math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 16

Speedup: 1.5×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× 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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub0-neg N/A

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

   11. cosh-undef N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. cosh-0 N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

   15. *-commutative N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   16. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   17. cosh-0 N/A

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

   18. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   19. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   20. exp-0 N/A

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

   21. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-lft-identity 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

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

6. Applied rewrites 100.0%

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

Alternative 2: 71.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right), re \cdot re, 1\right)\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (* 0.5 im)
       im
       (fma
        (fma (* im im) -0.08333333333333333 -0.16666666666666666)
        (* re re)
        1.0))
      re)
     (if (<= t_0 1.0)
       (* (fma (* 0.5 im) im 1.0) (sin re))
       (*
        (fma
         (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
         (* re re)
         (fma (fma (* 0.041666666666666664 im) im 0.5) (* im im) 1.0))
        re)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((0.5 * im), im, fma(fma((im * im), -0.08333333333333333, -0.16666666666666666), (re * re), 1.0)) * re;
	} else if (t_0 <= 1.0) {
		tmp = fma((0.5 * im), im, 1.0) * sin(re);
	} else {
		tmp = fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), fma(fma((0.041666666666666664 * im), im, 0.5), (im * im), 1.0)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(0.5 * im), im, fma(fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666), Float64(re * re), 1.0)) * re);
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re));
	else
		tmp = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), fma(fma(Float64(0.041666666666666664 * im), im, 0.5), Float64(im * im), 1.0)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(0.5 * im), $MachinePrecision] * im + N[(N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $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[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 19.4%

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

   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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 98.8

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

   5. Applied rewrites 98.8%

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

   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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) + {re}^{2} \cdot \left(\frac{1}{120} + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right) - \frac{1}{6}\right)\right)\right)} \]

   8. Applied rewrites 5.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00034722222222222224, im \cdot im, 0.004166666666666667\right), im \cdot im, 0.008333333333333333\right) \cdot re, re, \mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right), re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 5.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00034722222222222224, im \cdot im, 0.004166666666666667\right), im \cdot im, 0.008333333333333333\right) \cdot re, re, \mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right), re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} \cdot im, im, \frac{1}{2}\right), im \cdot im, 1\right)\right) \cdot re \]
   12. Step-by-step derivation

   13. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right), re \cdot re, 1\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right), re \cdot re, 1\right)\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (* 0.5 im)
       im
       (fma
        (fma (* im im) -0.08333333333333333 -0.16666666666666666)
        (* re re)
        1.0))
      re)
     (if (<= t_0 1.0)
       (* 1.0 (sin re))
       (*
        (fma
         (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
         (* re re)
         (fma (fma (* 0.041666666666666664 im) im 0.5) (* im im) 1.0))
        re)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((0.5 * im), im, fma(fma((im * im), -0.08333333333333333, -0.16666666666666666), (re * re), 1.0)) * re;
	} else if (t_0 <= 1.0) {
		tmp = 1.0 * sin(re);
	} else {
		tmp = fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), fma(fma((0.041666666666666664 * im), im, 0.5), (im * im), 1.0)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(0.5 * im), im, fma(fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666), Float64(re * re), 1.0)) * re);
	elseif (t_0 <= 1.0)
		tmp = Float64(1.0 * sin(re));
	else
		tmp = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), fma(fma(Float64(0.041666666666666664 * im), im, 0.5), Float64(im * im), 1.0)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(0.5 * im), $MachinePrecision] * im + N[(N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(1.0 * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 19.4%

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

   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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto 1 \cdot \sin \color{blue}{re} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.1%

      \[\leadsto 1 \cdot \sin \color{blue}{re} \]

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) + {re}^{2} \cdot \left(\frac{1}{120} + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right) - \frac{1}{6}\right)\right)\right)} \]

   8. Applied rewrites 5.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00034722222222222224, im \cdot im, 0.004166666666666667\right), im \cdot im, 0.008333333333333333\right) \cdot re, re, \mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right), re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 5.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00034722222222222224, im \cdot im, 0.004166666666666667\right), im \cdot im, 0.008333333333333333\right) \cdot re, re, \mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right), re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} \cdot im, im, \frac{1}{2}\right), im \cdot im, 1\right)\right) \cdot re \]
   12. Step-by-step derivation

   13. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right), re \cdot re, 1\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{4} \cdot re\right) \cdot \mathsf{fma}\left(0.00034722222222222224 \cdot \left(re \cdot re\right) - 0.006944444444444444, re \cdot re, 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 1.0)
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) (sin re))
   (*
    (* (pow im 4.0) re)
    (fma
     (- (* 0.00034722222222222224 (* re re)) 0.006944444444444444)
     (* re re)
     0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 1.0) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else {
		tmp = (pow(im, 4.0) * re) * fma(((0.00034722222222222224 * (re * re)) - 0.006944444444444444), (re * re), 0.041666666666666664);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 1.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	else
		tmp = Float64(Float64((im ^ 4.0) * re) * fma(Float64(Float64(0.00034722222222222224 * Float64(re * re)) - 0.006944444444444444), Float64(re * re), 0.041666666666666664));
	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], 1.0], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[im, 4.0], $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(0.00034722222222222224 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.006944444444444444), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.041666666666666664), $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))) < 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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.1

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

   7. Applied rewrites 91.1%

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

   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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) + {re}^{2} \cdot \left(\frac{1}{120} + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right) - \frac{1}{6}\right)\right)\right)} \]

   8. Applied rewrites 5.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00034722222222222224, im \cdot im, 0.004166666666666667\right), im \cdot im, 0.008333333333333333\right) \cdot re, re, \mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right), re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

      \[\leadsto {im}^{4} \cdot \left(re \cdot \color{blue}{\left(\frac{1}{24} + {re}^{2} \cdot \left(\frac{1}{2880} \cdot {re}^{2} - \frac{1}{144}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 63.4%

       \[\leadsto \left({im}^{4} \cdot re\right) \cdot \mathsf{fma}\left(0.00034722222222222224 \cdot \left(re \cdot re\right) - 0.006944444444444444, \color{blue}{re \cdot re}, 0.041666666666666664\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{4} \cdot re\right) \cdot \mathsf{fma}\left(0.00034722222222222224 \cdot \left(re \cdot re\right) - 0.006944444444444444, re \cdot re, 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 1.0)
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) (sin re))
   (*
    (fma
     (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
     (* re re)
     (fma (fma (* 0.041666666666666664 im) im 0.5) (* im im) 1.0))
    re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 1.0) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else {
		tmp = fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), fma(fma((0.041666666666666664 * im), im, 0.5), (im * im), 1.0)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 1.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	else
		tmp = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), fma(fma(Float64(0.041666666666666664 * im), im, 0.5), Float64(im * im), 1.0)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.1

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

   7. Applied rewrites 91.1%

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

   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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) + {re}^{2} \cdot \left(\frac{1}{120} + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right) - \frac{1}{6}\right)\right)\right)} \]

   8. Applied rewrites 5.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00034722222222222224, im \cdot im, 0.004166666666666667\right), im \cdot im, 0.008333333333333333\right) \cdot re, re, \mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right), re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 5.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00034722222222222224, im \cdot im, 0.004166666666666667\right), im \cdot im, 0.008333333333333333\right) \cdot re, re, \mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right), re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} \cdot im, im, \frac{1}{2}\right), im \cdot im, 1\right)\right) \cdot re \]
   12. Step-by-step derivation

   13. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right), re \cdot re, 1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.02)
   (*
    (fma
     (* 0.5 im)
     im
     (fma
      (fma (* im im) -0.08333333333333333 -0.16666666666666666)
      (* re re)
      1.0))
    re)
   (*
    (fma
     (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
     (* re re)
     (fma (fma (* 0.041666666666666664 im) im 0.5) (* im im) 1.0))
    re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = fma((0.5 * im), im, fma(fma((im * im), -0.08333333333333333, -0.16666666666666666), (re * re), 1.0)) * re;
	} else {
		tmp = fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), fma(fma((0.041666666666666664 * im), im, 0.5), (im * im), 1.0)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.02)
		tmp = Float64(fma(Float64(0.5 * im), im, fma(fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666), Float64(re * re), 1.0)) * re);
	else
		tmp = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), fma(fma(Float64(0.041666666666666664 * im), im, 0.5), Float64(im * im), 1.0)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(0.5 * im), $MachinePrecision] * im + N[(N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0200000000000000004

   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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.3%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 15.1%

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

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) + {re}^{2} \cdot \left(\frac{1}{120} + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right) - \frac{1}{6}\right)\right)\right)} \]

   8. Applied rewrites 46.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00034722222222222224, im \cdot im, 0.004166666666666667\right), im \cdot im, 0.008333333333333333\right) \cdot re, re, \mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right), re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00034722222222222224, im \cdot im, 0.004166666666666667\right), im \cdot im, 0.008333333333333333\right) \cdot re, re, \mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right), re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} \cdot im, im, \frac{1}{2}\right), im \cdot im, 1\right)\right) \cdot re \]
   12. Step-by-step derivation

   13. Applied rewrites 66.2%

       \[\leadsto \mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right), re \cdot re, 1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right), re \cdot re, 1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.02)
   (*
    (fma
     (* 0.5 im)
     im
     (fma
      (fma (* im im) -0.08333333333333333 -0.16666666666666666)
      (* re re)
      1.0))
    re)
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = fma((0.5 * im), im, fma(fma((im * im), -0.08333333333333333, -0.16666666666666666), (re * re), 1.0)) * re;
	} else {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.02)
		tmp = Float64(fma(Float64(0.5 * im), im, fma(fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666), Float64(re * re), 1.0)) * re);
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(0.5 * im), $MachinePrecision] * im + N[(N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0200000000000000004

   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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.3%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 15.1%

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

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.4%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right), re \cdot re, 1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.95:\\ \;\;\;\;\frac{re}{im} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot 0.5\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.95)
   (* (/ re im) im)
   (* (* (* im im) 0.5) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.95) {
		tmp = (re / im) * im;
	} else {
		tmp = ((im * im) * 0.5) * re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * sin(re)) * (exp(-im) + exp(im))) <= 0.95d0) then
        tmp = (re / im) * im
    else
        tmp = ((im * im) * 0.5d0) * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp(-im) + Math.exp(im))) <= 0.95) {
		tmp = (re / im) * im;
	} else {
		tmp = ((im * im) * 0.5) * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp(-im) + math.exp(im))) <= 0.95:
		tmp = (re / im) * im
	else:
		tmp = ((im * im) * 0.5) * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.95)
		tmp = Float64(Float64(re / im) * im);
	else
		tmp = Float64(Float64(Float64(im * im) * 0.5) * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.95)
		tmp = (re / im) * im;
	else
		tmp = ((im * im) * 0.5) * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.95], N[(N[(re / im), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.5), $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))) < 0.94999999999999996

   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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 32.2%

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

   12. Taylor expanded in im around 0

       \[\leadsto \frac{re}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 41.7%

       \[\leadsto \frac{re}{im} \cdot im \]

   if 0.94999999999999996 < (*.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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 37.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 37.5%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.95:\\ \;\;\;\;\frac{re}{im} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot 0.5\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (sin re)
  (+
   1.0
   (*
    (* im im)
    (fma
     (fma (* im im) 0.001388888888888889 0.041666666666666664)
     (* im im)
     0.5)))))

C

double code(double re, double im) {
	return sin(re) * (1.0 + ((im * im) * fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * im), 0.5)));
}

Julia

function code(re, im)
	return Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * im), 0.5))))
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision]), $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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub0-neg N/A

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

   11. cosh-undef N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. cosh-0 N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

   15. *-commutative N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   16. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   17. cosh-0 N/A

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

   18. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   19. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   20. exp-0 N/A

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

   21. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-lft-identity 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

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in im around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 91.9

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

9. Applied rewrites 91.9%

   \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 91.9%

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

12. Final simplification 91.9%

    \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right)\right) \]
13. Add Preprocessing

Alternative 10: 91.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (fma
   (fma
    (fma 0.001388888888888889 (* im im) 0.041666666666666664)
    (* im im)
    0.5)
   (* im im)
   1.0)
  (sin re)))

C

double code(double re, double im) {
	return fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * sin(re);
}

Julia

function code(re, im)
	return Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re))
end

Wolfram

code[re_, im_] := N[(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] * N[Sin[re], $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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub0-neg N/A

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

   11. cosh-undef N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. cosh-0 N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

   15. *-commutative N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   16. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   17. cosh-0 N/A

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

   18. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   19. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   20. exp-0 N/A

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

   21. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in im around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 91.9

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

7. Applied rewrites 91.9%

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

8. Final simplification 91.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re \]
9. Add Preprocessing

Alternative 11: 91.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (sin re)
  (fma (fma (* 0.001388888888888889 (* im im)) (* im im) 0.5) (* im im) 1.0)))

C

double code(double re, double im) {
	return sin(re) * fma(fma((0.001388888888888889 * (im * im)), (im * im), 0.5), (im * im), 1.0);
}

Julia

function code(re, im)
	return Float64(sin(re) * fma(fma(Float64(0.001388888888888889 * Float64(im * im)), Float64(im * im), 0.5), Float64(im * im), 1.0))
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub0-neg N/A

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

   11. cosh-undef N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. cosh-0 N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

   15. *-commutative N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   16. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   17. cosh-0 N/A

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

   18. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   19. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   20. exp-0 N/A

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

   21. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-lft-identity 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

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in im around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 91.9

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

9. Applied rewrites 91.9%

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

10. Taylor expanded in im around inf

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

12. Applied rewrites 91.5%

    \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \]
13. Add Preprocessing

Alternative 12: 55.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) re))

C

double code(double re, double im) {
	return fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * re;
}

Julia

function code(re, im)
	return Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * re)
end

Wolfram

code[re_, im_] := N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 84.4

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

5. Applied rewrites 84.4%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 59.9%

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

Alternative 13: 48.7% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(re, 1, re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma re 1.0 (* re (* (* im im) 0.5))))

C

double code(double re, double im) {
	return fma(re, 1.0, (re * ((im * im) * 0.5)));
}

Julia

function code(re, im)
	return fma(re, 1.0, Float64(re * Float64(Float64(im * im) * 0.5)))
end

Wolfram

code[re_, im_] := N[(re * 1.0 + N[(re * N[(N[(im * im), $MachinePrecision] * 0.5), $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. 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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 74.0

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

5. Applied rewrites 74.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 51.2%

   \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
9. Step-by-step derivation

10. Applied rewrites 51.2%

    \[\leadsto \mathsf{fma}\left(re, 1, re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)\right) \]
11. Add Preprocessing

Alternative 14: 48.7% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (fma (* im im) 0.5 1.0) re))

C

double code(double re, double im) {
	return fma((im * im), 0.5, 1.0) * re;
}

Julia

function code(re, im)
	return Float64(fma(Float64(im * im), 0.5, 1.0) * re)
end

Wolfram

code[re_, im_] := N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 74.0

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

5. Applied rewrites 74.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 51.2%

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

Alternative 15: 25.1% accurate, 19.8× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (* (* im im) 0.5) re))

C

double code(double re, double im) {
	return ((im * im) * 0.5) * re;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double re, double im) {
	return ((im * im) * 0.5) * re;
}

Python

def code(re, im):
	return ((im * im) * 0.5) * re

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 74.0

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

5. Applied rewrites 74.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 51.2%

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

9. Taylor expanded in im around inf

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

11. Applied rewrites 23.4%

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

12. Final simplification 23.4%

    \[\leadsto \left(\left(im \cdot im\right) \cdot 0.5\right) \cdot re \]
13. Add Preprocessing

Alternative 16: 19.2% accurate, 19.8× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (* (* im re) 0.5) im))

C

double code(double re, double im) {
	return ((im * re) * 0.5) * im;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double re, double im) {
	return ((im * re) * 0.5) * im;
}

Python

def code(re, im):
	return ((im * re) * 0.5) * im

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 74.0

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

5. Applied rewrites 74.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 51.2%

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

9. Taylor expanded in im around inf

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

11. Applied rewrites 31.4%

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

12. Taylor expanded in im around inf

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

14. Applied rewrites 17.6%

    \[\leadsto \left(\left(im \cdot re\right) \cdot 0.5\right) \cdot im \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024350 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))