Kahan p9 Example

Percentage Accurate: 68.6% → 100.0%

Time: 8.3s

Alternatives: 10

Speedup: 2.1×

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: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (/ (* (/ (- x y) (hypot x y)) (+ x y)) (hypot x y)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. add-sqr-sqrt 62.5%

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

   2. times-frac 63.2%

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

   3. hypot-def 63.2%

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

   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/ 100.0%

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

   2. +-commutative 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 93.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{x}{y} \cdot 1.5\right) - y}{\mathsf{hypot}\left(x, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (/ (- (* x (* (/ x y) 1.5)) y) (hypot x y)))))

C

y = abs(y);
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 * ((x / y) * 1.5)) - y) / hypot(x, y);
	}
	return tmp;
}

Java

y = Math.abs(y);
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 * ((x / y) * 1.5)) - y) / Math.hypot(x, y);
	}
	return tmp;
}

Python

y = abs(y)
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 * ((x / y) * 1.5)) - y) / math.hypot(x, y)
	return tmp

Julia

y = abs(y)
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(Float64(x * Float64(Float64(x / y) * 1.5)) - y) / hypot(x, y));
	end
	return tmp
end

MATLAB

y = abs(y)
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 * ((x / y) * 1.5)) - y) / hypot(x, y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
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[(N[(x * N[(N[(x / y), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $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. 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 y around inf 14.2%

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot x + \left(-1 \cdot y + \left(x + \frac{{x}^{2}}{y}\right)\right)\right) - -0.5 \cdot \frac{{x}^{2}}{y}}}{\mathsf{hypot}\left(x, y\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 14.2%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x + \left(-1 \cdot y + \left(x + \frac{{x}^{2}}{y}\right)\right)\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}}{\mathsf{hypot}\left(x, y\right)} \]

      2. associate-+r+ 14.2%

         \[\leadsto \frac{\left(-1 \cdot x + \color{blue}{\left(\left(-1 \cdot y + x\right) + \frac{{x}^{2}}{y}\right)}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      3. +-commutative 14.2%

         \[\leadsto \frac{\left(-1 \cdot x + \left(\color{blue}{\left(x + -1 \cdot y\right)} + \frac{{x}^{2}}{y}\right)\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      4. neg-mul-1 14.2%

         \[\leadsto \frac{\left(-1 \cdot x + \left(\left(x + \color{blue}{\left(-y\right)}\right) + \frac{{x}^{2}}{y}\right)\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      5. sub-neg 14.2%

         \[\leadsto \frac{\left(-1 \cdot x + \left(\color{blue}{\left(x - y\right)} + \frac{{x}^{2}}{y}\right)\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      6. associate-+r+ 14.2%

         \[\leadsto \frac{\color{blue}{\left(\left(-1 \cdot x + \left(x - y\right)\right) + \frac{{x}^{2}}{y}\right)} + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      7. sub-neg 14.2%

         \[\leadsto \frac{\left(\left(-1 \cdot x + \color{blue}{\left(x + \left(-y\right)\right)}\right) + \frac{{x}^{2}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      8. neg-mul-1 14.2%

         \[\leadsto \frac{\left(\left(-1 \cdot x + \left(x + \color{blue}{-1 \cdot y}\right)\right) + \frac{{x}^{2}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      9. associate-+r+ 14.4%

         \[\leadsto \frac{\left(\color{blue}{\left(\left(-1 \cdot x + x\right) + -1 \cdot y\right)} + \frac{{x}^{2}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      10. distribute-lft1-in 14.4%

          \[\leadsto \frac{\left(\left(\color{blue}{\left(-1 + 1\right) \cdot x} + -1 \cdot y\right) + \frac{{x}^{2}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      11. metadata-eval 14.4%

          \[\leadsto \frac{\left(\left(\color{blue}{0} \cdot x + -1 \cdot y\right) + \frac{{x}^{2}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      12. mul0-lft 14.4%

          \[\leadsto \frac{\left(\left(\color{blue}{0} + -1 \cdot y\right) + \frac{{x}^{2}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      13. neg-mul-1 14.4%

          \[\leadsto \frac{\left(\left(0 + \color{blue}{\left(-y\right)}\right) + \frac{{x}^{2}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      14. sub-neg 14.4%

          \[\leadsto \frac{\left(\color{blue}{\left(0 - y\right)} + \frac{{x}^{2}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      15. neg-sub0 14.4%

          \[\leadsto \frac{\left(\color{blue}{\left(-y\right)} + \frac{{x}^{2}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      16. unpow2 14.4%

          \[\leadsto \frac{\left(\left(-y\right) + \frac{\color{blue}{x \cdot x}}{y}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      17. associate-/l* 14.5%

          \[\leadsto \frac{\left(\left(-y\right) + \color{blue}{\frac{x}{\frac{y}{x}}}\right) + \left(--0.5\right) \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      18. metadata-eval 14.5%

          \[\leadsto \frac{\left(\left(-y\right) + \frac{x}{\frac{y}{x}}\right) + \color{blue}{0.5} \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      19. unpow2 14.5%

          \[\leadsto \frac{\left(\left(-y\right) + \frac{x}{\frac{y}{x}}\right) + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

      20. associate-/l* 14.6%

          \[\leadsto \frac{\left(\left(-y\right) + \frac{x}{\frac{y}{x}}\right) + 0.5 \cdot \color{blue}{\frac{x}{\frac{y}{x}}}}{\mathsf{hypot}\left(x, y\right)} \]

   8. Simplified 14.6%

      \[\leadsto \frac{\color{blue}{\left(\left(-y\right) + \frac{x}{\frac{y}{x}}\right) + 0.5 \cdot \frac{x}{\frac{y}{x}}}}{\mathsf{hypot}\left(x, y\right)} \]

   9. Taylor expanded in y around 0 14.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot y + \left(\frac{{x}^{2}}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)}}{\mathsf{hypot}\left(x, y\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt1-in 14.4%

          \[\leadsto \frac{-1 \cdot y + \color{blue}{\left(0.5 + 1\right) \cdot \frac{{x}^{2}}{y}}}{\mathsf{hypot}\left(x, y\right)} \]

       2. metadata-eval 14.4%

          \[\leadsto \frac{-1 \cdot y + \color{blue}{1.5} \cdot \frac{{x}^{2}}{y}}{\mathsf{hypot}\left(x, y\right)} \]

       3. +-commutative 14.4%

          \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{x}^{2}}{y} + -1 \cdot y}}{\mathsf{hypot}\left(x, y\right)} \]

       4. metadata-eval 14.4%

          \[\leadsto \frac{\color{blue}{\left(0.5 + 1\right)} \cdot \frac{{x}^{2}}{y} + -1 \cdot y}{\mathsf{hypot}\left(x, y\right)} \]

       5. unpow2 14.4%

          \[\leadsto \frac{\left(0.5 + 1\right) \cdot \frac{\color{blue}{x \cdot x}}{y} + -1 \cdot y}{\mathsf{hypot}\left(x, y\right)} \]

       6. associate-*r/ 14.6%

          \[\leadsto \frac{\left(0.5 + 1\right) \cdot \color{blue}{\left(x \cdot \frac{x}{y}\right)} + -1 \cdot y}{\mathsf{hypot}\left(x, y\right)} \]

       7. distribute-rgt1-in 14.6%

          \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{x}{y} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right)} + -1 \cdot y}{\mathsf{hypot}\left(x, y\right)} \]

       8. *-commutative 14.6%

          \[\leadsto \frac{\left(x \cdot \frac{x}{y} + \color{blue}{\left(x \cdot \frac{x}{y}\right) \cdot 0.5}\right) + -1 \cdot y}{\mathsf{hypot}\left(x, y\right)} \]

       9. fma-udef 14.6%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{x}{y}, \left(x \cdot \frac{x}{y}\right) \cdot 0.5\right)} + -1 \cdot y}{\mathsf{hypot}\left(x, y\right)} \]

       10. mul-1-neg 14.6%

           \[\leadsto \frac{\mathsf{fma}\left(x, \frac{x}{y}, \left(x \cdot \frac{x}{y}\right) \cdot 0.5\right) + \color{blue}{\left(-y\right)}}{\mathsf{hypot}\left(x, y\right)} \]

       11. unsub-neg 14.6%

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{x}{y}, \left(x \cdot \frac{x}{y}\right) \cdot 0.5\right) - y}}{\mathsf{hypot}\left(x, y\right)} \]

       12. fma-udef 14.6%

           \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{x}{y} + \left(x \cdot \frac{x}{y}\right) \cdot 0.5\right)} - y}{\mathsf{hypot}\left(x, y\right)} \]

       13. *-commutative 14.6%

           \[\leadsto \frac{\left(x \cdot \frac{x}{y} + \color{blue}{0.5 \cdot \left(x \cdot \frac{x}{y}\right)}\right) - y}{\mathsf{hypot}\left(x, y\right)} \]

       14. *-lft-identity 14.6%

           \[\leadsto \frac{\left(\color{blue}{1 \cdot \left(x \cdot \frac{x}{y}\right)} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right) - y}{\mathsf{hypot}\left(x, y\right)} \]

       15. distribute-rgt-out 14.6%

           \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{x}{y}\right) \cdot \left(1 + 0.5\right)} - y}{\mathsf{hypot}\left(x, y\right)} \]

       16. metadata-eval 14.6%

           \[\leadsto \frac{\left(x \cdot \frac{x}{y}\right) \cdot \color{blue}{1.5} - y}{\mathsf{hypot}\left(x, y\right)} \]

       17. associate-*l* 14.6%

           \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{x}{y} \cdot 1.5\right)} - y}{\mathsf{hypot}\left(x, y\right)} \]

   11. Simplified 14.6%

       \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{x}{y} \cdot 1.5\right) - y}}{\mathsf{hypot}\left(x, y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{x}{y} \cdot 1.5\right) - y}{\mathsf{hypot}\left(x, y\right)}\\ \end{array} \]

Alternative 3: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ t_1 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(t_1 + -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))
        (t_1 (* (/ x y) (/ x y))))
   (if (<= t_0 2.0) t_0 (+ t_1 (+ t_1 -1.0)))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double t_1 = (x / y) * (x / y);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (t_1 + -1.0);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    t_1 = (x / y) * (x / y)
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = t_1 + (t_1 + (-1.0d0))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double t_1 = (x / y) * (x / y);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (t_1 + -1.0);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	t_1 = (x / y) * (x / y)
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = t_1 + (t_1 + -1.0)
	return tmp

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	t_1 = Float64(Float64(x / y) * Float64(x / y))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(t_1 + -1.0));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	t_1 = (x / y) * (x / y);
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = t_1 + (t_1 + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
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]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(t$95$1 + N[(t$95$1 + -1.0), $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. 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 y around inf 53.1%

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

      1. associate--l+ 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

      4. times-frac 53.1%

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

      5. distribute-rgt1-in 53.1%

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

      6. metadata-eval 53.1%

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

      7. mul0-lft 53.1%

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

      8. neg-sub0 53.1%

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

      9. +-commutative 53.1%

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

      10. mul-1-neg 53.1%

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

      11. unsub-neg 53.1%

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

      12. unpow2 53.1%

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

      13. unpow2 53.1%

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

      14. times-frac 81.7%

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

   8. Simplified 81.7%

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

4. Final simplification 93.1%

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

Alternative 4: 92.9% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(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) (/ x y)) -1.0))))

C

y = abs(y);
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) * (x / y)) + -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
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) * (x / y)) + -1.0;
	}
	return tmp;
}

Python

y = abs(y)
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) * (x / y)) + -1.0
	return tmp

Julia

y = abs(y)
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(Float64(x / y) * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

y = abs(y)
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) * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
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[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $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. Taylor expanded in x around 0 0.0%

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

      1. unpow2 0.0%

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

   4. Simplified 0.0%

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

   5. Taylor expanded in x around 0 53.1%

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

      1. sub-neg 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

      4. times-frac 81.1%

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

      5. metadata-eval 81.1%

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

   7. Simplified 81.1%

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

4. Final simplification 92.9%

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

Alternative 5: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-180}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-131}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y} \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e-180)
   1.0
   (if (<= y 9.2e-150)
     (+ (* (/ x y) (/ x y)) -1.0)
     (if (<= y 5.5e-131)
       (+ 1.0 (* -2.0 (/ (* y y) (* x x))))
       (* (/ (+ x y) y) (+ (/ x y) -1.0))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-180) {
		tmp = 1.0;
	} else if (y <= 9.2e-150) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else if (y <= 5.5e-131) {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	} else {
		tmp = ((x + y) / y) * ((x / y) + -1.0);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-180) then
        tmp = 1.0d0
    else if (y <= 9.2d-150) then
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    else if (y <= 5.5d-131) then
        tmp = 1.0d0 + ((-2.0d0) * ((y * y) / (x * x)))
    else
        tmp = ((x + y) / y) * ((x / y) + (-1.0d0))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-180) {
		tmp = 1.0;
	} else if (y <= 9.2e-150) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else if (y <= 5.5e-131) {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	} else {
		tmp = ((x + y) / y) * ((x / y) + -1.0);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.3e-180:
		tmp = 1.0
	elif y <= 9.2e-150:
		tmp = ((x / y) * (x / y)) + -1.0
	elif y <= 5.5e-131:
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)))
	else:
		tmp = ((x + y) / y) * ((x / y) + -1.0)
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-180)
		tmp = 1.0;
	elseif (y <= 9.2e-150)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0);
	elseif (y <= 5.5e-131)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y * y) / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(x + y) / y) * Float64(Float64(x / y) + -1.0));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-180)
		tmp = 1.0;
	elseif (y <= 9.2e-150)
		tmp = ((x / y) * (x / y)) + -1.0;
	elseif (y <= 5.5e-131)
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	else
		tmp = ((x + y) / y) * ((x / y) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.3e-180], 1.0, If[LessEqual[y, 9.2e-150], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y, 5.5e-131], N[(1.0 + N[(-2.0 * N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.2999999999999999e-180

   1. Initial program 57.1%

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

      1. associate-*r/ 57.6%

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

      2. +-commutative 57.6%

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

      3. fma-def 57.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in x around inf 33.5%

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

   if 1.2999999999999999e-180 < y < 9.20000000000000011e-150

   1. Initial program 64.3%

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

   2. Taylor expanded in x around 0 22.9%

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

      1. unpow2 22.9%

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

   4. Simplified 22.9%

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

   5. Taylor expanded in x around 0 22.9%

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

      1. sub-neg 22.9%

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

      2. unpow2 22.9%

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

      3. unpow2 22.9%

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

      4. times-frac 57.4%

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

      5. metadata-eval 57.4%

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

   7. Simplified 57.4%

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

   if 9.20000000000000011e-150 < y < 5.4999999999999997e-131

   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/ 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 5.4999999999999997e-131 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 81.3%

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

      1. unpow2 81.3%

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

   4. Simplified 81.3%

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

      1. *-commutative 81.3%

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

      2. times-frac 81.3%

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

      3. div-sub 81.3%

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

      4. pow1 81.3%

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

      5. pow1 81.3%

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

      6. pow-div 81.3%

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

      7. metadata-eval 81.3%

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

      8. metadata-eval 81.3%

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

   6. Applied egg-rr 81.3%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-180}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-131}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y} \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]

Alternative 6: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := 1 + -2 \cdot \frac{\frac{y}{\frac{x}{y}}}{x}\\ \mathbf{if}\;y \leq 1.1 \cdot 10^{-179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y} \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -2.0 (/ (/ y (/ x y)) x)))))
   (if (<= y 1.1e-179)
     t_0
     (if (<= y 4.2e-149)
       (+ (* (/ x y) (/ x y)) -1.0)
       (if (<= y 6.8e-130) t_0 (* (/ (+ x y) y) (+ (/ x y) -1.0)))))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = 1.0 + (-2.0 * ((y / (x / y)) / x));
	double tmp;
	if (y <= 1.1e-179) {
		tmp = t_0;
	} else if (y <= 4.2e-149) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else if (y <= 6.8e-130) {
		tmp = t_0;
	} else {
		tmp = ((x + y) / y) * ((x / y) + -1.0);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 = 1.0d0 + ((-2.0d0) * ((y / (x / y)) / x))
    if (y <= 1.1d-179) then
        tmp = t_0
    else if (y <= 4.2d-149) then
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    else if (y <= 6.8d-130) then
        tmp = t_0
    else
        tmp = ((x + y) / y) * ((x / y) + (-1.0d0))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = 1.0 + (-2.0 * ((y / (x / y)) / x));
	double tmp;
	if (y <= 1.1e-179) {
		tmp = t_0;
	} else if (y <= 4.2e-149) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else if (y <= 6.8e-130) {
		tmp = t_0;
	} else {
		tmp = ((x + y) / y) * ((x / y) + -1.0);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	t_0 = 1.0 + (-2.0 * ((y / (x / y)) / x))
	tmp = 0
	if y <= 1.1e-179:
		tmp = t_0
	elif y <= 4.2e-149:
		tmp = ((x / y) * (x / y)) + -1.0
	elif y <= 6.8e-130:
		tmp = t_0
	else:
		tmp = ((x + y) / y) * ((x / y) + -1.0)
	return tmp

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / Float64(x / y)) / x)))
	tmp = 0.0
	if (y <= 1.1e-179)
		tmp = t_0;
	elseif (y <= 4.2e-149)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0);
	elseif (y <= 6.8e-130)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(x + y) / y) * Float64(Float64(x / y) + -1.0));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	t_0 = 1.0 + (-2.0 * ((y / (x / y)) / x));
	tmp = 0.0;
	if (y <= 1.1e-179)
		tmp = t_0;
	elseif (y <= 4.2e-149)
		tmp = ((x / y) * (x / y)) + -1.0;
	elseif (y <= 6.8e-130)
		tmp = t_0;
	else
		tmp = ((x + y) / y) * ((x / y) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-2.0 * N[(N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.1e-179], t$95$0, If[LessEqual[y, 4.2e-149], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y, 6.8e-130], t$95$0, N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.10000000000000002e-179 or 4.20000000000000022e-149 < y < 6.8000000000000001e-130

   1. Initial program 57.5%

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

      1. associate-*r/ 58.0%

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

      2. +-commutative 58.0%

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

      3. fma-def 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in y around 0 27.1%

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

      1. unpow2 27.1%

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

      2. unpow2 27.1%

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

   6. Simplified 27.1%

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

      1. associate-/r* 35.5%

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

      2. div-inv 35.5%

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

   8. Applied egg-rr 35.5%

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

      1. associate-*r/ 35.5%

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

      2. unpow2 35.5%

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

      3. *-rgt-identity 35.5%

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

      4. unpow2 35.5%

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

      5. associate-/l* 36.4%

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

   10. Simplified 36.4%

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

   if 1.10000000000000002e-179 < y < 4.20000000000000022e-149

   1. Initial program 64.3%

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

   2. Taylor expanded in x around 0 22.9%

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

      1. unpow2 22.9%

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

   4. Simplified 22.9%

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

   5. Taylor expanded in x around 0 22.9%

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

      1. sub-neg 22.9%

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

      2. unpow2 22.9%

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

      3. unpow2 22.9%

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

      4. times-frac 57.4%

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

      5. metadata-eval 57.4%

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

   7. Simplified 57.4%

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

   if 6.8000000000000001e-130 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 81.3%

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

      1. unpow2 81.3%

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

   4. Simplified 81.3%

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

      1. *-commutative 81.3%

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

      2. times-frac 81.3%

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

      3. div-sub 81.3%

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

      4. pow1 81.3%

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

      5. pow1 81.3%

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

      6. pow-div 81.3%

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

      7. metadata-eval 81.3%

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

      8. metadata-eval 81.3%

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

   6. Applied egg-rr 81.3%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-179}:\\ \;\;\;\;1 + -2 \cdot \frac{\frac{y}{\frac{x}{y}}}{x}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-130}:\\ \;\;\;\;1 + -2 \cdot \frac{\frac{y}{\frac{x}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y} \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]

Alternative 7: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-180}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-148}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-125}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1e-180)
   1.0
   (if (<= y 5.1e-148) -1.0 (if (<= y 2.25e-125) 1.0 (/ (- x y) y)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1e-180) {
		tmp = 1.0;
	} else if (y <= 5.1e-148) {
		tmp = -1.0;
	} else if (y <= 2.25e-125) {
		tmp = 1.0;
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d-180) then
        tmp = 1.0d0
    else if (y <= 5.1d-148) then
        tmp = -1.0d0
    else if (y <= 2.25d-125) then
        tmp = 1.0d0
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1e-180) {
		tmp = 1.0;
	} else if (y <= 5.1e-148) {
		tmp = -1.0;
	} else if (y <= 2.25e-125) {
		tmp = 1.0;
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1e-180:
		tmp = 1.0
	elif y <= 5.1e-148:
		tmp = -1.0
	elif y <= 2.25e-125:
		tmp = 1.0
	else:
		tmp = (x - y) / y
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1e-180)
		tmp = 1.0;
	elseif (y <= 5.1e-148)
		tmp = -1.0;
	elseif (y <= 2.25e-125)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e-180)
		tmp = 1.0;
	elseif (y <= 5.1e-148)
		tmp = -1.0;
	elseif (y <= 2.25e-125)
		tmp = 1.0;
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1e-180], 1.0, If[LessEqual[y, 5.1e-148], -1.0, If[LessEqual[y, 2.25e-125], 1.0, N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1e-180 or 5.1e-148 < y < 2.25000000000000006e-125

   1. Initial program 57.5%

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

      1. associate-*r/ 58.0%

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

      2. +-commutative 58.0%

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

      3. fma-def 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in x around inf 34.1%

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

   if 1e-180 < y < 5.1e-148

   1. Initial program 64.3%

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

      1. associate-*r/ 64.3%

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

      2. +-commutative 64.3%

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

      3. fma-def 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in x around 0 56.4%

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

   if 2.25000000000000006e-125 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 81.1%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-180}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-148}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-125}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 8: 82.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-180}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e-180) 1.0 (+ (* (/ x y) (/ x y)) -1.0)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-180) {
		tmp = 1.0;
	} else {
		tmp = ((x / y) * (x / y)) + -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.45d-180) then
        tmp = 1.0d0
    else
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-180) {
		tmp = 1.0;
	} else {
		tmp = ((x / y) * (x / y)) + -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.45e-180:
		tmp = 1.0
	else:
		tmp = ((x / y) * (x / y)) + -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-180)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.45e-180)
		tmp = 1.0;
	else
		tmp = ((x / y) * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.45e-180], 1.0, N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4499999999999999e-180

   1. Initial program 57.1%

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

      1. associate-*r/ 57.6%

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

      2. +-commutative 57.6%

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

      3. fma-def 57.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in x around inf 33.5%

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

   if 1.4499999999999999e-180 < y

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0 59.3%

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

      1. unpow2 59.3%

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

   4. Simplified 59.3%

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

   5. Taylor expanded in x around 0 59.3%

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

      1. sub-neg 59.3%

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

      2. unpow2 59.3%

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

      3. unpow2 59.3%

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

      4. times-frac 70.3%

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

      5. metadata-eval 70.3%

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

   7. Simplified 70.3%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-180}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 9: 82.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-149}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-179) 1.0 (if (<= y 1e-149) -1.0 (if (<= y 2e-132) 1.0 -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-179) {
		tmp = 1.0;
	} else if (y <= 1e-149) {
		tmp = -1.0;
	} else if (y <= 2e-132) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d-179) then
        tmp = 1.0d0
    else if (y <= 1d-149) then
        tmp = -1.0d0
    else if (y <= 2d-132) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-179) {
		tmp = 1.0;
	} else if (y <= 1e-149) {
		tmp = -1.0;
	} else if (y <= 2e-132) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.5e-179:
		tmp = 1.0
	elif y <= 1e-149:
		tmp = -1.0
	elif y <= 2e-132:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-179)
		tmp = 1.0;
	elseif (y <= 1e-149)
		tmp = -1.0;
	elseif (y <= 2e-132)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e-179)
		tmp = 1.0;
	elseif (y <= 1e-149)
		tmp = -1.0;
	elseif (y <= 2e-132)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.5e-179], 1.0, If[LessEqual[y, 1e-149], -1.0, If[LessEqual[y, 2e-132], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000003e-179 or 9.99999999999999979e-150 < y < 2e-132

   1. Initial program 57.3%

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

      1. associate-*r/ 57.8%

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

      2. +-commutative 57.8%

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

      3. fma-def 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around inf 33.8%

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

   if 1.50000000000000003e-179 < y < 9.99999999999999979e-150 or 2e-132 < y

   1. Initial program 88.3%

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

      1. associate-*r/ 88.1%

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

      2. +-commutative 88.1%

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

      3. fma-def 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in x around 0 70.7%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-149}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 66.6% accurate, 15.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

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

Python

y = abs(y)
def code(x, y):
	return -1.0

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := -1.0

Derivation

1. Initial program 62.5%

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

   1. associate-*r/ 62.9%

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

   2. +-commutative 62.9%

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

   3. fma-def 62.9%

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

3. Simplified 62.9%

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

4. Taylor expanded in x around 0 66.5%

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

5. Final simplification 66.5%

   \[\leadsto -1 \]

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 2023257 
(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))))