Kahan p9 Example

Percentage Accurate: 68.2% → 92.8%

Time: 3.1s

Alternatives: 9

Speedup: 0.5×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

C

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

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 - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

C

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

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 - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \left(x + y\right)\\ \mathbf{if}\;\frac{t\_0}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 2 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) (+ x y))))
   (if (<= (/ t_0 (+ (* x x) (* y y))) 2.0)
     (/ t_0 (fma y y (* x x)))
     (- (* (* (/ x y) (/ x y)) 2.0) 1.0))))

C

double code(double x, double y) {
	double t_0 = (x - y) * (x + y);
	double tmp;
	if ((t_0 / ((x * x) + (y * y))) <= 2.0) {
		tmp = t_0 / fma(y, y, (x * x));
	} else {
		tmp = (((x / y) * (x / y)) * 2.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * Float64(x + y))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(x * x) + Float64(y * y))) <= 2.0)
		tmp = Float64(t_0 / fma(y, y, Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(Float64(x / y) * Float64(x / y)) * 2.0) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 / N[(y * y + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{y}^{2}} \cdot 2 - 1 \]

      5. pow2 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 77.3

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

   7. Applied rewrites 77.3%

      \[\leadsto \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 2 - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 2 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 2}{y \cdot y} - 1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot y}{x \cdot x}, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 -0.5)
     (- (/ (* (* x x) 2.0) (* y y)) 1.0)
     (if (<= t_0 2.0)
       (fma (/ (* y y) (* x x)) -2.0 1.0)
       (* (- x y) (/ (+ (/ x y) 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (((x * x) * 2.0) / (y * y)) - 1.0;
	} else if (t_0 <= 2.0) {
		tmp = fma(((y * y) / (x * x)), -2.0, 1.0);
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(Float64(Float64(x * x) * 2.0) / Float64(y * y)) - 1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(Float64(y * y) / Float64(x * x)), -2.0, 1.0);
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{y}^{2}} \cdot 2 - 1 \]

      5. pow2 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{y \cdot y} \cdot 2 - 1 \]

      7. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \frac{{x}^{2}}{{y}^{2}} - 1 \]

      9. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {x}^{2}}{{y}^{2}} - 1 \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot {x}^{2}}{{y}^{2}} - 1 \]

      11. *-commutative N/A

          \[\leadsto \frac{{x}^{2} \cdot 2}{{y}^{2}} - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{{x}^{2} \cdot 2}{{y}^{2}} - 1 \]

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{y}^{2}} \cdot 2 - 1 \]

      5. pow2 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 4.9

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      2. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      4. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      5. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      6. +-commutative N/A

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

      7. *-commutative N/A

         \[\leadsto \frac{{y}^{2}}{{x}^{2}} \cdot -2 + 1 \]

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 99.5

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

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{y}{x}\right)}^{2}, -2, 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. frac-times N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. pow2 N/A

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

      18. lower-fma.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 3.1

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

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 76.6

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

   7. Applied rewrites 76.6%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{\frac{x}{y} + 1}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq -0.5:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 2}{y \cdot y} - 1\\ \mathbf{elif}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot y}{x \cdot x}, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 2}{y \cdot y} - 1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot y}{x \cdot x}, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 -0.5)
     (- (/ (* (* x x) 2.0) (* y y)) 1.0)
     (if (<= t_0 2.0) (fma (/ (* y y) (* x x)) -2.0 1.0) -1.0))))

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (((x * x) * 2.0) / (y * y)) - 1.0;
	} else if (t_0 <= 2.0) {
		tmp = fma(((y * y) / (x * x)), -2.0, 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(Float64(Float64(x * x) * 2.0) / Float64(y * y)) - 1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(Float64(y * y) / Float64(x * x)), -2.0, 1.0);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{y}^{2}} \cdot 2 - 1 \]

      5. pow2 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{y \cdot y} \cdot 2 - 1 \]

      7. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \frac{{x}^{2}}{{y}^{2}} - 1 \]

      9. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {x}^{2}}{{y}^{2}} - 1 \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot {x}^{2}}{{y}^{2}} - 1 \]

      11. *-commutative N/A

          \[\leadsto \frac{{x}^{2} \cdot 2}{{y}^{2}} - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{{x}^{2} \cdot 2}{{y}^{2}} - 1 \]

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{y}^{2}} \cdot 2 - 1 \]

      5. pow2 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 4.9

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      2. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      4. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      5. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      6. +-commutative N/A

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

      7. *-commutative N/A

         \[\leadsto \frac{{y}^{2}}{{x}^{2}} \cdot -2 + 1 \]

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 99.5

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

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{y}{x}\right)}^{2}, -2, 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. frac-times N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq -0.5:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 2}{y \cdot y} - 1\\ \mathbf{elif}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot y}{x \cdot x}, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \left(x + y\right)\\ t_1 := \frac{t\_0}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{t\_0}{y \cdot y}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot y}{x \cdot x}, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) (+ x y))) (t_1 (/ t_0 (+ (* x x) (* y y)))))
   (if (<= t_1 -0.5)
     (/ t_0 (* y y))
     (if (<= t_1 2.0) (fma (/ (* y y) (* x x)) -2.0 1.0) -1.0))))

C

double code(double x, double y) {
	double t_0 = (x - y) * (x + y);
	double t_1 = t_0 / ((x * x) + (y * y));
	double tmp;
	if (t_1 <= -0.5) {
		tmp = t_0 / (y * y);
	} else if (t_1 <= 2.0) {
		tmp = fma(((y * y) / (x * x)), -2.0, 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * Float64(x + y))
	t_1 = Float64(t_0 / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(t_0 / Float64(y * y));
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(Float64(y * y) / Float64(x * x)), -2.0, 1.0);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(t$95$0 / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{y}^{2}} \cdot 2 - 1 \]

      5. pow2 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 4.9

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      2. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      4. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      5. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      6. +-commutative N/A

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

      7. *-commutative N/A

         \[\leadsto \frac{{y}^{2}}{{x}^{2}} \cdot -2 + 1 \]

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 99.5

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

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{y}{x}\right)}^{2}, -2, 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. frac-times N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq -0.5:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{y \cdot y}\\ \mathbf{elif}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot y}{x \cdot x}, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot y}{x \cdot x}, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 2.0) (fma (/ (* y y) (* x x)) -2.0 1.0) -1.0))))

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 2.0) {
		tmp = fma(((y * y) / (x * x)), -2.0, 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(Float64(y * y) / Float64(x * x)), -2.0, 1.0);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 2.0], N[(N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -0.5 or 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 51.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.0%

      \[\leadsto \color{blue}{-1} \]

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{y}^{2}} \cdot 2 - 1 \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{y}^{2}} \cdot 2 - 1 \]

      5. pow2 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 4.9

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      2. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      4. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      5. pow2 N/A

         \[\leadsto 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      6. +-commutative N/A

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

      7. *-commutative N/A

         \[\leadsto \frac{{y}^{2}}{{x}^{2}} \cdot -2 + 1 \]

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 99.5

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

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{y}{x}\right)}^{2}, -2, 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. frac-times N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.5

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

   10. Applied rewrites 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{y \cdot y}{x \cdot x}, -2, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot y}{x \cdot x}, -2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-312}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 -1e-312) -1.0 (if (<= t_0 INFINITY) 1.0 -1.0))))

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= -1e-312) {
		tmp = -1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= -1e-312) {
		tmp = -1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= -1e-312:
		tmp = -1.0
	elif t_0 <= math.inf:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= -1e-312)
		tmp = -1.0;
	elseif (t_0 <= Inf)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (t_0 <= -1e-312)
		tmp = -1.0;
	elseif (t_0 <= Inf)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-312], -1.0, If[LessEqual[t$95$0, Infinity], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -9.9999999999847e-313 or +inf.0 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 51.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.0%

      \[\leadsto \color{blue}{-1} \]

   if -9.9999999999847e-313 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq -1 \cdot 10^{-312}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq \infty:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \left(x + y\right)\\ \mathbf{if}\;\frac{t\_0}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) (+ x y))))
   (if (<= (/ t_0 (+ (* x x) (* y y))) 2.0)
     (/ t_0 (fma y y (* x x)))
     (* (- x y) (/ (+ (/ x y) 1.0) y)))))

C

double code(double x, double y) {
	double t_0 = (x - y) * (x + y);
	double tmp;
	if ((t_0 / ((x * x) + (y * y))) <= 2.0) {
		tmp = t_0 / fma(y, y, (x * x));
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * Float64(x + y))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(x * x) + Float64(y * y))) <= 2.0)
		tmp = Float64(t_0 / fma(y, y, Float64(x * x)));
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 / N[(y * y + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. pow2 N/A

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

      18. lower-fma.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 3.1

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

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 76.6

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

   7. Applied rewrites 76.6%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{\frac{x}{y} + 1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\left(x - y\right) \cdot \frac{y + x}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) 2.0)
   (* (- x y) (/ (+ y x) (fma y y (* x x))))
   (* (- x y) (/ (+ (/ x y) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if ((((x - y) * (x + y)) / ((x * x) + (y * y))) <= 2.0) {
		tmp = (x - y) * ((y + x) / fma(y, y, (x * x)));
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y))) <= 2.0)
		tmp = Float64(Float64(x - y) * Float64(Float64(y + x) / fma(y, y, Float64(x * x))));
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(x - y), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / N[(y * y + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. pow2 N/A

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

      18. lower-fma.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 98.1

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

   4. Applied rewrites 98.1%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. pow2 N/A

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

      18. lower-fma.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 3.1

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

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 76.6

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

   7. Applied rewrites 76.6%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{\frac{x}{y} + 1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.4% accurate, 36.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

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 = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 63.7%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 63.7%

   \[\leadsto -1 \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;0.5 < t\_0 \land t\_0 < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (and (< 0.5 t_0) (< t_0 2.0))
     (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))
     (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = fabs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if ((0.5d0 < t_0) .and. (t_0 < 2.0d0)) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = 1.0d0 - (2.0d0 / (1.0d0 + ((x / y) * (x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs((x / y))
	tmp = 0
	if (0.5 < t_0) and (t_0 < 2.0):
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))))
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(2.0 / Float64(1.0 + Float64(Float64(x / y) * Float64(x / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs((x / y));
	tmp = 0.0;
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[And[Less[0.5, t$95$0], Less[t$95$0, 2.0]], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / N[(1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025051 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (! :herbie-platform default (if (< 1/2 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y)))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))