Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 11

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, \left|y - x\right|, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma 0.5 (fabs (- y x)) x))

C

double code(double x, double y) {
	return fma(0.5, fabs((y - x)), x);
}

Julia

function code(x, y)
	return fma(0.5, abs(Float64(y - x)), x)
end

Wolfram

code[x_, y_] := N[(0.5 * N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

   3. fabs-sub N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

   7. *-lft-identity N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

   10. cancel-sign-sub N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

   11. lower-fabs.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

   12. fp-cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

   14. *-lft-identity N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

   15. lower--.f64 99.9

       \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]

6. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(0.5, \left|y - x\right|, x\right) \]
7. Add Preprocessing

Alternative 2: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq -4 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) -4e-295) (* 0.5 x) (fma (- x y) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= -4e-295) {
		tmp = 0.5 * x;
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= -4e-295)
		tmp = Float64(0.5 * x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], -4e-295], N[(0.5 * x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < -4.00000000000000024e-295

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left|y - x\right|}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left|y - x\right|}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto x + \frac{\left|\color{blue}{y - x}\right|}{2} \]

      4. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   4. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + {x}^{3}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{3}}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2}} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2} \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      6. lift-*.f64 31.7

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   6. Applied rewrites 31.7%

      \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|x - y\right|} \]
   8. Step-by-step derivation

      1. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(x - \color{blue}{y}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. lift--.f64 94.8

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{1}{2} \cdot \color{blue}{x} \]
   11. Step-by-step derivation

       1. lower-*.f64 95.0

          \[\leadsto 0.5 \cdot x \]

   12. Applied rewrites 95.0%

       \[\leadsto 0.5 \cdot \color{blue}{x} \]

   if -4.00000000000000024e-295 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left|y - x\right|}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left|y - x\right|}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto x + \frac{\left|\color{blue}{y - x}\right|}{2} \]

      4. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + {x}^{3}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{3}}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2}} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2} \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      6. lift-*.f64 35.8

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|x - y\right|} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| + \color{blue}{x} \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} + x \]

      3. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \sqrt{x - y}\right) + x \]

      4. rem-square-sqrt N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - y, \color{blue}{\frac{1}{2}}, x\right) \]

      7. lift--.f64 68.6

         \[\leadsto \mathsf{fma}\left(x - y, 0.5, x\right) \]

   9. Applied rewrites 68.6%

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

Alternative 3: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8)
   (* (- x y) 0.5)
   (if (<= x 2.8e-66) (fma 0.5 (fabs (- y)) x) (fma 1.5 x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.8e-66) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma(1.5, x, (-0.5 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.8)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.8e-66)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(1.5, x, Float64(-0.5 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.8], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.8e-66], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.79999999999999982

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left|y - x\right|}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left|y - x\right|}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto x + \frac{\left|\color{blue}{y - x}\right|}{2} \]

      4. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   4. Applied rewrites 19.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + {x}^{3}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{3}}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2}} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2} \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      6. lift-*.f64 18.9

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   6. Applied rewrites 18.9%

      \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|x - y\right|} \]
   8. Step-by-step derivation

      1. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(x - \color{blue}{y}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. lift--.f64 85.3

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5} \]

   if -5.79999999999999982 < x < 2.8e-66

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(y\right)\right|, x\right) \]

      2. lower-neg.f64 84.3

         \[\leadsto \mathsf{fma}\left(0.5, \left|-y\right|, x\right) \]

   8. Applied rewrites 84.3%

      \[\leadsto \mathsf{fma}\left(0.5, \left|-y\right|, x\right) \]

   if 2.8e-66 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.7

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      7. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} - y}, x\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      9. lift--.f64 90.5

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 90.5%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot y + \color{blue}{\frac{3}{2} \cdot x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{3}{2} \cdot x + \frac{-1}{2} \cdot \color{blue}{y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{3}{2}, x, \frac{-1}{2} \cdot y\right) \]

      3. lower-*.f64 91.1

         \[\leadsto \mathsf{fma}\left(1.5, x, -0.5 \cdot y\right) \]

   10. Applied rewrites 91.1%

       \[\leadsto \mathsf{fma}\left(1.5, \color{blue}{x}, -0.5 \cdot y\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8)
   (* (- x y) 0.5)
   (if (<= x 2.8e-66) (fma 0.5 (fabs (- y)) x) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.8e-66) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.8)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.8e-66)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.8], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.8e-66], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.79999999999999982

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left|y - x\right|}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left|y - x\right|}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto x + \frac{\left|\color{blue}{y - x}\right|}{2} \]

      4. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   4. Applied rewrites 19.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + {x}^{3}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{3}}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2}} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2} \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      6. lift-*.f64 18.9

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   6. Applied rewrites 18.9%

      \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|x - y\right|} \]
   8. Step-by-step derivation

      1. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(x - \color{blue}{y}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. lift--.f64 85.3

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5} \]

   if -5.79999999999999982 < x < 2.8e-66

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(y\right)\right|, x\right) \]

      2. lower-neg.f64 84.3

         \[\leadsto \mathsf{fma}\left(0.5, \left|-y\right|, x\right) \]

   8. Applied rewrites 84.3%

      \[\leadsto \mathsf{fma}\left(0.5, \left|-y\right|, x\right) \]

   if 2.8e-66 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left|y - x\right|}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left|y - x\right|}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto x + \frac{\left|\color{blue}{y - x}\right|}{2} \]

      4. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   4. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + {x}^{3}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{3}}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2}} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2} \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      6. lift-*.f64 32.2

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   6. Applied rewrites 32.2%

      \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|x - y\right|} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| + \color{blue}{x} \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} + x \]

      3. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \sqrt{x - y}\right) + x \]

      4. rem-square-sqrt N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - y, \color{blue}{\frac{1}{2}}, x\right) \]

      7. lift--.f64 90.8

         \[\leadsto \mathsf{fma}\left(x - y, 0.5, x\right) \]

   9. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, 0.5, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-131}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \left|y - x\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e-131)
   (* (- x y) 0.5)
   (if (<= x 2.8e-66) (* 0.5 (fabs (- y x))) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-131) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.8e-66) {
		tmp = 0.5 * fabs((y - x));
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e-131)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.8e-66)
		tmp = Float64(0.5 * abs(Float64(y - x)));
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e-131], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.8e-66], N[(0.5 * N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e-131

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left|y - x\right|}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left|y - x\right|}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto x + \frac{\left|\color{blue}{y - x}\right|}{2} \]

      4. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   4. Applied rewrites 37.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + {x}^{3}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{3}}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2}} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2} \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      6. lift-*.f64 37.0

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   6. Applied rewrites 37.0%

      \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|x - y\right|} \]
   8. Step-by-step derivation

      1. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(x - \color{blue}{y}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. lift--.f64 79.0

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5} \]

   if -1.1e-131 < x < 2.8e-66

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|y - x\right|} \]

      2. fabs-sub N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      3. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - y\right| \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right| \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - y\right| \]

      6. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      7. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - 1 \cdot y\right| \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right| \]

      9. cancel-sign-sub N/A

         \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      10. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right| \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - 1 \cdot y\right| \]

      13. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      14. lower--.f64 87.8

          \[\leadsto 0.5 \cdot \left|x - y\right| \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]

   if 2.8e-66 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left|y - x\right|}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left|y - x\right|}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto x + \frac{\left|\color{blue}{y - x}\right|}{2} \]

      4. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   4. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + {x}^{3}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{3}}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2}} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2} \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      6. lift-*.f64 32.2

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   6. Applied rewrites 32.2%

      \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|x - y\right|} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| + \color{blue}{x} \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} + x \]

      3. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \sqrt{x - y}\right) + x \]

      4. rem-square-sqrt N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - y, \color{blue}{\frac{1}{2}}, x\right) \]

      7. lift--.f64 90.8

         \[\leadsto \mathsf{fma}\left(x - y, 0.5, x\right) \]

   9. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, 0.5, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-131}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \left|y - x\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 9.7 \cdot 10^{-178}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e+21) (* 0.5 x) (if (<= x 9.7e-178) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+21) {
		tmp = 0.5 * x;
	} else if (x <= 9.7e-178) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.5d+21)) then
        tmp = 0.5d0 * x
    else if (x <= 9.7d-178) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+21) {
		tmp = 0.5 * x;
	} else if (x <= 9.7e-178) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.5e+21:
		tmp = 0.5 * x
	elif x <= 9.7e-178:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e+21)
		tmp = Float64(0.5 * x);
	elseif (x <= 9.7e-178)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e+21)
		tmp = 0.5 * x;
	elseif (x <= 9.7e-178)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e+21], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 9.7e-178], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e21

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left|y - x\right|}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left|y - x\right|}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto x + \frac{\left|\color{blue}{y - x}\right|}{2} \]

      4. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   4. Applied rewrites 18.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + {x}^{3}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{3}}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2}} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2} \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      6. lift-*.f64 18.2

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   6. Applied rewrites 18.2%

      \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|x - y\right|} \]
   8. Step-by-step derivation

      1. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(x - \color{blue}{y}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. lift--.f64 84.5

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{1}{2} \cdot \color{blue}{x} \]
   11. Step-by-step derivation

       1. lower-*.f64 77.4

          \[\leadsto 0.5 \cdot x \]

   12. Applied rewrites 77.4%

       \[\leadsto 0.5 \cdot \color{blue}{x} \]

   if -7.5e21 < x < 9.7000000000000003e-178

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 100.0

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      7. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} - y}, x\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      9. lift--.f64 45.4

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 45.4%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lower-*.f64 45.0

         \[\leadsto -0.5 \cdot y \]

   10. Applied rewrites 45.0%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if 9.7000000000000003e-178 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.7

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      7. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} - y}, x\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      9. lift--.f64 85.8

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 85.8%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

      1. lower-*.f64 68.8

         \[\leadsto 1.5 \cdot x \]

   10. Applied rewrites 68.8%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-69}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.6e-69) (* (- x y) 0.5) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.6e-69) {
		tmp = (x - y) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.6d-69) then
        tmp = (x - y) * 0.5d0
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.6e-69) {
		tmp = (x - y) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.6e-69:
		tmp = (x - y) * 0.5
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.6e-69)
		tmp = Float64(Float64(x - y) * 0.5);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.6e-69)
		tmp = (x - y) * 0.5;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.6e-69], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.59999999999999999e-69

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left|y - x\right|}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left|y - x\right|}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto x + \frac{\left|\color{blue}{y - x}\right|}{2} \]

      4. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   4. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + {x}^{3}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{3}}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2}} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{{x}^{2} \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

      6. lift-*.f64 35.6

         \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right)} \cdot x}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   6. Applied rewrites 35.6%

      \[\leadsto \frac{{\left(\frac{\left|x - y\right|}{2}\right)}^{3} + \color{blue}{\left(x \cdot x\right) \cdot x}}{\mathsf{fma}\left(\frac{\left|x - y\right|}{2}, \frac{\left|x - y\right|}{2} - x, x \cdot x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|x - y\right|} \]
   8. Step-by-step derivation

      1. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(x - \color{blue}{y}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. lift--.f64 64.6

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5} \]

   if 1.59999999999999999e-69 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.7

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      7. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} - y}, x\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      9. lift--.f64 89.4

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 89.4%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

      1. lower-*.f64 75.0

         \[\leadsto 1.5 \cdot x \]

   10. Applied rewrites 75.0%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-30}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.5e-30) (* -0.5 y) (fma 0.5 (fabs x) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.5e-30) {
		tmp = -0.5 * y;
	} else {
		tmp = fma(0.5, fabs(x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.5e-30)
		tmp = Float64(-0.5 * y);
	else
		tmp = fma(0.5, abs(x), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.5e-30], N[(-0.5 * y), $MachinePrecision], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5000000000000003e-30

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      7. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} - y}, x\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      9. lift--.f64 88.8

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 88.8%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lower-*.f64 74.5

         \[\leadsto -0.5 \cdot y \]

   10. Applied rewrites 74.5%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -3.5000000000000003e-30 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 58.1%

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

Alternative 9: 46.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-67}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -4.5e-67) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e-67) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.5d-67)) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.5e-67) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5e-67:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e-67)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.5e-67)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e-67], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000015e-67

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      7. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} - y}, x\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      9. lift--.f64 88.7

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 88.7%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lower-*.f64 71.6

         \[\leadsto -0.5 \cdot y \]

   10. Applied rewrites 71.6%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -4.50000000000000015e-67 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      7. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} - y}, x\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      9. lift--.f64 33.7

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 33.7%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

      1. lower-*.f64 35.5

         \[\leadsto 1.5 \cdot x \]

   10. Applied rewrites 35.5%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 32.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-223}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1e-223) (* -0.5 y) x))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e-223) {
		tmp = -0.5 * y;
	} else {
		tmp = x;
	}
	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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1d-223)) then
        tmp = (-0.5d0) * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1e-223) {
		tmp = -0.5 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1e-223:
		tmp = -0.5 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e-223)
		tmp = Float64(-0.5 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e-223)
		tmp = -0.5 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e-223], N[(-0.5 * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999997e-224

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.8

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      7. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} - y}, x\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      9. lift--.f64 80.4

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 80.4%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lower-*.f64 58.8

         \[\leadsto -0.5 \cdot y \]

   10. Applied rewrites 58.8%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -9.9999999999999997e-224 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 12.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 11.4% accurate, 20.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Applied rewrites 11.0%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025043 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))