Kahan p9 Example

Percentage Accurate: 67.9% → 92.6%

Time: 10.0s

Alternatives: 10

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    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: 67.9% 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

real(8) function code(x, y)
    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: 92.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. difference-of-squares N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 81.7%

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

Alternative 2: 92.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.5%

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

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

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. difference-of-squares N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. associate-+l- N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. neg-sub0 N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 99.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 77.1%

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

Developer Target 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    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 2024228 
(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))))