Kahan p9 Example

Percentage Accurate: 68.6% → 92.7%

Time: 3.3s

Alternatives: 7

Speedup: 0.8×

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: 68.6% 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: 92.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.25 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{x}, \frac{y\_m}{x} \cdot y\_m, 1\right)\\ \mathbf{elif}\;y\_m \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{\mathsf{fma}\left(y\_m, y\_m, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.25e-164)
   (fma (/ -2.0 x) (* (/ y_m x) y_m) 1.0)
   (if (<= y_m 1.4e-24)
     (/ (* (- x y_m) (+ x y_m)) (fma y_m y_m (* x x)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.25e-164) {
		tmp = fma((-2.0 / x), ((y_m / x) * y_m), 1.0);
	} else if (y_m <= 1.4e-24) {
		tmp = ((x - y_m) * (x + y_m)) / fma(y_m, y_m, (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.25e-164)
		tmp = fma(Float64(-2.0 / x), Float64(Float64(y_m / x) * y_m), 1.0);
	elseif (y_m <= 1.4e-24)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / fma(y_m, y_m, Float64(x * x)));
	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, 1.25e-164], N[(N[(-2.0 / x), $MachinePrecision] * N[(N[(y$95$m / x), $MachinePrecision] * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 1.4e-24], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.2499999999999999e-164

   1. Initial program 68.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 41.9%

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

   if 1.2499999999999999e-164 < y < 1.4000000000000001e-24

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   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 1.4000000000000001e-24 < y

   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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{x}, \frac{y}{x} \cdot y, 1\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.2% accurate, 0.3× speedup

Math

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

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)
     (fma (/ x (* y_m y_m)) (+ x x) -1.0)
     (if (<= t_0 2.0)
       (fma (/ -2.0 x) (* (/ y_m x) y_m) 1.0)
       (fma (/ 2.0 y_m) (* (/ x y_m) x) -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 = fma((x / (y_m * y_m)), (x + x), -1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma((-2.0 / x), ((y_m / x) * y_m), 1.0);
	} else {
		tmp = fma((2.0 / y_m), ((x / y_m) * x), -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 = fma(Float64(x / Float64(y_m * y_m)), Float64(x + x), -1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(-2.0 / x), Float64(Float64(y_m / x) * y_m), 1.0);
	else
		tmp = fma(Float64(2.0 / y_m), Float64(Float64(x / y_m) * x), -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[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(x + x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(-2.0 / x), $MachinePrecision] * N[(N[(y$95$m / x), $MachinePrecision] * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(2.0 / y$95$m), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] * x), $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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

   10. Applied rewrites 99.3%

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

   12. Applied rewrites 99.3%

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

   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 inf

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

   5. Applied rewrites 99.5%

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

   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

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

   5. Applied rewrites 70.9%

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

4. Final simplification 92.0%

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

Alternative 3: 91.7% accurate, 0.3× speedup

Math

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

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)
     (fma (/ x (* y_m y_m)) (+ x x) -1.0)
     (if (<= t_0 2.0) (fma (/ -2.0 x) (* (/ y_m x) y_m) 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 = fma((x / (y_m * y_m)), (x + x), -1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma((-2.0 / x), ((y_m / x) * y_m), 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 = fma(Float64(x / Float64(y_m * y_m)), Float64(x + x), -1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(-2.0 / x), Float64(Float64(y_m / x) * y_m), 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[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(x + x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(-2.0 / x), $MachinePrecision] * N[(N[(y$95$m / x), $MachinePrecision] * y$95$m), $MachinePrecision] + 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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

   10. Applied rewrites 99.3%

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

   12. Applied rewrites 99.3%

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

   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 inf

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

   5. Applied rewrites 99.5%

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

   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 69.1%

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

4. Final simplification 91.6%

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

Alternative 4: 91.5% accurate, 0.3× speedup

Math

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

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)
     (fma (/ x (* y_m y_m)) (+ x x) -1.0)
     (if (<= t_0 2.0) (/ (* x x) (fma y_m y_m (* x x))) -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 = fma((x / (y_m * y_m)), (x + x), -1.0);
	} else if (t_0 <= 2.0) {
		tmp = (x * x) / fma(y_m, y_m, (x * x));
	} 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 = fma(Float64(x / Float64(y_m * y_m)), Float64(x + x), -1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(x * x) / fma(y_m, y_m, Float64(x * x)));
	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[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(x + x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]), $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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

   10. Applied rewrites 99.3%

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

   12. Applied rewrites 99.3%

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

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\left(1 + \color{blue}{0} \cdot \frac{y}{x}\right) \cdot {x}^{2}}{\mathsf{fma}\left(y, y, x \cdot x\right)} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\left(1 + \color{blue}{0}\right) \cdot {x}^{2}}{\mathsf{fma}\left(y, y, x \cdot x\right)} \]

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. swap-sqr N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. sqr-neg-rev N/A

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

      12. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   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. Taylor expanded in x around 0

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

   5. Applied rewrites 69.1%

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

4. Final simplification 91.5%

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

Alternative 5: 91.5% accurate, 0.4× speedup

Math

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

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)
     (fma (/ x (* y_m y_m)) (+ x x) -1.0)
     (if (<= t_0 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 = fma((x / (y_m * y_m)), (x + x), -1.0);
	} else if (t_0 <= 2.0) {
		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 = fma(Float64(x / Float64(y_m * y_m)), Float64(x + x), -1.0);
	elseif (t_0 <= 2.0)
		tmp = 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[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(x + x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, -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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

   10. Applied rewrites 99.3%

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

   12. Applied rewrites 99.3%

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

   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 inf

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

   5. Applied rewrites 99.1%

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

   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 69.1%

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

4. Final simplification 91.4%

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

Alternative 6: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

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 -4e-310) -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 <= -4e-310) {
		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 <= -4e-310) {
		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 <= -4e-310:
		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 <= -4e-310)
		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 <= -4e-310)
		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, -4e-310], -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))) < -3.999999999999988e-310 or +inf.0 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 62.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.8%

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

   if -3.999999999999988e-310 < (/.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 99.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq \infty:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 36.0× 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 74.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 60.2%

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

6. Final simplification 60.2%

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

Developer Target 1: 100.0% 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 2025017 
(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))))