984126984, (re * re), 0.008333333333333333)) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * (im * im)) * re;
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = re * 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(Float64(Float64(Float64(0.5 * fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * Float64(im * im)) * re);
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(re * 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[(N[(0.5 * N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(re * N[Cosh[im], $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 im around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 15.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right), re \cdot re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right) - \frac{1}{12}\right)\right)\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 49.4%

       \[\leadsto \left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re \]

   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

   5. Applied rewrites 99.3%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sub0-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. lift-exp.f64 N/A

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

      16. sub0-neg N/A

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

      17. cosh-undef-rev N/A

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

      18. lift-cosh.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      5. lower-*.f64 100.0

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   10. Step-by-step derivation

   11. Applied rewrites 72.5%

       \[\leadsto \color{blue}{re} \cdot \cosh im \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (-
         (*
          (* 0.5 (fma -0.0001984126984126984 (* re re) 0.008333333333333333))
          (* re re))
         0.08333333333333333)
        (* re re)
        0.5)
       (* im im))
      re)
     (if (<= t_0 1.0)
       (sin re)
       (*
        (* 0.5 re)
        (fma
         (fma
          (fma 0.002777777777777778 (* im im) 0.08333333333333333)
          (* im im)
          1.0)
         (* im im)
         2.0))))))

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((((0.5 * fma(-0.0001984126984126984, (re * re), 0.008333333333333333)) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * (im * im)) * re;
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	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(Float64(Float64(Float64(0.5 * fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * Float64(im * im)) * re);
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	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[(N[(0.5 * N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * 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]), $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 im around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 15.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right), re \cdot re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right) - \frac{1}{12}\right)\right)\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 49.4%

       \[\leadsto \left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re \]

   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

   5. Applied rewrites 99.3%

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

   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 im around 0

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.7%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 47.5%

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

   12. Taylor expanded in im around 0

       \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 66.9%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, -0.16666666666666666\right), re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.2)
   (*
    (fma
     (fma
      (fma -0.0001984126984126984 (* re re) 0.008333333333333333)
      (* re re)
      -0.16666666666666666)
     (* re re)
     1.0)
    re)
   (*
    (* 0.5 re)
    (fma
     (fma
      (fma 0.002777777777777778 (* im im) 0.08333333333333333)
      (* im im)
      1.0)
     (* im im)
     2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.2) {
		tmp = fma(fma(fma(-0.0001984126984126984, (re * re), 0.008333333333333333), (re * re), -0.16666666666666666), (re * re), 1.0) * re;
	} else {
		tmp = (0.5 * re) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.2)
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333), Float64(re * re), -0.16666666666666666), Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	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], 0.2], N[(N[(N[(N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * 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]), $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))) < 0.20000000000000001

   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

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right), re \cdot re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 46.5%

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

   if 0.20000000000000001 < (*.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

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.9%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 32.8%

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

   12. Taylor expanded in im around 0

       \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 45.5%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, -0.16666666666666666\right), re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, -0.16666666666666666\right), re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.2)
   (*
    (fma
     (fma
      (fma -0.0001984126984126984 (* re re) 0.008333333333333333)
      (* re re)
      -0.16666666666666666)
     (* re re)
     1.0)
    re)
   (* (* 0.5 re) (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.2) {
		tmp = fma(fma(fma(-0.0001984126984126984, (re * re), 0.008333333333333333), (re * re), -0.16666666666666666), (re * re), 1.0) * re;
	} else {
		tmp = (0.5 * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.2)
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333), Float64(re * re), -0.16666666666666666), Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	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], 0.2], N[(N[(N[(N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $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))) < 0.20000000000000001

   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

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right), re \cdot re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 46.5%

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

   if 0.20000000000000001 < (*.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

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.9%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 32.8%

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

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 44.5%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, -0.16666666666666666\right), re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot im\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.1)
   (* (* (* (fma (* re re) -0.08333333333333333 0.5) im) im) re)
   (* (* 0.5 re) (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = ((fma((re * re), -0.08333333333333333, 0.5) * im) * im) * re;
	} else {
		tmp = (0.5 * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * im) * im) * re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	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], -0.1], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $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))) < -0.10000000000000001

   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

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 10.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 32.2%

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

   if -0.10000000000000001 < (*.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

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.9%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 54.0%

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

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 61.6%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot im\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot im\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.1)
   (* (* (* (fma (* re re) -0.08333333333333333 0.5) im) im) re)
   (*
    (* 0.5 re)
    (+ 1.0 (fma (fma (fma 0.16666666666666666 im 0.5) im 1.0) im 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = ((fma((re * re), -0.08333333333333333, 0.5) * im) * im) * re;
	} else {
		tmp = (0.5 * re) * (1.0 + fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * im) * im) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 + fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0)));
	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], -0.1], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[(N[(N[(0.16666666666666666 * im + 0.5), $MachinePrecision] * im + 1.0), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $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))) < -0.10000000000000001

   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

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 10.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 32.2%

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

   if -0.10000000000000001 < (*.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

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.9%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 53.7%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot im\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.2)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (fma (* 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))) <= 0.2) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma((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))) <= 0.2)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(fma(Float64(0.5 * 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], 0.2], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 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))) < 0.20000000000000001

   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

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 41.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 44.4%

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

   if 0.20000000000000001 < (*.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

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.8%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \cosh im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sin re) (cosh im)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return math.sin(re) * math.cosh(im)

Julia

function code(re, im)
	return Float64(sin(re) * cosh(im))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[Cosh[im], $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub0-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. sub0-neg N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. distribute-rgt-in N/A

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

   14. +-commutative N/A

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

   15. lift-exp.f64 N/A

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

   16. sub0-neg N/A

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

   17. cosh-undef-rev N/A

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

   18. lift-cosh.f64 N/A

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

   19. lift-*.f64 N/A

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

6. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 100.0

      \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\cosh im \cdot \sin re} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   5. lower-*.f64 100.0

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
9. Add Preprocessing

Alternative 14: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (*
     (fma
      (-
       (*
        (* 0.5 (fma -0.0001984126984126984 (* re re) 0.008333333333333333))
        (* re re))
       0.08333333333333333)
      (* re re)
      0.5)
     (* im im))
    re)
   (*
    (* 0.5 re)
    (fma
     (fma
      (fma 0.002777777777777778 (* im im) 0.08333333333333333)
      (* im im)
      1.0)
     (* im im)
     2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma((((0.5 * fma(-0.0001984126984126984, (re * re), 0.008333333333333333)) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * (im * im)) * re;
	} else {
		tmp = (0.5 * re) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(0.5 * fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333)) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * Float64(im * im)) * re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	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[(0.5 * N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * 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]), $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

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 15.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right), re \cdot re, \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right), re \cdot re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right)\right) - \frac{1}{12}\right)\right)\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 24.2%

       \[\leadsto \left(\mathsf{fma}\left(\left(0.5 \cdot \mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right)\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re \]

   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 im around 0

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.5%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 54.2%

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

   12. Taylor expanded in im around 0

       \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 65.4%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 48.6% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* (fma (* re re) -0.08333333333333333 0.5) im) im) re)
   (* (fma (* 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) * im) * im) * re;
	} else {
		tmp = fma((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(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * im) * im) * re);
	else
		tmp = Float64(fma(Float64(0.5 * 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[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 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

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 13.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.3%

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

   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 im around 0

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.2%

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

Alternative 16: 48.5% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* (fma (* re re) -0.08333333333333333 0.5) re) im) im)
   (* (fma (* 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) * im) * im;
	} else {
		tmp = fma((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(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * im) * im);
	else
		tmp = Float64(fma(Float64(0.5 * 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[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 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

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 13.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.2%

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

   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 im around 0

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.2%

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

Alternative 17: 34.5% 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

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

5. Applied rewrites 73.3%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 27.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

9. Taylor expanded in im around 0

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

11. Applied rewrites 32.1%

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

Alternative 18: 26.6% 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 im around 0

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

5. Applied rewrites 49.9%

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

6. Taylor expanded in re around 0

   \[\leadsto re \]
7. Step-by-step derivation

8. Applied rewrites 23.3%

   \[\leadsto re \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025019 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))