Kahan p9 Example

Percentage Accurate: 68.0% → 92.0%

Time: 3.2s

Alternatives: 6

Speedup: 0.7×

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: 68.0% 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: 92.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}, -2, 1\right)\\ \mathbf{elif}\;y\_m \leq 3 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{x + x}{y\_m} \cdot x}{y\_m} - 1\\ \mathbf{elif}\;y\_m \leq 4 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2e-212)
   (fma (* (/ y_m x) (/ y_m x)) -2.0 1.0)
   (if (<= y_m 3e-190)
     (- (/ (* (/ (+ x x) y_m) x) y_m) 1.0)
     (if (<= y_m 4e-56)
       (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
       -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2e-212) {
		tmp = fma(((y_m / x) * (y_m / x)), -2.0, 1.0);
	} else if (y_m <= 3e-190) {
		tmp = ((((x + x) / y_m) * x) / y_m) - 1.0;
	} else if (y_m <= 4e-56) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 2e-212)
		tmp = fma(Float64(Float64(y_m / x) * Float64(y_m / x)), -2.0, 1.0);
	elseif (y_m <= 3e-190)
		tmp = Float64(Float64(Float64(Float64(Float64(x + x) / y_m) * x) / y_m) - 1.0);
	elseif (y_m <= 4e-56)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 2e-212], N[(N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 3e-190], N[(N[(N[(N[(N[(x + x), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision] / y$95$m), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[y$95$m, 4e-56], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.99999999999999991e-212

   1. Initial program 51.6%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 51.6

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

   4. Applied rewrites 51.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 85.0

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

   6. Applied rewrites 85.0%

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

   if 1.99999999999999991e-212 < y < 2.9999999999999998e-190

   1. Initial program 51.8%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 1.6

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

   4. Applied rewrites 1.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 39.5

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

   6. Applied rewrites 39.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 39.5

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

   8. Applied rewrites 39.5%

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

   if 2.9999999999999998e-190 < y < 4.0000000000000002e-56

   1. Initial program 89.4%

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

   if 4.0000000000000002e-56 < y

   1. Initial program 64.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.3% accurate, 0.3× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))))
   (if (<= t_0 -0.5)
     (- (/ (* (+ x x) x) (* y_m y_m)) 1.0)
     (if (<= t_0 2.0)
       (fma (/ (* y_m y_m) (* x x)) -2.0 1.0)
       (- (/ (* (/ (+ x x) y_m) x) y_m) 1.0)))))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x + x), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision] / y$95$m), $MachinePrecision] - 1.0), $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. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

   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. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 76.4

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

   6. Applied rewrites 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 76.4

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

   8. Applied rewrites 76.4%

      \[\leadsto \frac{\frac{x + x}{y} \cdot x}{y} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.1% accurate, 0.3× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))))
   (if (<= t_0 -0.5)
     (- (/ (* (+ x x) x) (* y_m y_m)) 1.0)
     (if (<= t_0 2.0) (fma (/ (* y_m y_m) (* x x)) -2.0 1.0) -1.0))))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(y$95$m * y$95$m), $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. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

   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. Taylor expanded in x around 0

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

   4. Applied rewrites 74.5%

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

Alternative 4: 91.0% accurate, 0.3× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 2.0) (fma (/ (* y_m y_m) (* x x)) -2.0 1.0) -1.0))))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 2.0], N[(N[(N[(y$95$m * y$95$m), $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 56.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.4%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 5: 89.4% accurate, 0.4× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))))
   (if (<= t_0 -0.5) -1.0 (if (<= t_0 INFINITY) 1.0 -1.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -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))) < -0.5 or +inf.0 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 56.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.4%

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

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < +inf.0

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.1%

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

Alternative 6: 66.0% accurate, 21.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

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

Fortran

y_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = -1.0d0
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return -1.0;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

y_m = abs(y)
function code(x, y_m)
	return -1.0
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 68.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 66.0%

   \[\leadsto \color{blue}{-1} \]
5. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× 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 2025106 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (! :herbie-platform c (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))))