Kahan p9 Example

Percentage Accurate: 67.8% → 99.9%

Time: 11.5s

Alternatives: 11

Speedup: 0.9×

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: 67.8% 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: 99.9% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ (* (- x y_m) (/ (+ x y_m) (hypot y_m x))) (hypot y_m x)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[y$95$m ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

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

   1. fma-def 62.7%

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

   2. add-sqr-sqrt 62.7%

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

   3. times-frac 63.4%

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

   4. fma-def 63.4%

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

   5. hypot-def 63.4%

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

   6. fma-def 63.4%

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

   7. hypot-def 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

   2. +-commutative 99.9%

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

   3. hypot-udef 63.4%

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

   4. +-commutative 63.4%

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

   5. hypot-def 99.9%

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

   6. hypot-udef 63.4%

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

   7. +-commutative 63.4%

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

   8. hypot-def 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* (/ (- x y_m) (hypot x y_m)) (/ (+ x y_m) (hypot x y_m))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

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

   1. fma-def 62.7%

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

   2. add-sqr-sqrt 62.7%

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

   3. times-frac 63.4%

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

   4. fma-def 63.4%

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

   5. hypot-def 63.4%

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

   6. fma-def 63.4%

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

   7. hypot-def 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.85 \cdot 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y\_m}{x}}{\frac{x}{y\_m}}, -2, 1\right)\\ \mathbf{elif}\;y\_m \leq 1.62 \cdot 10^{-167}:\\ \;\;\;\;\frac{x - y\_m}{\mathsf{hypot}\left(x, y\_m\right)} \cdot \left(\frac{x}{y\_m} + 1\right)\\ \mathbf{elif}\;y\_m \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.85e-192)
   (fma (/ (/ y_m x) (/ x y_m)) -2.0 1.0)
   (if (<= y_m 1.62e-167)
     (* (/ (- x y_m) (hypot x y_m)) (+ (/ x y_m) 1.0))
     (if (<= y_m 4e-10)
       (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
       -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.85e-192) {
		tmp = fma(((y_m / x) / (x / y_m)), -2.0, 1.0);
	} else if (y_m <= 1.62e-167) {
		tmp = ((x - y_m) / hypot(x, y_m)) * ((x / y_m) + 1.0);
	} else if (y_m <= 4e-10) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.85e-192)
		tmp = fma(Float64(Float64(y_m / x) / Float64(x / y_m)), -2.0, 1.0);
	elseif (y_m <= 1.62e-167)
		tmp = Float64(Float64(Float64(x - y_m) / hypot(x, y_m)) * Float64(Float64(x / y_m) + 1.0));
	elseif (y_m <= 4e-10)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.85e-192], N[(N[(N[(y$95$m / x), $MachinePrecision] / N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 1.62e-167], N[(N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 4e-10], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.85e-192

   1. Initial program 56.2%

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

      1. fma-def 56.2%

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

      2. add-sqr-sqrt 56.2%

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

      3. times-frac 57.1%

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

      4. fma-def 57.1%

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

      5. hypot-def 57.1%

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

      6. fma-def 57.1%

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

      7. hypot-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. hypot-udef 57.1%

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

      4. +-commutative 57.1%

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

      5. hypot-def 100.0%

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

      6. hypot-udef 57.1%

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

      7. +-commutative 57.1%

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

      8. hypot-def 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 25.5%

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

      1. +-commutative 25.5%

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

      2. *-commutative 25.5%

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

      3. fma-def 25.5%

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

      4. unpow2 25.5%

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

      5. unpow2 25.5%

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

      6. times-frac 37.3%

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

      7. unpow2 37.3%

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

   9. Simplified 37.3%

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

       1. unpow2 37.3%

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

       2. clear-num 37.3%

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

       3. un-div-inv 37.3%

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

   11. Applied egg-rr 37.3%

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

   if 1.85e-192 < y < 1.62e-167

   1. Initial program 30.0%

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

      1. fma-def 30.0%

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

      2. add-sqr-sqrt 30.0%

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

      3. times-frac 32.2%

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

      4. fma-def 32.2%

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

      5. hypot-def 32.2%

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

      6. fma-def 32.2%

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

      7. hypot-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 71.4%

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

   if 1.62e-167 < y < 4.00000000000000015e-10

   1. Initial program 98.8%

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

   if 4.00000000000000015e-10 < y

   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. +-commutative 100.0%

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

      2. associate-*r/ 99.2%

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

      3. +-commutative 99.2%

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

      4. fma-def 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{x}}{\frac{x}{y}}, -2, 1\right)\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-167}:\\ \;\;\;\;\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 6.8 \cdot 10^{-188}:\\ \;\;\;\;\left(1 + \frac{y\_m}{x} \cdot \left(1 - \frac{y\_m}{x}\right)\right) - \frac{y\_m}{x}\\ \mathbf{elif}\;y\_m \leq 2.8 \cdot 10^{-165}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{x}{y\_m}}{\frac{y\_m}{x}}, -1\right)\\ \mathbf{elif}\;y\_m \leq 10^{-5}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 6.8e-188)
   (- (+ 1.0 (* (/ y_m x) (- 1.0 (/ y_m x)))) (/ y_m x))
   (if (<= y_m 2.8e-165)
     (fma 2.0 (/ (/ x y_m) (/ y_m x)) -1.0)
     (if (<= y_m 1e-5)
       (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
       -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 6.8e-188) {
		tmp = (1.0 + ((y_m / x) * (1.0 - (y_m / x)))) - (y_m / x);
	} else if (y_m <= 2.8e-165) {
		tmp = fma(2.0, ((x / y_m) / (y_m / x)), -1.0);
	} else if (y_m <= 1e-5) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 6.8e-188)
		tmp = Float64(Float64(1.0 + Float64(Float64(y_m / x) * Float64(1.0 - Float64(y_m / x)))) - Float64(y_m / x));
	elseif (y_m <= 2.8e-165)
		tmp = fma(2.0, Float64(Float64(x / y_m) / Float64(y_m / x)), -1.0);
	elseif (y_m <= 1e-5)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 6.8e-188], N[(N[(1.0 + N[(N[(y$95$m / x), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 2.8e-165], N[(2.0 * N[(N[(x / y$95$m), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y$95$m, 1e-5], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.80000000000000055e-188

   1. Initial program 56.1%

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

      1. fma-def 56.1%

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

      2. add-sqr-sqrt 56.1%

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

      3. times-frac 57.1%

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

      4. fma-def 57.1%

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

      5. hypot-def 57.1%

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

      6. fma-def 57.1%

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

      7. hypot-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 37.6%

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

      1. mul-1-neg 37.6%

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

      2. unsub-neg 37.6%

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

   7. Simplified 37.6%

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

   8. Taylor expanded in x around inf 37.1%

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

      1. +-commutative 37.1%

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

   10. Simplified 37.1%

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

       1. distribute-rgt-in 36.3%

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

       2. *-un-lft-identity 36.3%

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

       3. associate-+r- 36.3%

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

   12. Applied egg-rr 36.3%

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

   if 6.80000000000000055e-188 < y < 2.7999999999999999e-165

   1. Initial program 25.0%

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

      1. fma-def 25.0%

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

      2. add-sqr-sqrt 25.0%

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

      3. times-frac 27.3%

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

      4. fma-def 27.3%

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

      5. hypot-def 27.3%

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

      6. fma-def 27.3%

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

      7. hypot-def 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. hypot-udef 27.3%

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

      4. +-commutative 27.3%

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

      5. hypot-def 99.8%

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

      6. hypot-udef 27.3%

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

      7. +-commutative 27.3%

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

      8. hypot-def 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 0.8%

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

      1. fma-neg 0.8%

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

      2. unpow2 0.8%

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

      3. unpow2 0.8%

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

      4. times-frac 76.2%

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

      5. unpow2 76.2%

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

      6. metadata-eval 76.2%

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

   9. Simplified 76.2%

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

       1. unpow2 76.2%

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

       2. clear-num 76.2%

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

       3. un-div-inv 76.2%

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

   11. Applied egg-rr 76.2%

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

   if 2.7999999999999999e-165 < y < 1.00000000000000008e-5

   1. Initial program 98.8%

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

   if 1.00000000000000008e-5 < y

   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. +-commutative 100.0%

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

      2. associate-*r/ 99.2%

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

      3. +-commutative 99.2%

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

      4. fma-def 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-188}:\\ \;\;\;\;\left(1 + \frac{y}{x} \cdot \left(1 - \frac{y}{x}\right)\right) - \frac{y}{x}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-165}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{x}{y}}{\frac{y}{x}}, -1\right)\\ \mathbf{elif}\;y \leq 10^{-5}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 7.2 \cdot 10^{-188}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y\_m}{x}}{\frac{x}{y\_m}}, -2, 1\right)\\ \mathbf{elif}\;y\_m \leq 1.5 \cdot 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{x}{y\_m}}{\frac{y\_m}{x}}, -1\right)\\ \mathbf{elif}\;y\_m \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 7.2e-188)
   (fma (/ (/ y_m x) (/ x y_m)) -2.0 1.0)
   (if (<= y_m 1.5e-167)
     (fma 2.0 (/ (/ x y_m) (/ y_m x)) -1.0)
     (if (<= y_m 2e-6)
       (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
       -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 7.2e-188) {
		tmp = fma(((y_m / x) / (x / y_m)), -2.0, 1.0);
	} else if (y_m <= 1.5e-167) {
		tmp = fma(2.0, ((x / y_m) / (y_m / x)), -1.0);
	} else if (y_m <= 2e-6) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 7.2e-188)
		tmp = fma(Float64(Float64(y_m / x) / Float64(x / y_m)), -2.0, 1.0);
	elseif (y_m <= 1.5e-167)
		tmp = fma(2.0, Float64(Float64(x / y_m) / Float64(y_m / x)), -1.0);
	elseif (y_m <= 2e-6)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 7.2e-188], N[(N[(N[(y$95$m / x), $MachinePrecision] / N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 1.5e-167], N[(2.0 * N[(N[(x / y$95$m), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y$95$m, 2e-6], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 7.1999999999999994e-188

   1. Initial program 56.1%

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

      1. fma-def 56.1%

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

      2. add-sqr-sqrt 56.1%

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

      3. times-frac 57.1%

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

      4. fma-def 57.1%

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

      5. hypot-def 57.1%

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

      6. fma-def 57.1%

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

      7. hypot-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. hypot-udef 57.1%

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

      4. +-commutative 57.1%

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

      5. hypot-def 100.0%

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

      6. hypot-udef 57.1%

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

      7. +-commutative 57.1%

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

      8. hypot-def 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 25.7%

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

      1. +-commutative 25.7%

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

      2. *-commutative 25.7%

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

      3. fma-def 25.7%

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

      4. unpow2 25.7%

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

      5. unpow2 25.7%

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

      6. times-frac 37.5%

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

      7. unpow2 37.5%

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

   9. Simplified 37.5%

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

       1. unpow2 37.5%

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

       2. clear-num 37.5%

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

       3. un-div-inv 37.5%

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

   11. Applied egg-rr 37.5%

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

   if 7.1999999999999994e-188 < y < 1.4999999999999999e-167

   1. Initial program 25.0%

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

      1. fma-def 25.0%

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

      2. add-sqr-sqrt 25.0%

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

      3. times-frac 27.3%

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

      4. fma-def 27.3%

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

      5. hypot-def 27.3%

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

      6. fma-def 27.3%

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

      7. hypot-def 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. hypot-udef 27.3%

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

      4. +-commutative 27.3%

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

      5. hypot-def 99.8%

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

      6. hypot-udef 27.3%

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

      7. +-commutative 27.3%

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

      8. hypot-def 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 0.8%

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

      1. fma-neg 0.8%

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

      2. unpow2 0.8%

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

      3. unpow2 0.8%

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

      4. times-frac 76.2%

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

      5. unpow2 76.2%

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

      6. metadata-eval 76.2%

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

   9. Simplified 76.2%

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

       1. unpow2 76.2%

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

       2. clear-num 76.2%

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

       3. un-div-inv 76.2%

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

   11. Applied egg-rr 76.2%

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

   if 1.4999999999999999e-167 < y < 1.99999999999999991e-6

   1. Initial program 98.8%

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

   if 1.99999999999999991e-6 < y

   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. +-commutative 100.0%

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

      2. associate-*r/ 99.2%

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

      3. +-commutative 99.2%

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

      4. fma-def 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-188}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{x}}{\frac{x}{y}}, -2, 1\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{x}{y}}{\frac{y}{x}}, -1\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 7.2 \cdot 10^{-188} \lor \neg \left(y\_m \leq 5.7 \cdot 10^{-173}\right) \land y\_m \leq 1.92 \cdot 10^{-143}:\\ \;\;\;\;\left(1 + \frac{y\_m}{x} \cdot \left(1 - \frac{y\_m}{x}\right)\right) - \frac{y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} + 1\right) \cdot \left(\frac{x}{y\_m} + -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 7.2e-188) (and (not (<= y_m 5.7e-173)) (<= y_m 1.92e-143)))
   (- (+ 1.0 (* (/ y_m x) (- 1.0 (/ y_m x)))) (/ y_m x))
   (* (+ (/ x y_m) 1.0) (+ (/ x y_m) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 7.2e-188) || (!(y_m <= 5.7e-173) && (y_m <= 1.92e-143))) {
		tmp = (1.0 + ((y_m / x) * (1.0 - (y_m / x)))) - (y_m / x);
	} else {
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m <= 7.2d-188) .or. (.not. (y_m <= 5.7d-173)) .and. (y_m <= 1.92d-143)) then
        tmp = (1.0d0 + ((y_m / x) * (1.0d0 - (y_m / x)))) - (y_m / x)
    else
        tmp = ((x / y_m) + 1.0d0) * ((x / y_m) + (-1.0d0))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 7.2e-188) || (!(y_m <= 5.7e-173) && (y_m <= 1.92e-143))) {
		tmp = (1.0 + ((y_m / x) * (1.0 - (y_m / x)))) - (y_m / x);
	} else {
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m <= 7.2e-188) or (not (y_m <= 5.7e-173) and (y_m <= 1.92e-143)):
		tmp = (1.0 + ((y_m / x) * (1.0 - (y_m / x)))) - (y_m / x)
	else:
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m <= 7.2e-188) || (!(y_m <= 5.7e-173) && (y_m <= 1.92e-143)))
		tmp = Float64(Float64(1.0 + Float64(Float64(y_m / x) * Float64(1.0 - Float64(y_m / x)))) - Float64(y_m / x));
	else
		tmp = Float64(Float64(Float64(x / y_m) + 1.0) * Float64(Float64(x / y_m) + -1.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m <= 7.2e-188) || (~((y_m <= 5.7e-173)) && (y_m <= 1.92e-143)))
		tmp = (1.0 + ((y_m / x) * (1.0 - (y_m / x)))) - (y_m / x);
	else
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[Or[LessEqual[y$95$m, 7.2e-188], And[N[Not[LessEqual[y$95$m, 5.7e-173]], $MachinePrecision], LessEqual[y$95$m, 1.92e-143]]], N[(N[(1.0 + N[(N[(y$95$m / x), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.1999999999999994e-188 or 5.7000000000000001e-173 < y < 1.91999999999999995e-143

   1. Initial program 56.9%

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

      1. fma-def 56.9%

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

      2. add-sqr-sqrt 56.9%

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

      3. times-frac 57.7%

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

      4. fma-def 57.7%

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

      5. hypot-def 57.8%

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

      6. fma-def 57.8%

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

      7. hypot-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 38.6%

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

      1. mul-1-neg 38.6%

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

      2. unsub-neg 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in x around inf 38.1%

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

      1. +-commutative 38.1%

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

   10. Simplified 38.1%

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

       1. distribute-rgt-in 37.3%

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

       2. *-un-lft-identity 37.3%

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

       3. associate-+r- 37.3%

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

   12. Applied egg-rr 37.3%

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

   if 7.1999999999999994e-188 < y < 5.7000000000000001e-173 or 1.91999999999999995e-143 < y

   1. Initial program 89.0%

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

      1. fma-def 89.0%

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

      2. add-sqr-sqrt 88.9%

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

      3. times-frac 89.3%

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

      4. fma-def 89.3%

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

      5. hypot-def 89.3%

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

      6. fma-def 89.3%

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

      7. hypot-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 72.6%

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

   6. Taylor expanded in x around 0 72.1%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-188} \lor \neg \left(y \leq 5.7 \cdot 10^{-173}\right) \land y \leq 1.92 \cdot 10^{-143}:\\ \;\;\;\;\left(1 + \frac{y}{x} \cdot \left(1 - \frac{y}{x}\right)\right) - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 7.2 \cdot 10^{-188}:\\ \;\;\;\;\left(1 + \frac{y\_m}{x} \cdot \left(1 - \frac{y\_m}{x}\right)\right) - \frac{y\_m}{x}\\ \mathbf{elif}\;y\_m \leq 1.48 \cdot 10^{-170}:\\ \;\;\;\;\left(\frac{x}{y\_m} + 1\right) \cdot \left(\frac{x}{y\_m} + -1\right)\\ \mathbf{elif}\;y\_m \leq 10^{-7}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 7.2e-188)
   (- (+ 1.0 (* (/ y_m x) (- 1.0 (/ y_m x)))) (/ y_m x))
   (if (<= y_m 1.48e-170)
     (* (+ (/ x y_m) 1.0) (+ (/ x y_m) -1.0))
     (if (<= y_m 1e-7)
       (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
       -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 7.2e-188) {
		tmp = (1.0 + ((y_m / x) * (1.0 - (y_m / x)))) - (y_m / x);
	} else if (y_m <= 1.48e-170) {
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0);
	} else if (y_m <= 1e-7) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 7.2d-188) then
        tmp = (1.0d0 + ((y_m / x) * (1.0d0 - (y_m / x)))) - (y_m / x)
    else if (y_m <= 1.48d-170) then
        tmp = ((x / y_m) + 1.0d0) * ((x / y_m) + (-1.0d0))
    else if (y_m <= 1d-7) then
        tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 7.2e-188) {
		tmp = (1.0 + ((y_m / x) * (1.0 - (y_m / x)))) - (y_m / x);
	} else if (y_m <= 1.48e-170) {
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0);
	} else if (y_m <= 1e-7) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 7.2e-188:
		tmp = (1.0 + ((y_m / x) * (1.0 - (y_m / x)))) - (y_m / x)
	elif y_m <= 1.48e-170:
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0)
	elif y_m <= 1e-7:
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 7.2e-188)
		tmp = Float64(Float64(1.0 + Float64(Float64(y_m / x) * Float64(1.0 - Float64(y_m / x)))) - Float64(y_m / x));
	elseif (y_m <= 1.48e-170)
		tmp = Float64(Float64(Float64(x / y_m) + 1.0) * Float64(Float64(x / y_m) + -1.0));
	elseif (y_m <= 1e-7)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 7.2e-188)
		tmp = (1.0 + ((y_m / x) * (1.0 - (y_m / x)))) - (y_m / x);
	elseif (y_m <= 1.48e-170)
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0);
	elseif (y_m <= 1e-7)
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 7.2e-188], N[(N[(1.0 + N[(N[(y$95$m / x), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.48e-170], N[(N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1e-7], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 7.1999999999999994e-188

   1. Initial program 56.1%

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

      1. fma-def 56.1%

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

      2. add-sqr-sqrt 56.1%

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

      3. times-frac 57.1%

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

      4. fma-def 57.1%

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

      5. hypot-def 57.1%

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

      6. fma-def 57.1%

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

      7. hypot-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 37.6%

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

      1. mul-1-neg 37.6%

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

      2. unsub-neg 37.6%

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

   7. Simplified 37.6%

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

   8. Taylor expanded in x around inf 37.1%

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

      1. +-commutative 37.1%

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

   10. Simplified 37.1%

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

       1. distribute-rgt-in 36.3%

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

       2. *-un-lft-identity 36.3%

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

       3. associate-+r- 36.3%

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

   12. Applied egg-rr 36.3%

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

   if 7.1999999999999994e-188 < y < 1.48000000000000005e-170

   1. Initial program 25.0%

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

      1. fma-def 25.0%

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

      2. add-sqr-sqrt 25.0%

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

      3. times-frac 27.3%

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

      4. fma-def 27.3%

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

      5. hypot-def 27.3%

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

      6. fma-def 27.3%

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

      7. hypot-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 76.0%

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

   6. Taylor expanded in x around 0 75.6%

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

   if 1.48000000000000005e-170 < y < 9.9999999999999995e-8

   1. Initial program 98.8%

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

   if 9.9999999999999995e-8 < y

   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. +-commutative 100.0%

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

      2. associate-*r/ 99.2%

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

      3. +-commutative 99.2%

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

      4. fma-def 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-188}:\\ \;\;\;\;\left(1 + \frac{y}{x} \cdot \left(1 - \frac{y}{x}\right)\right) - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.48 \cdot 10^{-170}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{elif}\;y \leq 10^{-7}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 7.2 \cdot 10^{-188} \lor \neg \left(y\_m \leq 6.5 \cdot 10^{-174}\right) \land y\_m \leq 1.6 \cdot 10^{-143}:\\ \;\;\;\;\left(1 - \frac{y\_m}{x}\right) \cdot \left(\frac{y\_m}{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 7.2e-188) (and (not (<= y_m 6.5e-174)) (<= y_m 1.6e-143)))
   (* (- 1.0 (/ y_m x)) (+ (/ y_m x) 1.0))
   -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 7.2e-188) || (!(y_m <= 6.5e-174) && (y_m <= 1.6e-143))) {
		tmp = (1.0 - (y_m / x)) * ((y_m / x) + 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m <= 7.2d-188) .or. (.not. (y_m <= 6.5d-174)) .and. (y_m <= 1.6d-143)) then
        tmp = (1.0d0 - (y_m / x)) * ((y_m / x) + 1.0d0)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 7.2e-188) || (!(y_m <= 6.5e-174) && (y_m <= 1.6e-143))) {
		tmp = (1.0 - (y_m / x)) * ((y_m / x) + 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m <= 7.2e-188) or (not (y_m <= 6.5e-174) and (y_m <= 1.6e-143)):
		tmp = (1.0 - (y_m / x)) * ((y_m / x) + 1.0)
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m <= 7.2e-188) || (!(y_m <= 6.5e-174) && (y_m <= 1.6e-143)))
		tmp = Float64(Float64(1.0 - Float64(y_m / x)) * Float64(Float64(y_m / x) + 1.0));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m <= 7.2e-188) || (~((y_m <= 6.5e-174)) && (y_m <= 1.6e-143)))
		tmp = (1.0 - (y_m / x)) * ((y_m / x) + 1.0);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[Or[LessEqual[y$95$m, 7.2e-188], And[N[Not[LessEqual[y$95$m, 6.5e-174]], $MachinePrecision], LessEqual[y$95$m, 1.6e-143]]], N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 7.1999999999999994e-188 or 6.50000000000000009e-174 < y < 1.5999999999999999e-143

   1. Initial program 56.9%

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

      1. fma-def 56.9%

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

      2. add-sqr-sqrt 56.9%

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

      3. times-frac 57.7%

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

      4. fma-def 57.7%

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

      5. hypot-def 57.8%

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

      6. fma-def 57.8%

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

      7. hypot-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 38.6%

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

      1. mul-1-neg 38.6%

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

      2. unsub-neg 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in x around inf 38.1%

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

      1. +-commutative 38.1%

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

   10. Simplified 38.1%

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

   if 7.1999999999999994e-188 < y < 6.50000000000000009e-174 or 1.5999999999999999e-143 < y

   1. Initial program 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-*r/ 89.0%

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

      3. +-commutative 89.0%

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

      4. fma-def 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 70.4%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-188} \lor \neg \left(y \leq 6.5 \cdot 10^{-174}\right) \land y \leq 1.6 \cdot 10^{-143}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(\frac{y}{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 7.2 \cdot 10^{-188} \lor \neg \left(y\_m \leq 7.2 \cdot 10^{-174}\right) \land y\_m \leq 1.35 \cdot 10^{-143}:\\ \;\;\;\;\left(1 - \frac{y\_m}{x}\right) \cdot \left(\frac{y\_m}{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} + 1\right) \cdot \left(\frac{x}{y\_m} + -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 7.2e-188) (and (not (<= y_m 7.2e-174)) (<= y_m 1.35e-143)))
   (* (- 1.0 (/ y_m x)) (+ (/ y_m x) 1.0))
   (* (+ (/ x y_m) 1.0) (+ (/ x y_m) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 7.2e-188) || (!(y_m <= 7.2e-174) && (y_m <= 1.35e-143))) {
		tmp = (1.0 - (y_m / x)) * ((y_m / x) + 1.0);
	} else {
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m <= 7.2d-188) .or. (.not. (y_m <= 7.2d-174)) .and. (y_m <= 1.35d-143)) then
        tmp = (1.0d0 - (y_m / x)) * ((y_m / x) + 1.0d0)
    else
        tmp = ((x / y_m) + 1.0d0) * ((x / y_m) + (-1.0d0))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 7.2e-188) || (!(y_m <= 7.2e-174) && (y_m <= 1.35e-143))) {
		tmp = (1.0 - (y_m / x)) * ((y_m / x) + 1.0);
	} else {
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m <= 7.2e-188) or (not (y_m <= 7.2e-174) and (y_m <= 1.35e-143)):
		tmp = (1.0 - (y_m / x)) * ((y_m / x) + 1.0)
	else:
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m <= 7.2e-188) || (!(y_m <= 7.2e-174) && (y_m <= 1.35e-143)))
		tmp = Float64(Float64(1.0 - Float64(y_m / x)) * Float64(Float64(y_m / x) + 1.0));
	else
		tmp = Float64(Float64(Float64(x / y_m) + 1.0) * Float64(Float64(x / y_m) + -1.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m <= 7.2e-188) || (~((y_m <= 7.2e-174)) && (y_m <= 1.35e-143)))
		tmp = (1.0 - (y_m / x)) * ((y_m / x) + 1.0);
	else
		tmp = ((x / y_m) + 1.0) * ((x / y_m) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[Or[LessEqual[y$95$m, 7.2e-188], And[N[Not[LessEqual[y$95$m, 7.2e-174]], $MachinePrecision], LessEqual[y$95$m, 1.35e-143]]], N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.1999999999999994e-188 or 7.19999999999999997e-174 < y < 1.35000000000000005e-143

   1. Initial program 56.9%

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

      1. fma-def 56.9%

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

      2. add-sqr-sqrt 56.9%

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

      3. times-frac 57.7%

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

      4. fma-def 57.7%

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

      5. hypot-def 57.8%

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

      6. fma-def 57.8%

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

      7. hypot-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 38.6%

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

      1. mul-1-neg 38.6%

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

      2. unsub-neg 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in x around inf 38.1%

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

      1. +-commutative 38.1%

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

   10. Simplified 38.1%

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

   if 7.1999999999999994e-188 < y < 7.19999999999999997e-174 or 1.35000000000000005e-143 < y

   1. Initial program 89.0%

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

      1. fma-def 89.0%

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

      2. add-sqr-sqrt 88.9%

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

      3. times-frac 89.3%

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

      4. fma-def 89.3%

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

      5. hypot-def 89.3%

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

      6. fma-def 89.3%

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

      7. hypot-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 72.6%

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

   6. Taylor expanded in x around 0 72.1%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-188} \lor \neg \left(y \leq 7.2 \cdot 10^{-174}\right) \land y \leq 1.35 \cdot 10^{-143}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(\frac{y}{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;y\_m \leq 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y\_m \leq 2.25 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 6.2e-188)
   1.0
   (if (<= y_m 1e-173) -1.0 (if (<= y_m 2.25e-143) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 6.2e-188) {
		tmp = 1.0;
	} else if (y_m <= 1e-173) {
		tmp = -1.0;
	} else if (y_m <= 2.25e-143) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 6.2d-188) then
        tmp = 1.0d0
    else if (y_m <= 1d-173) then
        tmp = -1.0d0
    else if (y_m <= 2.25d-143) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 6.2e-188) {
		tmp = 1.0;
	} else if (y_m <= 1e-173) {
		tmp = -1.0;
	} else if (y_m <= 2.25e-143) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 6.2e-188:
		tmp = 1.0
	elif y_m <= 1e-173:
		tmp = -1.0
	elif y_m <= 2.25e-143:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 6.2e-188)
		tmp = 1.0;
	elseif (y_m <= 1e-173)
		tmp = -1.0;
	elseif (y_m <= 2.25e-143)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 6.2e-188)
		tmp = 1.0;
	elseif (y_m <= 1e-173)
		tmp = -1.0;
	elseif (y_m <= 2.25e-143)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 6.2e-188], 1.0, If[LessEqual[y$95$m, 1e-173], -1.0, If[LessEqual[y$95$m, 2.25e-143], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.2000000000000004e-188 or 1e-173 < y < 2.25e-143

   1. Initial program 56.9%

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

      1. +-commutative 56.9%

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

      2. associate-*r/ 57.5%

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

      3. +-commutative 57.5%

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

      4. fma-def 57.5%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in x around inf 36.5%

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

   if 6.2000000000000004e-188 < y < 1e-173 or 2.25e-143 < y

   1. Initial program 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-*r/ 89.0%

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

      3. +-commutative 89.0%

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

      4. fma-def 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 70.4%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.2% accurate, 15.0× 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 = abs(y)
real(8) function code(x, y_m)
    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 62.7%

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

   1. +-commutative 62.7%

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

   2. associate-*r/ 63.2%

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

   3. +-commutative 63.2%

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

   4. fma-def 63.2%

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

3. Simplified 63.2%

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

5. Taylor expanded in x around 0 64.6%

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

6. Final simplification 64.6%

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

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 2024036 
(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))))