math.sin on complex, real part

Percentage Accurate: 99.9% → 99.9%

Time: 4.2s

Alternatives: 15

Speedup: 1.2×

• 48.4% 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: 99.9% 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: 99.9% 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 99.9%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
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 99.9%

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

Alternative 2: 75.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.08333333333333333\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + e^{im}\right)\\ \end{array} \end{array} \]

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) (fma im im 2.0))
     (if (<= t_0 50000.0)
       (* (* (sin re) 0.5) (fma im im 2.0))
       (* (* 0.5 re) (+ 1.0 (exp im)))))))

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) * fma(im, im, 2.0);
	} else if (t_0 <= 50000.0) {
		tmp = (sin(re) * 0.5) * fma(im, im, 2.0);
	} else {
		tmp = (0.5 * re) * (1.0 + exp(im));
	}
	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) * fma(im, im, 2.0));
	elseif (t_0 <= 50000.0)
		tmp = Float64(Float64(sin(re) * 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[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[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 50000.0], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 48.4

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{12}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{12}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 23.4

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

   10. Applied rewrites 23.4%

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

   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))) < 5e4

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if 5e4 < (*.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 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 50.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 38.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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 75.3%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} + e^{im}\right) \]
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. lift-+.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 75.3

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

6. Applied rewrites 75.3%

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

Alternative 4: 65.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.08333333333333333\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + e^{im}\right)\\ \end{array} \end{array} \]

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) (fma im im 2.0))
     (if (<= t_0 50000.0) (sin re) (* (* 0.5 re) (+ 1.0 (exp im)))))))

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) * fma(im, im, 2.0);
	} else if (t_0 <= 50000.0) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * (1.0 + exp(im));
	}
	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) * fma(im, im, 2.0));
	elseif (t_0 <= 50000.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[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[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 50000.0], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 48.4

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{12}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{12}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 23.4

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

   10. Applied rewrites 23.4%

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

   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))) < 5e4

   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 98.6

         \[\leadsto \sin re \]

   4. Applied rewrites 98.6%

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

   if 5e4 < (*.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 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 50.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 38.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 49.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0 - im} + e^{im}\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot t\_0 \leq -0.02:\\ \;\;\;\;\left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (exp (- 0.0 im)) (exp im))))
   (if (<= (* (* 0.5 (sin re)) t_0) -0.02)
     (* (* 0.5 (* (* (* re re) re) -0.16666666666666666)) (* im im))
     (* (* 0.5 re) t_0))))

C

double code(double re, double im) {
	double t_0 = exp((0.0 - im)) + exp(im);
	double tmp;
	if (((0.5 * sin(re)) * t_0) <= -0.02) {
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * im);
	} else {
		tmp = (0.5 * re) * t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = exp((0.0d0 - im)) + exp(im)
    if (((0.5d0 * sin(re)) * t_0) <= (-0.02d0)) then
        tmp = (0.5d0 * (((re * re) * re) * (-0.16666666666666666d0))) * (im * im)
    else
        tmp = (0.5d0 * re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp((0.0 - im)) + Math.exp(im);
	double tmp;
	if (((0.5 * Math.sin(re)) * t_0) <= -0.02) {
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * im);
	} else {
		tmp = (0.5 * re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp((0.0 - im)) + math.exp(im)
	tmp = 0
	if ((0.5 * math.sin(re)) * t_0) <= -0.02:
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * im)
	else:
		tmp = (0.5 * re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(0.0 - im)) + exp(im))
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * t_0) <= -0.02)
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64(re * re) * re) * -0.16666666666666666)) * Float64(im * im));
	else
		tmp = Float64(Float64(0.5 * re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp((0.0 - im)) + exp(im);
	tmp = 0.0;
	if (((0.5 * sin(re)) * t_0) <= -0.02)
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * im);
	else
		tmp = (0.5 * re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], -0.02], N[(N[(0.5 * N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in re around inf

       \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{6} \cdot \color{blue}{{re}^{3}}\right)\right) \cdot \left(im \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{3} \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{3} \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot im\right) \]

       3. unpow3 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 15.8

          \[\leadsto \left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot im\right) \]

   13. Applied rewrites 15.8%

       \[\leadsto \left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right) \cdot \left(im \cdot im\right) \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 70.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 49.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.02)
   (* (* 0.5 (* (* (* re re) re) -0.16666666666666666)) (* im 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))) <= -0.02) {
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * 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))) <= (-0.02d0)) then
        tmp = (0.5d0 * (((re * re) * re) * (-0.16666666666666666d0))) * (im * 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))) <= -0.02) {
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * 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))) <= -0.02:
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * 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))) <= -0.02)
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64(re * re) * re) * -0.16666666666666666)) * Float64(im * 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))) <= -0.02)
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * 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], -0.02], N[(N[(0.5 * N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $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))) < -0.0200000000000000004

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in re around inf

       \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{6} \cdot \color{blue}{{re}^{3}}\right)\right) \cdot \left(im \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{3} \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{3} \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot im\right) \]

       3. unpow3 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 15.8

          \[\leadsto \left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot im\right) \]

   13. Applied rewrites 15.8%

       \[\leadsto \left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right) \cdot \left(im \cdot im\right) \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 70.2

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

   4. Applied rewrites 70.2%

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

Alternative 7: 47.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 1e-6)
   (* (* (fma re (* re -0.08333333333333333) 0.5) re) (fma im im 2.0))
   (* (* 0.5 re) (+ 1.0 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 1e-6) {
		tmp = (fma(re, (re * -0.08333333333333333), 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = (0.5 * re) * (1.0 + exp(im));
	}
	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))) <= 1e-6)
		tmp = Float64(Float64(fma(re, Float64(re * -0.08333333333333333), 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 + exp(im)));
	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], 1e-6], N[(N[(N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 9.99999999999999955e-7

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 59.0

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

   7. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 59.0

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

   9. Applied rewrites 59.0%

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

   if 9.99999999999999955e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in 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 66.8%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 26.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 42.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.02)
   (* (* 0.5 (* (* (* re re) re) -0.16666666666666666)) (* im im))
   (* (* 0.5 re) (+ 1.0 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= -0.02) {
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * im);
	} else {
		tmp = (0.5 * re) * (1.0 + exp(im));
	}
	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))) <= (-0.02d0)) then
        tmp = (0.5d0 * (((re * re) * re) * (-0.16666666666666666d0))) * (im * im)
    else
        tmp = (0.5d0 * re) * (1.0d0 + exp(im))
    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))) <= -0.02) {
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * im);
	} else {
		tmp = (0.5 * re) * (1.0 + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))) <= -0.02:
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * im)
	else:
		tmp = (0.5 * re) * (1.0 + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.02)
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64(re * re) * re) * -0.16666666666666666)) * Float64(im * im));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 + exp(im)));
	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))) <= -0.02)
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * im);
	else
		tmp = (0.5 * re) * (1.0 + exp(im));
	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], -0.02], N[(N[(0.5 * N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in re around inf

       \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{6} \cdot \color{blue}{{re}^{3}}\right)\right) \cdot \left(im \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{3} \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{3} \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot im\right) \]

       3. unpow3 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 15.8

          \[\leadsto \left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot im\right) \]

   13. Applied rewrites 15.8%

       \[\leadsto \left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right) \cdot \left(im \cdot im\right) \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in 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 79.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 55.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 42.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.02)
   (* (* 0.5 (* (* (* re re) re) -0.16666666666666666)) (* im im))
   (* (* 0.5 re) (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= -0.02) {
		tmp = (0.5 * (((re * re) * re) * -0.16666666666666666)) * (im * im);
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.02)
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64(re * re) * re) * -0.16666666666666666)) * Float64(im * im));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(0.5 * N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in re around inf

       \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{6} \cdot \color{blue}{{re}^{3}}\right)\right) \cdot \left(im \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{3} \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{3} \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot im\right) \]

       3. unpow3 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 15.8

          \[\leadsto \left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot im\right) \]

   13. Applied rewrites 15.8%

       \[\leadsto \left(0.5 \cdot \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right) \cdot \left(im \cdot im\right) \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.8

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.5%

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

Alternative 10: 41.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 32.2

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

   7. Applied rewrites 32.2%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{12}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{12}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 15.9

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

   10. Applied rewrites 15.9%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.8

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.5%

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

Alternative 11: 41.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 1e-6)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (* 0.5 re) (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 1e-6) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	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))) <= 1e-6)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 9.99999999999999955e-7

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. lift-sin.f64 61.4

         \[\leadsto \sin re \]

   4. Applied rewrites 61.4%

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

   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 47.3

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

   7. Applied rewrites 47.3%

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

   if 9.99999999999999955e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 68.3

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 31.2%

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

Alternative 12: 41.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;t\_0 \leq 0.92:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 -0.02)
     (* (* (* re re) re) -0.16666666666666666)
     (if (<= t_0 0.92) re (* (* 0.5 re) (* im im))))))

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 <= -0.02) {
		tmp = ((re * re) * re) * -0.16666666666666666;
	} else if (t_0 <= 0.92) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (im * im);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
    if (t_0 <= (-0.02d0)) then
        tmp = ((re * re) * re) * (-0.16666666666666666d0)
    else if (t_0 <= 0.92d0) then
        tmp = re
    else
        tmp = (0.5d0 * re) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = ((re * re) * re) * -0.16666666666666666;
	} else if (t_0 <= 0.92) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))
	tmp = 0
	if t_0 <= -0.02:
		tmp = ((re * re) * re) * -0.16666666666666666
	elif t_0 <= 0.92:
		tmp = re
	else:
		tmp = (0.5 * re) * (im * im)
	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 <= -0.02)
		tmp = Float64(Float64(Float64(re * re) * re) * -0.16666666666666666);
	elseif (t_0 <= 0.92)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = ((re * re) * re) * -0.16666666666666666;
	elseif (t_0 <= 0.92)
		tmp = re;
	else
		tmp = (0.5 * re) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[t$95$0, 0.92], re, N[(N[(0.5 * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. lift-sin.f64 36.4

         \[\leadsto \sin re \]

   4. Applied rewrites 36.4%

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

   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 13.0

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

   7. Applied rewrites 13.0%

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

   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. lower-*.f64 N/A

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

      5. lift-*.f64 12.6

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

   10. Applied rewrites 12.6%

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

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.92000000000000004

   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 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 72.9%

      \[\leadsto re \]

   if 0.92000000000000004 < (*.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 im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 57.7

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 46.0

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

   7. Applied rewrites 46.0%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 40.2%

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

Alternative 13: 40.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.42:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.42)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (* 0.5 re) (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.42) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = (0.5 * re) * (im * im);
	}
	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))) <= 0.42)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im * im));
	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], 0.42], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.419999999999999984

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0

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

      1. lift-sin.f64 63.5

         \[\leadsto \sin re \]

   4. Applied rewrites 63.5%

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

   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 45.2

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

   7. Applied rewrites 45.2%

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

   if 0.419999999999999984 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 64.9

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 38.8

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

   7. Applied rewrites 38.8%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 35.8

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

   10. Applied rewrites 35.8%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 33.9%

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

Alternative 14: 30.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		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)) <= (-0.01d0)) 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)) <= -0.01) {
		tmp = ((re * re) * re) * -0.16666666666666666;
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.01:
		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)) <= -0.01)
		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)) <= -0.01)
		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], -0.01], 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)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lift-sin.f64 53.1

         \[\leadsto \sin re \]

   4. Applied rewrites 53.1%

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

   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 18.1

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

   7. Applied rewrites 18.1%

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

   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. lower-*.f64 N/A

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

      5. lift-*.f64 17.9

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

   10. Applied rewrites 17.9%

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

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

   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 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 35.3%

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

Alternative 15: 27.1% 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 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 62.5

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in im around 0

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

7. Applied rewrites 27.1%

   \[\leadsto re \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))