math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

Alternatives: 11

Speedup: 1.2×

• 49.1% 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, 1.2× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (sin(re) * (2.0 * cosh(im))) * 0.5;
}

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) * (2.0d0 * cosh(im))) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   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. lift-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. sub0-neg N/A

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

   8. mul-1-neg N/A

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

   9. lift-exp.f64 N/A

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

   10. +-commutative N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 2: 74.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[Cosh[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * -0.08333333333333333), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * 0.5), $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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left({re}^{3} \cdot \left(e^{im} + \frac{1}{e^{im}}\right)\right) \cdot \frac{-1}{12} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot \left(e^{im} + \frac{1}{e^{im}}\right)\right) \cdot \frac{-1}{12} \]

      3. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot \left(e^{im} + \frac{1}{e^{im}}\right)\right) \cdot \frac{-1}{12} \]

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. rec-exp N/A

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

      10. cosh-undef-rev N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cosh.f64 14.3

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

   7. Applied rewrites 14.3%

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

   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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.4

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

   4. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 75.4

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

   6. Applied rewrites 75.4%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 62.8

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

   4. Applied rewrites 62.8%

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

Alternative 3: 74.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (* (* re re) re) (* (cosh im) 2.0)) -0.08333333333333333)
     (if (<= t_0 1.0)
       (* (* (sin re) 0.5) 2.0)
       (* (* (* 2.0 (cosh im)) re) 0.5)))))

C

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

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (((re * re) * re) * (Math.cosh(im) * 2.0)) * -0.08333333333333333;
	} else if (t_0 <= 1.0) {
		tmp = (Math.sin(re) * 0.5) * 2.0;
	} else {
		tmp = ((2.0 * Math.cosh(im)) * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (((re * re) * re) * (math.cosh(im) * 2.0)) * -0.08333333333333333
	elif t_0 <= 1.0:
		tmp = (math.sin(re) * 0.5) * 2.0
	else:
		tmp = ((2.0 * math.cosh(im)) * re) * 0.5
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (((re * re) * re) * (cosh(im) * 2.0)) * -0.08333333333333333;
	elseif (t_0 <= 1.0)
		tmp = (sin(re) * 0.5) * 2.0;
	else
		tmp = ((2.0 * cosh(im)) * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[Cosh[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * -0.08333333333333333), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * 0.5), $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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left({re}^{3} \cdot \left(e^{im} + \frac{1}{e^{im}}\right)\right) \cdot \frac{-1}{12} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot \left(e^{im} + \frac{1}{e^{im}}\right)\right) \cdot \frac{-1}{12} \]

      3. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot \left(e^{im} + \frac{1}{e^{im}}\right)\right) \cdot \frac{-1}{12} \]

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. rec-exp N/A

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

      10. cosh-undef-rev N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cosh.f64 14.3

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

   7. Applied rewrites 14.3%

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

   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. Taylor expanded in im around 0

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

   4. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 51.4

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

   6. Applied rewrites 51.4%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 62.8

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

   4. Applied rewrites 62.8%

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

Alternative 4: 62.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \cosh im\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.02:\\ \;\;\;\;\left(t\_0 \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 2.0 (cosh im))))
   (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.02)
     (* (* t_0 (fma re (* re -0.08333333333333333) 0.5)) re)
     (* (* t_0 re) 0.5))))

C

double code(double re, double im) {
	double t_0 = 2.0 * cosh(im);
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.02) {
		tmp = (t_0 * fma(re, (re * -0.08333333333333333), 0.5)) * re;
	} else {
		tmp = (t_0 * re) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(2.0 * cosh(im))
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.02)
		tmp = Float64(Float64(t_0 * fma(re, Float64(re * -0.08333333333333333), 0.5)) * re);
	else
		tmp = Float64(Float64(t_0 * re) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(t$95$0 * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(t$95$0 * re), $MachinePrecision] * 0.5), $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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 62.7

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

   6. Applied rewrites 62.7%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 62.8

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

   4. Applied rewrites 62.8%

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

Alternative 5: 55.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.02)
   (* (* (fma (* re re) -0.08333333333333333 0.5) (fma im im 2.0)) re)
   (* (* (* 2.0 (cosh im)) re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.02) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * fma(im, im, 2.0)) * re;
	} else {
		tmp = ((2.0 * cosh(im)) * re) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.02)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * fma(im, im, 2.0)) * re);
	else
		tmp = Float64(Float64(Float64(2.0 * cosh(im)) * re) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * 0.5), $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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in im around 0

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

      1. distribute-rgt-out N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lift-fma.f64 49.4

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

   7. Applied rewrites 49.4%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 62.8

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

   4. Applied rewrites 62.8%

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

Alternative 6: 48.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.02)
   (* (* (fma (* re re) -0.08333333333333333 0.5) (fma im im 2.0)) re)
   (fma (* (* im im) re) 0.5 re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.02) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * fma(im, im, 2.0)) * re;
	} else {
		tmp = fma(((im * im) * re), 0.5, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.02)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * fma(im, im, 2.0)) * re);
	else
		tmp = fma(Float64(Float64(im * im) * re), 0.5, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * 0.5 + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in im around 0

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

      1. distribute-rgt-out N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lift-fma.f64 49.4

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

   7. Applied rewrites 49.4%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 62.8

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

Alternative 7: 47.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot \left(im \cdot im\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.01)
   (* (* (fma (* re re) -0.08333333333333333 0.5) (* im im)) re)
   (fma (* (* im im) re) 0.5 re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * (im * im)) * re;
	} else {
		tmp = fma(((im * im) * re), 0.5, re);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * 0.5 + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in im around 0

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

      1. distribute-rgt-out N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lift-fma.f64 49.4

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 25.6

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

   10. Applied rewrites 25.6%

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

   if -0.0100000000000000002 < (*.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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 62.8

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

Alternative 8: 46.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.02:\\ \;\;\;\;\left(re + re\right) \cdot \mathsf{fma}\left(-0.08333333333333333 \cdot re, re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.02)
   (* (+ re re) (fma (* -0.08333333333333333 re) re 0.5))
   (fma (* (* im im) re) 0.5 re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.02) {
		tmp = (re + re) * fma((-0.08333333333333333 * re), re, 0.5);
	} else {
		tmp = fma(((im * im) * re), 0.5, re);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(re + re), $MachinePrecision] * N[(N[(-0.08333333333333333 * re), $MachinePrecision] * re + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * 0.5 + 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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 34.8

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

   7. Applied rewrites 34.8%

      \[\leadsto \left(re + re\right) \cdot \color{blue}{\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 34.8

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

   9. Applied rewrites 34.8%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 62.8

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

Alternative 9: 40.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (* re re) (* re -0.16666666666666666))
   (fma (* (* im im) re) 0.5 re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = (re * re) * (re * -0.16666666666666666);
	} else {
		tmp = fma(((im * im) * re), 0.5, re);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(re * re), $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * 0.5 + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 34.8

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto {re}^{3} \cdot \frac{-1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {re}^{3} \cdot \frac{-1}{6} \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 10.8

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

   10. Applied rewrites 10.8%

       \[\leadsto \left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 10.7

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

   12. Applied rewrites 10.7%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 62.8

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

Alternative 10: 30.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (* re re) (* re -0.16666666666666666))
   (* (+ re re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = (re * re) * (re * -0.16666666666666666);
	} else {
		tmp = (re + re) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.005d0)) then
        tmp = (re * re) * (re * (-0.16666666666666666d0))
    else
        tmp = (re + re) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.005) {
		tmp = (re * re) * (re * -0.16666666666666666);
	} else {
		tmp = (re + re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.005:
		tmp = (re * re) * (re * -0.16666666666666666)
	else:
		tmp = (re + re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(Float64(re * re) * Float64(re * -0.16666666666666666));
	else
		tmp = Float64(Float64(re + re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.005)
		tmp = (re * re) * (re * -0.16666666666666666);
	else
		tmp = (re + re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(re * re), $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(re + re), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 34.8

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto {re}^{3} \cdot \frac{-1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {re}^{3} \cdot \frac{-1}{6} \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 10.8

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

   10. Applied rewrites 10.8%

       \[\leadsto \left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 10.7

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

   12. Applied rewrites 10.7%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 62.8

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in im around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 27.1

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

   7. Applied rewrites 27.1%

      \[\leadsto \left(re + re\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 27.1% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. cosh-undef N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cosh.f64 62.8

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in im around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f64 27.1

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

7. Applied rewrites 27.1%

   \[\leadsto \left(re + re\right) \cdot 0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))