Kahan p9 Example

Percentage Accurate: 67.7% → 99.9%

Time: 8.3s

Alternatives: 9

Speedup: 1.6×

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: 67.7% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.2%

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

   2. times-frac 68.0%

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

   3. hypot-def 68.0%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. frac-times 100.0%

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

   3. *-un-lft-identity 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.2%

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

   2. times-frac 68.0%

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

   3. hypot-def 68.0%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 92.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + y) * (x - y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = ((x / y) * (x / y)) + ((-1.0d0) - ((x * (x / -y)) / y))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 3.1%

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

      2. +-commutative 3.1%

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

      3. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around inf 59.5%

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

      1. associate--l+ 59.5%

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

      2. unpow2 59.5%

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

      3. unpow2 59.5%

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

      4. times-frac 59.5%

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

      5. distribute-rgt1-in 59.5%

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

      6. metadata-eval 59.5%

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

      7. mul0-lft 59.5%

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

      8. +-commutative 59.5%

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

      9. associate--r+ 59.5%

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

      10. metadata-eval 59.5%

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

      11. unpow2 59.5%

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

      12. unpow2 59.5%

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

      13. associate-*r/ 59.5%

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

      14. neg-mul-1 59.5%

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

      15. associate-/r* 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in x around 0 86.8%

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

      1. *-commutative 86.8%

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

      2. unpow2 86.8%

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

      3. *-lft-identity 86.8%

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

      4. *-lft-identity 86.8%

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

      5. associate-*r/ 87.8%

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

      6. associate-*l* 87.8%

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

      7. metadata-eval 87.8%

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

      8. times-frac 87.8%

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

      9. *-rgt-identity 87.8%

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

      10. *-commutative 87.8%

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

      11. neg-mul-1 87.8%

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

   9. Simplified 87.8%

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

4. Final simplification 96.0%

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

Alternative 4: 92.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + y) * (x - y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      1. associate-*r/ 3.1%

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

      2. +-commutative 3.1%

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

      3. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around inf 3.1%

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

      1. unpow2 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in x around 0 59.5%

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

      1. sub-neg 59.5%

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

      2. unpow2 59.5%

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

      3. unpow2 59.5%

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

      4. metadata-eval 59.5%

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

   9. Simplified 59.5%

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

       1. frac-times 86.8%

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

   11. Applied egg-rr 86.8%

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

4. Final simplification 95.7%

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

Alternative 5: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot \frac{x}{y} + -1\\ t_1 := 1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-216}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* (/ x y) (/ x y)) -1.0))
        (t_1 (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))))
   (if (<= y -3.8e-191)
     t_0
     (if (<= y 5e-222)
       t_1
       (if (<= y 8e-216) -1.0 (if (<= y 5.7e-157) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = ((x / y) * (x / y)) + -1.0;
	double t_1 = 1.0 + (-2.0 * ((y / x) * (y / x)));
	double tmp;
	if (y <= -3.8e-191) {
		tmp = t_0;
	} else if (y <= 5e-222) {
		tmp = t_1;
	} else if (y <= 8e-216) {
		tmp = -1.0;
	} else if (y <= 5.7e-157) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((x / y) * (x / y)) + (-1.0d0)
    t_1 = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    if (y <= (-3.8d-191)) then
        tmp = t_0
    else if (y <= 5d-222) then
        tmp = t_1
    else if (y <= 8d-216) then
        tmp = -1.0d0
    else if (y <= 5.7d-157) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x / y) * (x / y)) + -1.0;
	double t_1 = 1.0 + (-2.0 * ((y / x) * (y / x)));
	double tmp;
	if (y <= -3.8e-191) {
		tmp = t_0;
	} else if (y <= 5e-222) {
		tmp = t_1;
	} else if (y <= 8e-216) {
		tmp = -1.0;
	} else if (y <= 5.7e-157) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x / y) * (x / y)) + -1.0
	t_1 = 1.0 + (-2.0 * ((y / x) * (y / x)))
	tmp = 0
	if y <= -3.8e-191:
		tmp = t_0
	elif y <= 5e-222:
		tmp = t_1
	elif y <= 8e-216:
		tmp = -1.0
	elif y <= 5.7e-157:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0)
	t_1 = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))))
	tmp = 0.0
	if (y <= -3.8e-191)
		tmp = t_0;
	elseif (y <= 5e-222)
		tmp = t_1;
	elseif (y <= 8e-216)
		tmp = -1.0;
	elseif (y <= 5.7e-157)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x / y) * (x / y)) + -1.0;
	t_1 = 1.0 + (-2.0 * ((y / x) * (y / x)));
	tmp = 0.0;
	if (y <= -3.8e-191)
		tmp = t_0;
	elseif (y <= 5e-222)
		tmp = t_1;
	elseif (y <= 8e-216)
		tmp = -1.0;
	elseif (y <= 5.7e-157)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e-191], t$95$0, If[LessEqual[y, 5e-222], t$95$1, If[LessEqual[y, 8e-216], -1.0, If[LessEqual[y, 5.7e-157], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7999999999999998e-191 or 5.69999999999999998e-157 < y

   1. Initial program 70.1%

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

      1. associate-*r/ 70.7%

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

      2. +-commutative 70.7%

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

      3. fma-def 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in y around inf 59.1%

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

      1. unpow2 59.1%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in x around 0 86.1%

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

      1. sub-neg 86.1%

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

      2. unpow2 86.1%

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

      3. unpow2 86.1%

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

      4. metadata-eval 86.1%

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

   9. Simplified 86.1%

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

       1. frac-times 88.3%

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

   11. Applied egg-rr 88.3%

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

   if -3.7999999999999998e-191 < y < 5.00000000000000008e-222 or 8.0000000000000003e-216 < y < 5.69999999999999998e-157

   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/ 64.8%

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

      2. +-commutative 64.8%

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

      3. fma-def 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in y around 0 64.3%

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

      1. unpow2 64.3%

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

      2. unpow2 64.3%

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

   6. Simplified 64.3%

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

      1. times-frac 80.1%

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

   8. Applied egg-rr 80.1%

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

   if 5.00000000000000008e-222 < y < 8.0000000000000003e-216

   1. Initial program 0.0%

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

      1. associate-*r/ 3.1%

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

      2. +-commutative 3.1%

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

      3. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 100.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-191}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-222}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-216}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-157}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 6: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-192} \lor \neg \left(y \leq 4.9 \cdot 10^{-222} \lor \neg \left(y \leq 7.4 \cdot 10^{-190}\right) \land y \leq 2.9 \cdot 10^{-157}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.4e-192)
         (not
          (or (<= y 4.9e-222) (and (not (<= y 7.4e-190)) (<= y 2.9e-157)))))
   (+ (* (/ x y) (/ x y)) -1.0)
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e-192) || !((y <= 4.9e-222) || (!(y <= 7.4e-190) && (y <= 2.9e-157)))) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.4d-192)) .or. (.not. (y <= 4.9d-222) .or. (.not. (y <= 7.4d-190)) .and. (y <= 2.9d-157))) then
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e-192) || !((y <= 4.9e-222) || (!(y <= 7.4e-190) && (y <= 2.9e-157)))) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.4e-192) or not ((y <= 4.9e-222) or (not (y <= 7.4e-190) and (y <= 2.9e-157))):
		tmp = ((x / y) * (x / y)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.4e-192) || !((y <= 4.9e-222) || (!(y <= 7.4e-190) && (y <= 2.9e-157))))
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.4e-192) || ~(((y <= 4.9e-222) || (~((y <= 7.4e-190)) && (y <= 2.9e-157)))))
		tmp = ((x / y) * (x / y)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.4e-192], N[Not[Or[LessEqual[y, 4.9e-222], And[N[Not[LessEqual[y, 7.4e-190]], $MachinePrecision], LessEqual[y, 2.9e-157]]]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000002e-192 or 4.9e-222 < y < 7.4000000000000004e-190 or 2.89999999999999988e-157 < y

   1. Initial program 68.1%

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

      1. associate-*r/ 68.7%

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

      2. +-commutative 68.7%

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

      3. fma-def 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in y around inf 57.0%

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

      1. unpow2 57.0%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in x around 0 82.9%

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

      1. sub-neg 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. metadata-eval 82.9%

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

   9. Simplified 82.9%

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

       1. frac-times 88.3%

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

   11. Applied egg-rr 88.3%

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

   if -1.40000000000000002e-192 < y < 4.9e-222 or 7.4000000000000004e-190 < y < 2.89999999999999988e-157

   1. Initial program 64.7%

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

      1. associate-*r/ 65.1%

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

      2. +-commutative 65.1%

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

      3. fma-def 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in x around inf 79.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-192} \lor \neg \left(y \leq 4.9 \cdot 10^{-222} \lor \neg \left(y \leq 7.4 \cdot 10^{-190}\right) \land y \leq 2.9 \cdot 10^{-157}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-191} \lor \neg \left(y \leq 2.2 \cdot 10^{-222} \lor \neg \left(y \leq 6.5 \cdot 10^{-188}\right) \land y \leq 4.2 \cdot 10^{-157}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.65e-191)
         (not
          (or (<= y 2.2e-222) (and (not (<= y 6.5e-188)) (<= y 4.2e-157)))))
   (+ (* (/ x y) (/ x y)) -1.0)
   (+ (/ y x) (- 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.65e-191) || !((y <= 2.2e-222) || (!(y <= 6.5e-188) && (y <= 4.2e-157)))) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else {
		tmp = (y / x) + (1.0 - (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.65d-191)) .or. (.not. (y <= 2.2d-222) .or. (.not. (y <= 6.5d-188)) .and. (y <= 4.2d-157))) then
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    else
        tmp = (y / x) + (1.0d0 - (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.65e-191) || !((y <= 2.2e-222) || (!(y <= 6.5e-188) && (y <= 4.2e-157)))) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else {
		tmp = (y / x) + (1.0 - (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.65e-191) or not ((y <= 2.2e-222) or (not (y <= 6.5e-188) and (y <= 4.2e-157))):
		tmp = ((x / y) * (x / y)) + -1.0
	else:
		tmp = (y / x) + (1.0 - (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.65e-191) || !((y <= 2.2e-222) || (!(y <= 6.5e-188) && (y <= 4.2e-157))))
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0);
	else
		tmp = Float64(Float64(y / x) + Float64(1.0 - Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.65e-191) || ~(((y <= 2.2e-222) || (~((y <= 6.5e-188)) && (y <= 4.2e-157)))))
		tmp = ((x / y) * (x / y)) + -1.0;
	else
		tmp = (y / x) + (1.0 - (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.65e-191], N[Not[Or[LessEqual[y, 2.2e-222], And[N[Not[LessEqual[y, 6.5e-188]], $MachinePrecision], LessEqual[y, 4.2e-157]]]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(y / x), $MachinePrecision] + N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.64999999999999993e-191 or 2.2e-222 < y < 6.4999999999999998e-188 or 4.2e-157 < y

   1. Initial program 68.1%

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

      1. associate-*r/ 68.7%

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

      2. +-commutative 68.7%

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

      3. fma-def 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in y around inf 57.0%

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

      1. unpow2 57.0%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in x around 0 82.9%

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

      1. sub-neg 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. metadata-eval 82.9%

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

   9. Simplified 82.9%

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

       1. frac-times 88.3%

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

   11. Applied egg-rr 88.3%

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

   if -2.64999999999999993e-191 < y < 2.2e-222 or 6.4999999999999998e-188 < y < 4.2e-157

   1. Initial program 64.7%

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

      1. associate-*r/ 65.1%

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

      2. +-commutative 65.1%

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

      3. fma-def 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in x around inf 79.6%

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

      1. mul-1-neg 79.6%

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

      2. unsub-neg 79.6%

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

   6. Simplified 79.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-191} \lor \neg \left(y \leq 2.2 \cdot 10^{-222} \lor \neg \left(y \leq 6.5 \cdot 10^{-188}\right) \land y \leq 4.2 \cdot 10^{-157}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \end{array} \]

Alternative 8: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-191}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{-215}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e-191)
   -1.0
   (if (<= y 5e-222)
     1.0
     (if (<= y 1.76e-215) -1.0 (if (<= y 5e-157) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e-191) {
		tmp = -1.0;
	} else if (y <= 5e-222) {
		tmp = 1.0;
	} else if (y <= 1.76e-215) {
		tmp = -1.0;
	} else if (y <= 5e-157) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2d-191)) then
        tmp = -1.0d0
    else if (y <= 5d-222) then
        tmp = 1.0d0
    else if (y <= 1.76d-215) then
        tmp = -1.0d0
    else if (y <= 5d-157) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2e-191) {
		tmp = -1.0;
	} else if (y <= 5e-222) {
		tmp = 1.0;
	} else if (y <= 1.76e-215) {
		tmp = -1.0;
	} else if (y <= 5e-157) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2e-191:
		tmp = -1.0
	elif y <= 5e-222:
		tmp = 1.0
	elif y <= 1.76e-215:
		tmp = -1.0
	elif y <= 5e-157:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e-191)
		tmp = -1.0;
	elseif (y <= 5e-222)
		tmp = 1.0;
	elseif (y <= 1.76e-215)
		tmp = -1.0;
	elseif (y <= 5e-157)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2e-191)
		tmp = -1.0;
	elseif (y <= 5e-222)
		tmp = 1.0;
	elseif (y <= 1.76e-215)
		tmp = -1.0;
	elseif (y <= 5e-157)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e-191], -1.0, If[LessEqual[y, 5e-222], 1.0, If[LessEqual[y, 1.76e-215], -1.0, If[LessEqual[y, 5e-157], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2e-191 or 5.00000000000000008e-222 < y < 1.75999999999999992e-215 or 5.0000000000000002e-157 < y

   1. Initial program 68.3%

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

      1. associate-*r/ 68.8%

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

      2. +-commutative 68.8%

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

      3. fma-def 68.8%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in x around 0 88.1%

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

   if -2e-191 < y < 5.00000000000000008e-222 or 1.75999999999999992e-215 < y < 5.0000000000000002e-157

   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/ 64.8%

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

      2. +-commutative 64.8%

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

      3. fma-def 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in x around inf 78.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-191}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{-215}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9: 66.1% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 67.2%

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

   1. associate-*r/ 67.7%

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

   2. +-commutative 67.7%

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

   3. fma-def 67.7%

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

3. Simplified 67.7%

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

4. Taylor expanded in x around 0 69.5%

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

5. Final simplification 69.5%

   \[\leadsto -1 \]

Developer target: 99.8% 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 2023174 
(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))))