Kahan p9 Example

Percentage Accurate: 67.3% → 100.0%

Time: 9.5s

Alternatives: 12

Speedup: 2.9×

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: 67.3% 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: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

   1. associate-/l* 66.9%

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

   2. fma-def 66.9%

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

3. Simplified 66.9%

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

   1. fma-def 66.9%

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

   2. div-inv 66.7%

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

   3. add-sqr-sqrt 66.7%

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

   4. associate-*l* 66.8%

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

   5. hypot-def 66.8%

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

   6. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 97.9%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(x, y\right) \cdot \frac{1}{x + y}\right)\right)}} \]

   2. expm1-udef 97.9%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(x, y\right) \cdot \frac{1}{x + y}\right)} - 1\right)}} \]

   3. un-div-inv 98.0%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\right)} - 1\right)} \]

7. Applied egg-rr 98.0%

   \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{hypot}\left(x, y\right)}{x + y}\right)} - 1\right)}} \]
8. Step-by-step derivation

   1. expm1-def 98.0%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{hypot}\left(x, y\right)}{x + y}\right)\right)}} \]

   2. expm1-log1p 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 92.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. associate-*r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

      1. associate-*r/ 100.0%

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

      2. fma-udef 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. difference-of-squares 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. pow2 100.0%

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

      8. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   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. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-def 3.1%

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

      4. hypot-def 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 56.3%

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

      1. fma-neg 56.3%

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

      2. unpow2 56.3%

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

      3. unpow2 56.3%

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

      4. times-frac 83.3%

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

      5. unpow2 83.3%

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

      6. metadata-eval 83.3%

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

   8. Simplified 83.3%

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

4. Final simplification 94.3%

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

Alternative 3: 92.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = fma(2.0, pow((x / y), 2.0), -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 <= 2.0)
		tmp = t_0;
	else
		tmp = fma(2.0, (Float64(x / y) ^ 2.0), -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, 2.0], t$95$0, N[(2.0 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   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. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-def 3.1%

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

      4. hypot-def 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 56.3%

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

      1. fma-neg 56.3%

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

      2. unpow2 56.3%

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

      3. unpow2 56.3%

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

      4. times-frac 83.3%

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

      5. unpow2 83.3%

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

      6. metadata-eval 83.3%

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

   8. Simplified 83.3%

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

4. Final simplification 94.3%

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

Alternative 4: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

   1. add-sqr-sqrt 66.0%

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

   2. times-frac 67.0%

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

   3. hypot-def 67.0%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 5: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(x + y), $MachinePrecision] * N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

   1. add-sqr-sqrt 66.0%

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

   2. times-frac 67.0%

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

   3. hypot-def 67.0%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 6: 92.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (x - y) * ((1.0 / y) + ((x / y) / 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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (x - y) * ((1.0d0 / y) + ((x / y) / y))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   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. Step-by-step derivation

      1. associate-*r/ 3.1%

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

      2. +-commutative 3.1%

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

      3. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 57.3%

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

      1. unpow2 57.3%

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

   6. Simplified 57.3%

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

      1. *-un-lft-identity 57.3%

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

      2. times-frac 82.4%

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

   8. Applied egg-rr 82.4%

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

      1. associate-*l/ 82.4%

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

      2. *-lft-identity 82.4%

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

   10. Simplified 82.4%

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

4. Final simplification 94.0%

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

Alternative 7: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-149}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-190}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\frac{1}{y} + \frac{\frac{x}{y}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.32e-149)
   (+ -1.0 (/ (* x x) (* y y)))
   (if (<= y 3.6e-190)
     (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))
     (* (- x y) (+ (/ 1.0 y) (/ (/ x y) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.32e-149) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else if (y <= 3.6e-190) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = (x - y) * ((1.0 / y) + ((x / y) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.32d-149)) then
        tmp = (-1.0d0) + ((x * x) / (y * y))
    else if (y <= 3.6d-190) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    else
        tmp = (x - y) * ((1.0d0 / y) + ((x / y) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.32e-149) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else if (y <= 3.6e-190) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = (x - y) * ((1.0 / y) + ((x / y) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.32e-149:
		tmp = -1.0 + ((x * x) / (y * y))
	elif y <= 3.6e-190:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	else:
		tmp = (x - y) * ((1.0 / y) + ((x / y) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.32e-149)
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	elseif (y <= 3.6e-190)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 / y) + Float64(Float64(x / y) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.32e-149)
		tmp = -1.0 + ((x * x) / (y * y));
	elseif (y <= 3.6e-190)
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	else
		tmp = (x - y) * ((1.0 / y) + ((x / y) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.32e-149], N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e-190], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3200000000000001e-149

   1. Initial program 59.8%

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

      1. associate-*r/ 60.8%

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

      2. +-commutative 60.8%

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

      3. fma-def 60.8%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. unpow2 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in x around 0 93.8%

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

      1. sub-neg 93.8%

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

      2. unpow2 93.8%

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

      3. unpow2 93.8%

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

      4. metadata-eval 93.8%

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

   9. Simplified 93.8%

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

   if -1.3200000000000001e-149 < y < 3.60000000000000007e-190

   1. Initial program 58.4%

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

      1. associate-*r/ 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-def 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in y around 0 57.2%

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

      1. unpow2 57.2%

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

      2. unpow2 57.2%

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

   6. Simplified 57.2%

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

      1. times-frac 76.7%

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

   8. Applied egg-rr 76.7%

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

   if 3.60000000000000007e-190 < y

   1. Initial program 89.4%

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

      1. associate-*r/ 89.3%

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

      2. +-commutative 89.3%

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

      3. fma-def 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in x around 0 61.6%

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

      1. unpow2 61.6%

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

   6. Simplified 61.6%

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

      1. *-un-lft-identity 61.6%

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

      2. times-frac 71.8%

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

   8. Applied egg-rr 71.8%

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

      1. associate-*l/ 71.8%

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

      2. *-lft-identity 71.8%

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

   10. Simplified 71.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-149}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-190}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\frac{1}{y} + \frac{\frac{x}{y}}{y}\right)\\ \end{array} \]

Alternative 8: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-150} \lor \neg \left(y \leq 5.3 \cdot 10^{-151}\right):\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.8e-150) (not (<= y 5.3e-151)))
   (+ -1.0 (/ (* x x) (* y y)))
   (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e-150) || !(y <= 5.3e-151)) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.8d-150)) .or. (.not. (y <= 5.3d-151))) then
        tmp = (-1.0d0) + ((x * x) / (y * y))
    else
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e-150) || !(y <= 5.3e-151)) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.8e-150) or not (y <= 5.3e-151):
		tmp = -1.0 + ((x * x) / (y * y))
	else:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.8e-150) || !(y <= 5.3e-151))
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	else
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.8e-150) || ~((y <= 5.3e-151)))
		tmp = -1.0 + ((x * x) / (y * y));
	else
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.8e-150], N[Not[LessEqual[y, 5.3e-151]], $MachinePrecision]], N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8000000000000004e-150 or 5.29999999999999978e-151 < y

   1. Initial program 70.8%

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

      1. associate-*r/ 71.4%

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

      2. +-commutative 71.4%

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

      3. fma-def 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around 0 88.6%

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

      1. unpow2 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in x around 0 88.8%

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

      1. sub-neg 88.8%

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

      2. unpow2 88.8%

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

      3. unpow2 88.8%

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

      4. metadata-eval 88.8%

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

   9. Simplified 88.8%

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

   if -7.8000000000000004e-150 < y < 5.29999999999999978e-151

   1. Initial program 56.8%

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

      1. associate-*r/ 57.7%

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

      2. +-commutative 57.7%

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

      3. fma-def 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in y around 0 55.7%

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

      1. unpow2 55.7%

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

      2. unpow2 55.7%

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

   6. Simplified 55.7%

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

      1. times-frac 73.3%

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

   8. Applied egg-rr 73.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-150} \lor \neg \left(y \leq 5.3 \cdot 10^{-151}\right):\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 9: 84.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{-149} \lor \neg \left(y \leq 4.6 \cdot 10^{-149}\right):\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.85e-149) (not (<= y 4.6e-149)))
   (+ -1.0 (/ (* x x) (* y y)))
   (- 1.0 (* (/ y x) (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.85e-149) || !(y <= 4.6e-149)) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else {
		tmp = 1.0 - ((y / x) * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.85d-149)) .or. (.not. (y <= 4.6d-149))) then
        tmp = (-1.0d0) + ((x * x) / (y * y))
    else
        tmp = 1.0d0 - ((y / x) * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.85e-149) || !(y <= 4.6e-149)) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else {
		tmp = 1.0 - ((y / x) * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.85e-149) or not (y <= 4.6e-149):
		tmp = -1.0 + ((x * x) / (y * y))
	else:
		tmp = 1.0 - ((y / x) * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.85e-149) || !(y <= 4.6e-149))
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(Float64(y / x) * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.85e-149) || ~((y <= 4.6e-149)))
		tmp = -1.0 + ((x * x) / (y * y));
	else
		tmp = 1.0 - ((y / x) * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.85e-149], N[Not[LessEqual[y, 4.6e-149]], $MachinePrecision]], N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8499999999999999e-149 or 4.5999999999999999e-149 < y

   1. Initial program 70.8%

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

      1. associate-*r/ 71.4%

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

      2. +-commutative 71.4%

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

      3. fma-def 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around 0 88.6%

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

      1. unpow2 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in x around 0 88.8%

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

      1. sub-neg 88.8%

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

      2. unpow2 88.8%

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

      3. unpow2 88.8%

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

      4. metadata-eval 88.8%

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

   9. Simplified 88.8%

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

   if -2.8499999999999999e-149 < y < 4.5999999999999999e-149

   1. Initial program 56.8%

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

      1. associate-*r/ 57.7%

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

      2. +-commutative 57.7%

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

      3. fma-def 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in x around inf 56.8%

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

      1. +-commutative 56.8%

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

      2. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in y around 0 55.7%

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

      1. associate-*r/ 55.7%

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

      2. mul-1-neg 55.7%

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

      3. unpow2 55.7%

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

      4. unpow2 55.7%

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

   9. Simplified 55.7%

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

       1. distribute-lft-neg-in 55.7%

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

       2. times-frac 72.6%

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

   11. Applied egg-rr 72.6%

       \[\leadsto 1 + \color{blue}{\frac{-y}{x} \cdot \frac{y}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{-149} \lor \neg \left(y \leq 4.6 \cdot 10^{-149}\right):\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 10: 82.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-149}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e-149)
   (+ -1.0 (/ (* x x) (* y y)))
   (if (<= y 3.5e-190) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-149) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else if (y <= 3.5e-190) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.05d-149)) then
        tmp = (-1.0d0) + ((x * x) / (y * y))
    else if (y <= 3.5d-190) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-149) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else if (y <= 3.5e-190) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05e-149:
		tmp = -1.0 + ((x * x) / (y * y))
	elif y <= 3.5e-190:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e-149)
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	elseif (y <= 3.5e-190)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.05e-149)
		tmp = -1.0 + ((x * x) / (y * y));
	elseif (y <= 3.5e-190)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e-149], N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-190], 1.0, -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000005e-149

   1. Initial program 59.8%

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

      1. associate-*r/ 60.8%

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

      2. +-commutative 60.8%

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

      3. fma-def 60.8%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. unpow2 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in x around 0 93.8%

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

      1. sub-neg 93.8%

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

      2. unpow2 93.8%

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

      3. unpow2 93.8%

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

      4. metadata-eval 93.8%

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

   9. Simplified 93.8%

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

   if -1.05000000000000005e-149 < y < 3.4999999999999999e-190

   1. Initial program 58.4%

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

      1. associate-*r/ 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-def 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in x around inf 74.6%

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

   if 3.4999999999999999e-190 < y

   1. Initial program 89.4%

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

      1. associate-*r/ 89.3%

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

      2. +-commutative 89.3%

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

      3. fma-def 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in x around 0 70.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-149}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11: 82.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.8e-150) -1.0 (if (<= y 3e-190) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.8e-150) {
		tmp = -1.0;
	} else if (y <= 3e-190) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.8d-150)) then
        tmp = -1.0d0
    else if (y <= 3d-190) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.8e-150) {
		tmp = -1.0;
	} else if (y <= 3e-190) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.8e-150:
		tmp = -1.0
	elif y <= 3e-190:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.8e-150)
		tmp = -1.0;
	elseif (y <= 3e-190)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.8e-150)
		tmp = -1.0;
	elseif (y <= 3e-190)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.8e-150], -1.0, If[LessEqual[y, 3e-190], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -8.7999999999999997e-150 or 2.9999999999999998e-190 < y

   1. Initial program 69.2%

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

      1. associate-*r/ 69.9%

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

      2. +-commutative 69.9%

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

      3. fma-def 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in x around 0 86.1%

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

   if -8.7999999999999997e-150 < y < 2.9999999999999998e-190

   1. Initial program 58.4%

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

      1. associate-*r/ 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-def 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in x around inf 74.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12: 66.2% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 66.0%

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

   1. associate-*r/ 66.7%

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

   2. +-commutative 66.7%

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

   3. fma-def 66.7%

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

3. Simplified 66.7%

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

4. Taylor expanded in x around 0 67.1%

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

5. Final simplification 67.1%

   \[\leadsto -1 \]

Developer target: 99.9% accurate, 0.1× 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 2023171 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :herbie-target
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

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