math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 15

Speedup: 1.2×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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. lift-exp.f64 N/A

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

   8. distribute-rgt-in N/A

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

   9. sub0-neg N/A

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

   10. distribute-rgt-in N/A

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

   11. +-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

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: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.08333333333333333\right) \cdot \left(1 + e^{im}\right)\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, re\right) \cdot 0.5\\ \end{array} \end{array} \]

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) -0.08333333333333333) (+ 1.0 (exp im)))
     (if (<= t_0 10.0)
       (* (* (sin re) (fma im im 2.0)) 0.5)
       (* (fma (exp im) re 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) * -0.08333333333333333) * (1.0 + exp(im));
	} else if (t_0 <= 10.0) {
		tmp = (sin(re) * fma(im, im, 2.0)) * 0.5;
	} else {
		tmp = fma(exp(im), re, 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) * -0.08333333333333333) * Float64(1.0 + exp(im)));
	elseif (t_0 <= 10.0)
		tmp = Float64(Float64(sin(re) * fma(im, im, 2.0)) * 0.5);
	else
		tmp = Float64(fma(exp(im), re, 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] * -0.08333333333333333), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 10.0], N[(N[(N[Sin[re], $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[im], $MachinePrecision] * re + 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 im around 0

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.4

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

   7. Applied rewrites 46.4%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 20.7

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

   10. Applied rewrites 20.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 10

   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. lift-exp.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. sub0-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

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

      1. cosh-undef-rev N/A

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

      2. rec-exp N/A

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

      3. flip-+ N/A

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

      4. exp-sum N/A

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

      5. rec-exp N/A

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

      6. rec-exp N/A

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

      7. exp-lft-sqr N/A

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

      8. rec-exp N/A

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

      9. sinh-undef-rev N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 99.0

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

   6. Applied rewrites 99.0%

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

   if 10 < (*.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 99.9%

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

   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 72.8

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

   4. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. *-commutative N/A

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

      5. cosh-undef-rev N/A

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

      6. rec-exp N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-neg.f64 72.8

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

   6. Applied rewrites 72.8%

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

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 37.1%

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

Alternative 3: 64.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.08333333333333333\right) \cdot \left(1 + e^{im}\right)\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, re\right) \cdot 0.5\\ \end{array} \end{array} \]

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) -0.08333333333333333) (+ 1.0 (exp im)))
     (if (<= t_0 10.0) (sin re) (* (fma (exp im) re 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) * -0.08333333333333333) * (1.0 + exp(im));
	} else if (t_0 <= 10.0) {
		tmp = sin(re);
	} else {
		tmp = fma(exp(im), re, 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) * -0.08333333333333333) * Float64(1.0 + exp(im)));
	elseif (t_0 <= 10.0)
		tmp = sin(re);
	else
		tmp = Float64(fma(exp(im), re, 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] * -0.08333333333333333), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 10.0], N[Sin[re], $MachinePrecision], N[(N[(N[Exp[im], $MachinePrecision] * re + 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 im around 0

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.4

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

   7. Applied rewrites 46.4%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 20.7

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

   10. Applied rewrites 20.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 10

   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

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

      1. lift-sin.f64 98.4

         \[\leadsto \sin re \]

   4. Applied rewrites 98.4%

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

   if 10 < (*.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 99.9%

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

   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 72.8

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

   4. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. *-commutative N/A

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

      5. cosh-undef-rev N/A

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

      6. rec-exp N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-neg.f64 72.8

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

   6. Applied rewrites 72.8%

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

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 37.1%

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

Alternative 4: 63.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(1 + e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (+ 1.0 (exp im)) (* (sin re) 0.5)))

C

double code(double re, double im) {
	return (1.0 + exp(im)) * (sin(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 = (1.0d0 + exp(im)) * (sin(re) * 0.5d0)
end function

Java

public static double code(double re, double im) {
	return (1.0 + Math.exp(im)) * (Math.sin(re) * 0.5);
}

Python

def code(re, im):
	return (1.0 + math.exp(im)) * (math.sin(re) * 0.5)

Julia

function code(re, im)
	return Float64(Float64(1.0 + exp(im)) * Float64(sin(re) * 0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = (1.0 + exp(im)) * (sin(re) * 0.5);
end

Wolfram

code[re_, im_] := N[(N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $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 im around 0

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

4. Applied rewrites 74.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} + e^{im}\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 74.0

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

6. Applied rewrites 74.0%

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

Alternative 5: 62.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \cosh im\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0004:\\ \;\;\;\;\left(t\_0 \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 2.0 (cosh im))))
   (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.0004)
     (* (* 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))) <= 0.0004) {
		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))) <= 0.0004)
		tmp = Float64(Float64(t_0 * fma(Float64(re * 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], 0.0004], N[(N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 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))) < 4.00000000000000019e-4

   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 70.4%

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

   if 4.00000000000000019e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   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 50.2

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

   4. Applied rewrites 50.2%

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

Alternative 6: 56.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.08333333333333333\right) \cdot \left(1 + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.005)
   (* (* (* (* re re) re) -0.08333333333333333) (+ 1.0 (exp im)))
   (* (* (* 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))) <= -0.005) {
		tmp = (((re * re) * re) * -0.08333333333333333) * (1.0 + exp(im));
	} else {
		tmp = ((2.0 * cosh(im)) * 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)) * (exp((0.0d0 - im)) + exp(im))) <= (-0.005d0)) then
        tmp = (((re * re) * re) * (-0.08333333333333333d0)) * (1.0d0 + exp(im))
    else
        tmp = ((2.0d0 * cosh(im)) * 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)) * (Math.exp((0.0 - im)) + Math.exp(im))) <= -0.005) {
		tmp = (((re * re) * re) * -0.08333333333333333) * (1.0 + Math.exp(im));
	} else {
		tmp = ((2.0 * Math.cosh(im)) * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))) <= -0.005:
		tmp = (((re * re) * re) * -0.08333333333333333) * (1.0 + math.exp(im))
	else:
		tmp = ((2.0 * math.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))) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(re * re) * re) * -0.08333333333333333) * Float64(1.0 + exp(im)));
	else
		tmp = Float64(Float64(Float64(2.0 * cosh(im)) * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= -0.005)
		tmp = (((re * re) * re) * -0.08333333333333333) * (1.0 + exp(im));
	else
		tmp = ((2.0 * cosh(im)) * re) * 0.5;
	end
	tmp_2 = 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], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $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))) < -0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.5

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

   7. Applied rewrites 31.5%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.4

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

   10. Applied rewrites 14.4%

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

   if -0.0050000000000000001 < (*.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 99.9%

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

   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 69.7

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

   4. Applied rewrites 69.7%

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

Alternative 7: 48.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.005)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (fma im im 2.0))
   (* (* (* 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))) <= -0.005) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0);
	} 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))) <= -0.005)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * 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_] := 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], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $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 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))) < -0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.5

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

   7. Applied rewrites 31.5%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 34.2

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

   10. Applied rewrites 34.2%

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

   if -0.0050000000000000001 < (*.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 99.9%

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

   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 69.7

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

   4. Applied rewrites 69.7%

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

Alternative 8: 47.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0004:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.0004)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (fma im im 2.0))
   (* (fma (exp im) re re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.0004) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = fma(exp(im), re, 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))) <= 0.0004)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(exp(im), re, 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], 0.0004], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[im], $MachinePrecision] * re + 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))) < 4.00000000000000019e-4

   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

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

   4. Applied rewrites 78.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.6

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

   7. Applied rewrites 57.6%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 59.8

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

   10. Applied rewrites 59.8%

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

   if 4.00000000000000019e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   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 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. *-commutative N/A

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

      5. cosh-undef-rev N/A

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

      6. rec-exp N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-neg.f64 50.2

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

   6. Applied rewrites 50.2%

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

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 26.2%

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

Alternative 9: 44.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.0004)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (fma (exp im) re re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.0004) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma(exp(im), re, 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))) <= 0.0004)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(fma(exp(im), re, 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], 0.0004], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[Exp[im], $MachinePrecision] * re + 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))) < 4.00000000000000019e-4

   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

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

      1. lift-sin.f64 60.3

         \[\leadsto \sin re \]

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 46.5

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

   7. Applied rewrites 46.5%

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

   if 4.00000000000000019e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   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 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. *-commutative N/A

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

      5. cosh-undef-rev N/A

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

      6. rec-exp N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-neg.f64 50.2

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

   6. Applied rewrites 50.2%

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

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 26.2%

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

Alternative 10: 40.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot re\right) \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.0004)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (* (* (* im im) (* im im)) re) 0.041666666666666664)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.0004) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = (((im * im) * (im * im)) * re) * 0.041666666666666664;
	}
	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))) <= 0.0004)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(Float64(Float64(im * im) * Float64(im * im)) * re) * 0.041666666666666664);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0004], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * 0.041666666666666664), $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))) < 4.00000000000000019e-4

   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

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

      1. lift-sin.f64 60.3

         \[\leadsto \sin re \]

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 46.5

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

   7. Applied rewrites 46.5%

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

   if 4.00000000000000019e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   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 50.2

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 36.4

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto \left({im}^{4} \cdot re\right) \cdot \frac{1}{24} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({im}^{4} \cdot re\right) \cdot \frac{1}{24} \]

      3. lower-*.f64 N/A

         \[\leadsto \left({im}^{4} \cdot re\right) \cdot \frac{1}{24} \]

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 40.6

          \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot re\right) \cdot 0.041666666666666664 \]

   10. Applied rewrites 40.6%

       \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot re\right) \cdot 0.041666666666666664 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 38.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.005)
   (* (* (* 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)) * (exp((0.0 - im)) + exp(im))) <= -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(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.005)
		tmp = Float64(Float64(Float64(re * re) * 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[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $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))) < -0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lift-sin.f64 35.4

         \[\leadsto \sin re \]

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 12.5

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

   7. Applied rewrites 12.5%

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

   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 12.1

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

   10. Applied rewrites 12.1%

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

   if -0.0050000000000000001 < (*.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 99.9%

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

   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 69.7

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

   4. Applied rewrites 69.7%

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

   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. unpow2 N/A

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

      6. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

Alternative 12: 33.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + im, re, re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.0004)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (fma (+ 1.0 im) re re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.0004) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma((1.0 + im), re, 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))) <= 0.0004)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(fma(Float64(1.0 + im), re, 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], 0.0004], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(1.0 + im), $MachinePrecision] * re + 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))) < 4.00000000000000019e-4

   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

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

      1. lift-sin.f64 60.3

         \[\leadsto \sin re \]

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 46.5

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

   7. Applied rewrites 46.5%

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

   if 4.00000000000000019e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   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 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. *-commutative N/A

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

      5. cosh-undef-rev N/A

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

      6. rec-exp N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-neg.f64 50.2

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

   6. Applied rewrites 50.2%

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

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 26.2%

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

   10. Taylor expanded in im around 0

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

       1. lower-+.f64 11.0

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

   12. Applied rewrites 11.0%

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

Alternative 13: 32.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.002:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + im, re, re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.002)
   (* (* (* re re) re) -0.16666666666666666)
   (* (fma (+ 1.0 im) re re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.002) {
		tmp = ((re * re) * re) * -0.16666666666666666;
	} else {
		tmp = fma((1.0 + im), re, re) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.002)
		tmp = Float64(Float64(Float64(re * re) * re) * -0.16666666666666666);
	else
		tmp = Float64(fma(Float64(1.0 + im), re, re) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(N[(N[(1.0 + im), $MachinePrecision] * re + re), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -2e-3

   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

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

      1. lift-sin.f64 52.2

         \[\leadsto \sin re \]

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 17.7

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

   7. Applied rewrites 17.7%

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

   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 17.5

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

   10. Applied rewrites 17.5%

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

   if -2e-3 < (*.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 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. *-commutative N/A

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

      5. cosh-undef-rev N/A

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

      6. rec-exp N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-neg.f64 74.9

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 53.6%

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

   10. Taylor expanded in im around 0

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

       1. lower-+.f64 37.7

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

   12. Applied rewrites 37.7%

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

Alternative 14: 29.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.002:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.002)
   (* (* (* re re) re) -0.16666666666666666)
   re))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.002) {
		tmp = ((re * re) * re) * -0.16666666666666666;
	} else {
		tmp = re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.002d0)) then
        tmp = ((re * re) * re) * (-0.16666666666666666d0)
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.002) {
		tmp = ((re * re) * re) * -0.16666666666666666;
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.002:
		tmp = ((re * re) * re) * -0.16666666666666666
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.002)
		tmp = Float64(Float64(Float64(re * re) * re) * -0.16666666666666666);
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.002)
		tmp = ((re * re) * re) * -0.16666666666666666;
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -2e-3

   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

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

      1. lift-sin.f64 52.2

         \[\leadsto \sin re \]

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 17.7

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

   7. Applied rewrites 17.7%

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

   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 17.5

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

   10. Applied rewrites 17.5%

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

   if -2e-3 < (*.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 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 34.0%

      \[\leadsto re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 26.3% accurate, 64.3× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

2. 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 62.7

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

4. Applied rewrites 62.7%

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

5. Taylor expanded in im around 0

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

7. Applied rewrites 26.3%

   \[\leadsto re \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))