Kahan p9 Example

Percentage Accurate: 68.5% → 99.9%

Time: 14.8s

Alternatives: 16

Speedup: 0.1×

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: 68.5% 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

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

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 65.6%

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

   1. fma-define 65.6%

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

   2. add-sqr-sqrt 65.6%

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

   3. times-frac 65.6%

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

   4. fma-define 65.6%

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

   5. hypot-define 65.6%

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

   6. fma-define 65.6%

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

   7. hypot-define 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 47.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;1 + \frac{-1}{{\left(\frac{x}{y}\right)}^{2} \cdot 0.5}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-175}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(\frac{x}{y} \cdot 2 + -1\right)}\\ \mathbf{elif}\;y \leq 10^{-22}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\mathsf{fma}\left(x, x, {y}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e-205)
   (+ 1.0 (/ -1.0 (* (pow (/ x y) 2.0) 0.5)))
   (if (<= y 2.25e-175)
     (/ (- x y) (+ y (* x (+ (* (/ x y) 2.0) -1.0))))
     (if (<= y 1e-22) (/ (* (- x y) (+ x y)) (fma x x (pow y 2.0))) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 + (-1.0 / (pow((x / y), 2.0) * 0.5));
	} else if (y <= 2.25e-175) {
		tmp = (x - y) / (y + (x * (((x / y) * 2.0) + -1.0)));
	} else if (y <= 1e-22) {
		tmp = ((x - y) * (x + y)) / fma(x, x, pow(y, 2.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e-205)
		tmp = Float64(1.0 + Float64(-1.0 / Float64((Float64(x / y) ^ 2.0) * 0.5)));
	elseif (y <= 2.25e-175)
		tmp = Float64(Float64(x - y) / Float64(y + Float64(x * Float64(Float64(Float64(x / y) * 2.0) + -1.0))));
	elseif (y <= 1e-22)
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / fma(x, x, (y ^ 2.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e-205], N[(1.0 + N[(-1.0 / N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e-175], N[(N[(x - y), $MachinePrecision] / N[(y + N[(x * N[(N[(N[(x / y), $MachinePrecision] * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-22], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.99999999999999912e-205

   1. Initial program 58.4%

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

      1. associate-/l* 58.3%

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

      2. fma-define 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around -inf 36.3%

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

      1. mul-1-neg 36.3%

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

      2. unsub-neg 36.3%

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

   7. Simplified 36.3%

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

      1. clear-num 36.3%

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

      2. inv-pow 36.3%

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

      3. *-un-lft-identity 36.3%

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

      4. associate-/l* 36.3%

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

      5. times-frac 36.3%

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

      6. metadata-eval 36.3%

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

      7. *-un-lft-identity 36.3%

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

      8. *-commutative 36.3%

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

      9. unpow2 36.3%

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

      10. associate-/l* 36.9%

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

      11. frac-times 36.9%

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

      12. clear-num 36.9%

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

      13. pow2 36.9%

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

   9. Applied egg-rr 36.9%

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

       1. unpow-1 36.9%

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

       2. *-commutative 36.9%

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

   11. Simplified 36.9%

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

   if 8.99999999999999912e-205 < y < 2.24999999999999999e-175

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

      1. *-un-lft-identity 17.0%

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

      2. add-sqr-sqrt 17.0%

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

      3. times-frac 17.0%

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

      4. fma-define 17.0%

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

      5. hypot-define 17.0%

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

      6. fma-define 17.0%

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

      7. hypot-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 0.1%

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

      1. associate-+r+ 0.1%

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

      2. mul-1-neg 0.1%

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

      3. unpow2 0.1%

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

      4. unpow2 0.1%

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

      5. times-frac 85.0%

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

      6. unpow2 85.0%

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

      7. sub-neg 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.1%

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

       2. un-div-inv 85.8%

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

       3. +-commutative 85.8%

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

   11. Applied egg-rr 85.8%

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

   12. Taylor expanded in x around 0 86.6%

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

   if 2.24999999999999999e-175 < y < 1e-22

   1. Initial program 97.7%

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

      1. fma-define 97.8%

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

      2. pow2 97.8%

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

   4. Applied egg-rr 97.8%

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

   if 1e-22 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.8%

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

      2. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;1 + \frac{-1}{{\left(\frac{x}{y}\right)}^{2} \cdot 0.5}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-175}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(\frac{x}{y} \cdot 2 + -1\right)}\\ \mathbf{elif}\;y \leq 10^{-22}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\mathsf{fma}\left(x, x, {y}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;1 + \frac{-1}{{\left(\frac{x}{y}\right)}^{2} \cdot 0.5}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(\frac{x}{y} \cdot 2 + -1\right)}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e-205)
   (+ 1.0 (/ -1.0 (* (pow (/ x y) 2.0) 0.5)))
   (if (<= y 6.8e-176)
     (/ (- x y) (+ y (* x (+ (* (/ x y) 2.0) -1.0))))
     (if (<= y 3.1e-19) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 + (-1.0 / (pow((x / y), 2.0) * 0.5));
	} else if (y <= 6.8e-176) {
		tmp = (x - y) / (y + (x * (((x / y) * 2.0) + -1.0)));
	} else if (y <= 3.1e-19) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} 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 <= 9d-205) then
        tmp = 1.0d0 + ((-1.0d0) / (((x / y) ** 2.0d0) * 0.5d0))
    else if (y <= 6.8d-176) then
        tmp = (x - y) / (y + (x * (((x / y) * 2.0d0) + (-1.0d0))))
    else if (y <= 3.1d-19) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 + (-1.0 / (Math.pow((x / y), 2.0) * 0.5));
	} else if (y <= 6.8e-176) {
		tmp = (x - y) / (y + (x * (((x / y) * 2.0) + -1.0)));
	} else if (y <= 3.1e-19) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e-205:
		tmp = 1.0 + (-1.0 / (math.pow((x / y), 2.0) * 0.5))
	elif y <= 6.8e-176:
		tmp = (x - y) / (y + (x * (((x / y) * 2.0) + -1.0)))
	elif y <= 3.1e-19:
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e-205)
		tmp = Float64(1.0 + Float64(-1.0 / Float64((Float64(x / y) ^ 2.0) * 0.5)));
	elseif (y <= 6.8e-176)
		tmp = Float64(Float64(x - y) / Float64(y + Float64(x * Float64(Float64(Float64(x / y) * 2.0) + -1.0))));
	elseif (y <= 3.1e-19)
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = 1.0 + (-1.0 / (((x / y) ^ 2.0) * 0.5));
	elseif (y <= 6.8e-176)
		tmp = (x - y) / (y + (x * (((x / y) * 2.0) + -1.0)));
	elseif (y <= 3.1e-19)
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e-205], N[(1.0 + N[(-1.0 / N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e-176], N[(N[(x - y), $MachinePrecision] / N[(y + N[(x * N[(N[(N[(x / y), $MachinePrecision] * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e-19], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.99999999999999912e-205

   1. Initial program 58.4%

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

      1. associate-/l* 58.3%

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

      2. fma-define 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around -inf 36.3%

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

      1. mul-1-neg 36.3%

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

      2. unsub-neg 36.3%

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

   7. Simplified 36.3%

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

      1. clear-num 36.3%

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

      2. inv-pow 36.3%

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

      3. *-un-lft-identity 36.3%

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

      4. associate-/l* 36.3%

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

      5. times-frac 36.3%

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

      6. metadata-eval 36.3%

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

      7. *-un-lft-identity 36.3%

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

      8. *-commutative 36.3%

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

      9. unpow2 36.3%

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

      10. associate-/l* 36.9%

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

      11. frac-times 36.9%

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

      12. clear-num 36.9%

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

      13. pow2 36.9%

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

   9. Applied egg-rr 36.9%

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

       1. unpow-1 36.9%

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

       2. *-commutative 36.9%

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

   11. Simplified 36.9%

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

   if 8.99999999999999912e-205 < y < 6.7999999999999994e-176

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

      1. *-un-lft-identity 17.0%

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

      2. add-sqr-sqrt 17.0%

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

      3. times-frac 17.0%

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

      4. fma-define 17.0%

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

      5. hypot-define 17.0%

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

      6. fma-define 17.0%

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

      7. hypot-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 0.1%

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

      1. associate-+r+ 0.1%

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

      2. mul-1-neg 0.1%

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

      3. unpow2 0.1%

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

      4. unpow2 0.1%

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

      5. times-frac 85.0%

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

      6. unpow2 85.0%

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

      7. sub-neg 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.1%

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

       2. un-div-inv 85.8%

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

       3. +-commutative 85.8%

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

   11. Applied egg-rr 85.8%

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

   12. Taylor expanded in x around 0 86.6%

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

   if 6.7999999999999994e-176 < y < 3.0999999999999999e-19

   1. Initial program 97.7%

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

   if 3.0999999999999999e-19 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.8%

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

      2. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;1 + \frac{-1}{{\left(\frac{x}{y}\right)}^{2} \cdot 0.5}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(\frac{x}{y} \cdot 2 + -1\right)}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(1 + \frac{y}{x}\right)}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-175}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e-205)
   (/ 1.0 (/ x (* (- x y) (+ 1.0 (/ y x)))))
   (if (<= y 9e-175)
     (/ (* (- x y) (+ 1.0 (/ x y))) y)
     (if (<= y 5e-20) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 / (x / ((x - y) * (1.0 + (y / x))));
	} else if (y <= 9e-175) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else if (y <= 5e-20) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} 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 <= 9d-205) then
        tmp = 1.0d0 / (x / ((x - y) * (1.0d0 + (y / x))))
    else if (y <= 9d-175) then
        tmp = ((x - y) * (1.0d0 + (x / y))) / y
    else if (y <= 5d-20) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 / (x / ((x - y) * (1.0 + (y / x))));
	} else if (y <= 9e-175) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else if (y <= 5e-20) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e-205:
		tmp = 1.0 / (x / ((x - y) * (1.0 + (y / x))))
	elif y <= 9e-175:
		tmp = ((x - y) * (1.0 + (x / y))) / y
	elif y <= 5e-20:
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e-205)
		tmp = Float64(1.0 / Float64(x / Float64(Float64(x - y) * Float64(1.0 + Float64(y / x)))));
	elseif (y <= 9e-175)
		tmp = Float64(Float64(Float64(x - y) * Float64(1.0 + Float64(x / y))) / y);
	elseif (y <= 5e-20)
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = 1.0 / (x / ((x - y) * (1.0 + (y / x))));
	elseif (y <= 9e-175)
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	elseif (y <= 5e-20)
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e-205], N[(1.0 / N[(x / N[(N[(x - y), $MachinePrecision] * N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-175], N[(N[(N[(x - y), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 5e-20], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.99999999999999912e-205

   1. Initial program 58.4%

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

      1. associate-/l* 58.3%

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

      2. fma-define 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around inf 36.6%

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

      1. associate-*r/ 36.8%

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

      2. clear-num 36.8%

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

   7. Applied egg-rr 36.8%

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

   if 8.99999999999999912e-205 < y < 8.99999999999999996e-175

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in y around inf 83.5%

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

      1. associate-*r/ 84.0%

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

   7. Applied egg-rr 84.0%

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

   if 8.99999999999999996e-175 < y < 4.9999999999999999e-20

   1. Initial program 97.7%

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

   if 4.9999999999999999e-20 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.8%

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

      2. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(1 + \frac{y}{x}\right)}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-175}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(1 + \frac{y}{x}\right)}}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-175}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(\frac{x}{y} \cdot 2 + -1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e-205)
   (/ 1.0 (/ x (* (- x y) (+ 1.0 (/ y x)))))
   (if (<= y 5.4e-175)
     (/ (- x y) (+ y (* x (+ (* (/ x y) 2.0) -1.0))))
     (if (<= y 2e-19) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 / (x / ((x - y) * (1.0 + (y / x))));
	} else if (y <= 5.4e-175) {
		tmp = (x - y) / (y + (x * (((x / y) * 2.0) + -1.0)));
	} else if (y <= 2e-19) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} 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 <= 9d-205) then
        tmp = 1.0d0 / (x / ((x - y) * (1.0d0 + (y / x))))
    else if (y <= 5.4d-175) then
        tmp = (x - y) / (y + (x * (((x / y) * 2.0d0) + (-1.0d0))))
    else if (y <= 2d-19) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 / (x / ((x - y) * (1.0 + (y / x))));
	} else if (y <= 5.4e-175) {
		tmp = (x - y) / (y + (x * (((x / y) * 2.0) + -1.0)));
	} else if (y <= 2e-19) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e-205:
		tmp = 1.0 / (x / ((x - y) * (1.0 + (y / x))))
	elif y <= 5.4e-175:
		tmp = (x - y) / (y + (x * (((x / y) * 2.0) + -1.0)))
	elif y <= 2e-19:
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e-205)
		tmp = Float64(1.0 / Float64(x / Float64(Float64(x - y) * Float64(1.0 + Float64(y / x)))));
	elseif (y <= 5.4e-175)
		tmp = Float64(Float64(x - y) / Float64(y + Float64(x * Float64(Float64(Float64(x / y) * 2.0) + -1.0))));
	elseif (y <= 2e-19)
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = 1.0 / (x / ((x - y) * (1.0 + (y / x))));
	elseif (y <= 5.4e-175)
		tmp = (x - y) / (y + (x * (((x / y) * 2.0) + -1.0)));
	elseif (y <= 2e-19)
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e-205], N[(1.0 / N[(x / N[(N[(x - y), $MachinePrecision] * N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e-175], N[(N[(x - y), $MachinePrecision] / N[(y + N[(x * N[(N[(N[(x / y), $MachinePrecision] * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e-19], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.99999999999999912e-205

   1. Initial program 58.4%

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

      1. associate-/l* 58.3%

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

      2. fma-define 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around inf 36.6%

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

      1. associate-*r/ 36.8%

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

      2. clear-num 36.8%

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

   7. Applied egg-rr 36.8%

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

   if 8.99999999999999912e-205 < y < 5.39999999999999998e-175

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

      1. *-un-lft-identity 17.0%

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

      2. add-sqr-sqrt 17.0%

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

      3. times-frac 17.0%

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

      4. fma-define 17.0%

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

      5. hypot-define 17.0%

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

      6. fma-define 17.0%

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

      7. hypot-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 0.1%

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

      1. associate-+r+ 0.1%

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

      2. mul-1-neg 0.1%

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

      3. unpow2 0.1%

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

      4. unpow2 0.1%

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

      5. times-frac 85.0%

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

      6. unpow2 85.0%

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

      7. sub-neg 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.1%

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

       2. un-div-inv 85.8%

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

       3. +-commutative 85.8%

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

   11. Applied egg-rr 85.8%

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

   12. Taylor expanded in x around 0 86.6%

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

   if 5.39999999999999998e-175 < y < 2e-19

   1. Initial program 97.7%

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

   if 2e-19 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.8%

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

      2. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(1 + \frac{y}{x}\right)}}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-175}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(\frac{x}{y} \cdot 2 + -1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)\\ t_1 := 1 + \frac{y}{x}\\ \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot t\_1}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-175}:\\ \;\;\;\;\frac{t\_0}{y}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{x - y}{\frac{x}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) (+ 1.0 (/ x y)))) (t_1 (+ 1.0 (/ y x))))
   (if (<= y 9e-205)
     (/ 1.0 (/ x (* (- x y) t_1)))
     (if (<= y 8e-175)
       (/ t_0 y)
       (if (<= y 4.8e-168) (/ (- x y) (/ x t_1)) (/ 1.0 (/ y t_0)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) * (1.0 + (x / y));
	double t_1 = 1.0 + (y / x);
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 / (x / ((x - y) * t_1));
	} else if (y <= 8e-175) {
		tmp = t_0 / y;
	} else if (y <= 4.8e-168) {
		tmp = (x - y) / (x / t_1);
	} else {
		tmp = 1.0 / (y / 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) * (1.0d0 + (x / y))
    t_1 = 1.0d0 + (y / x)
    if (y <= 9d-205) then
        tmp = 1.0d0 / (x / ((x - y) * t_1))
    else if (y <= 8d-175) then
        tmp = t_0 / y
    else if (y <= 4.8d-168) then
        tmp = (x - y) / (x / t_1)
    else
        tmp = 1.0d0 / (y / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) * (1.0 + (x / y));
	double t_1 = 1.0 + (y / x);
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 / (x / ((x - y) * t_1));
	} else if (y <= 8e-175) {
		tmp = t_0 / y;
	} else if (y <= 4.8e-168) {
		tmp = (x - y) / (x / t_1);
	} else {
		tmp = 1.0 / (y / t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) * (1.0 + (x / y))
	t_1 = 1.0 + (y / x)
	tmp = 0
	if y <= 9e-205:
		tmp = 1.0 / (x / ((x - y) * t_1))
	elif y <= 8e-175:
		tmp = t_0 / y
	elif y <= 4.8e-168:
		tmp = (x - y) / (x / t_1)
	else:
		tmp = 1.0 / (y / t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * Float64(1.0 + Float64(x / y)))
	t_1 = Float64(1.0 + Float64(y / x))
	tmp = 0.0
	if (y <= 9e-205)
		tmp = Float64(1.0 / Float64(x / Float64(Float64(x - y) * t_1)));
	elseif (y <= 8e-175)
		tmp = Float64(t_0 / y);
	elseif (y <= 4.8e-168)
		tmp = Float64(Float64(x - y) / Float64(x / t_1));
	else
		tmp = Float64(1.0 / Float64(y / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) * (1.0 + (x / y));
	t_1 = 1.0 + (y / x);
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = 1.0 / (x / ((x - y) * t_1));
	elseif (y <= 8e-175)
		tmp = t_0 / y;
	elseif (y <= 4.8e-168)
		tmp = (x - y) / (x / t_1);
	else
		tmp = 1.0 / (y / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9e-205], N[(1.0 / N[(x / N[(N[(x - y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e-175], N[(t$95$0 / y), $MachinePrecision], If[LessEqual[y, 4.8e-168], N[(N[(x - y), $MachinePrecision] / N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.99999999999999912e-205

   1. Initial program 58.4%

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

      1. associate-/l* 58.3%

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

      2. fma-define 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around inf 36.6%

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

      1. associate-*r/ 36.8%

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

      2. clear-num 36.8%

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

   7. Applied egg-rr 36.8%

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

   if 8.99999999999999912e-205 < y < 8e-175

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in y around inf 83.5%

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

      1. associate-*r/ 84.0%

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

   7. Applied egg-rr 84.0%

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

   if 8e-175 < y < 4.7999999999999999e-168

   1. Initial program 66.7%

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

      1. associate-/l* 67.2%

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

      2. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in x around inf 69.3%

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

      1. clear-num 69.3%

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

      2. un-div-inv 69.8%

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

   7. Applied egg-rr 69.8%

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

   if 4.7999999999999999e-168 < 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* 98.8%

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

      2. fma-define 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around inf 70.2%

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

      1. associate-*r/ 70.2%

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

      2. clear-num 70.2%

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

   7. Applied egg-rr 70.2%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(1 + \frac{y}{x}\right)}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-175}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{x - y}{\frac{x}{1 + \frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-175} \lor \neg \left(y \leq 4.7 \cdot 10^{-168}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e-205)
   1.0
   (if (or (<= y 2.5e-175) (not (<= y 4.7e-168)))
     (* (- x y) (/ (+ 1.0 (/ x y)) y))
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0;
	} else if ((y <= 2.5e-175) || !(y <= 4.7e-168)) {
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	} 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 <= 9d-205) then
        tmp = 1.0d0
    else if ((y <= 2.5d-175) .or. (.not. (y <= 4.7d-168))) then
        tmp = (x - y) * ((1.0d0 + (x / y)) / y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0;
	} else if ((y <= 2.5e-175) || !(y <= 4.7e-168)) {
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e-205:
		tmp = 1.0
	elif (y <= 2.5e-175) or not (y <= 4.7e-168):
		tmp = (x - y) * ((1.0 + (x / y)) / y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e-205)
		tmp = 1.0;
	elseif ((y <= 2.5e-175) || !(y <= 4.7e-168))
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(x / y)) / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = 1.0;
	elseif ((y <= 2.5e-175) || ~((y <= 4.7e-168)))
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e-205], 1.0, If[Or[LessEqual[y, 2.5e-175], N[Not[LessEqual[y, 4.7e-168]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 8.99999999999999912e-205 or 2.5e-175 < y < 4.70000000000000026e-168

   1. Initial program 58.6%

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

      1. associate-/l* 58.4%

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

      2. fma-define 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in x around inf 35.5%

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

   if 8.99999999999999912e-205 < y < 2.5e-175 or 4.70000000000000026e-168 < y

   1. Initial program 89.6%

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

      1. associate-/l* 88.9%

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

      2. fma-define 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 71.8%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-175} \lor \neg \left(y \leq 4.7 \cdot 10^{-168}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205} \lor \neg \left(y \leq 8.2 \cdot 10^{-176}\right) \land y \leq 5.2 \cdot 10^{-168}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 9e-205) (and (not (<= y 8.2e-176)) (<= y 5.2e-168)))
   (* (- x y) (/ (+ 1.0 (/ y x)) x))
   (* (- x y) (/ (+ 1.0 (/ x y)) y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 9e-205) || (!(y <= 8.2e-176) && (y <= 5.2e-168))) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 9d-205) .or. (.not. (y <= 8.2d-176)) .and. (y <= 5.2d-168)) then
        tmp = (x - y) * ((1.0d0 + (y / x)) / x)
    else
        tmp = (x - y) * ((1.0d0 + (x / y)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 9e-205) || (!(y <= 8.2e-176) && (y <= 5.2e-168))) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 9e-205) or (not (y <= 8.2e-176) and (y <= 5.2e-168)):
		tmp = (x - y) * ((1.0 + (y / x)) / x)
	else:
		tmp = (x - y) * ((1.0 + (x / y)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 9e-205) || (!(y <= 8.2e-176) && (y <= 5.2e-168)))
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(y / x)) / x));
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(x / y)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 9e-205) || (~((y <= 8.2e-176)) && (y <= 5.2e-168)))
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	else
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 9e-205], And[N[Not[LessEqual[y, 8.2e-176]], $MachinePrecision], LessEqual[y, 5.2e-168]]], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.99999999999999912e-205 or 8.2000000000000005e-176 < y < 5.2000000000000002e-168

   1. Initial program 58.6%

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

      1. associate-/l* 58.4%

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

      2. fma-define 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in x around inf 37.1%

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

   if 8.99999999999999912e-205 < y < 8.2000000000000005e-176 or 5.2000000000000002e-168 < y

   1. Initial program 89.6%

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

      1. associate-/l* 88.9%

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

      2. fma-define 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 71.8%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205} \lor \neg \left(y \leq 8.2 \cdot 10^{-176}\right) \land y \leq 5.2 \cdot 10^{-168}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-175} \lor \neg \left(y \leq 4.4 \cdot 10^{-168}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e-205)
   (* (- x y) (/ (/ (+ x y) x) x))
   (if (or (<= y 2.75e-175) (not (<= y 4.4e-168)))
     (/ (* (- x y) (+ 1.0 (/ x y))) y)
     (* (- x y) (/ (+ 1.0 (/ y x)) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = (x - y) * (((x + y) / x) / x);
	} else if ((y <= 2.75e-175) || !(y <= 4.4e-168)) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else {
		tmp = (x - y) * ((1.0 + (y / x)) / 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 <= 9d-205) then
        tmp = (x - y) * (((x + y) / x) / x)
    else if ((y <= 2.75d-175) .or. (.not. (y <= 4.4d-168))) then
        tmp = ((x - y) * (1.0d0 + (x / y))) / y
    else
        tmp = (x - y) * ((1.0d0 + (y / x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = (x - y) * (((x + y) / x) / x);
	} else if ((y <= 2.75e-175) || !(y <= 4.4e-168)) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e-205:
		tmp = (x - y) * (((x + y) / x) / x)
	elif (y <= 2.75e-175) or not (y <= 4.4e-168):
		tmp = ((x - y) * (1.0 + (x / y))) / y
	else:
		tmp = (x - y) * ((1.0 + (y / x)) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e-205)
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x + y) / x) / x));
	elseif ((y <= 2.75e-175) || !(y <= 4.4e-168))
		tmp = Float64(Float64(Float64(x - y) * Float64(1.0 + Float64(x / y))) / y);
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(y / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = (x - y) * (((x + y) / x) / x);
	elseif ((y <= 2.75e-175) || ~((y <= 4.4e-168)))
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	else
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e-205], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 2.75e-175], N[Not[LessEqual[y, 4.4e-168]], $MachinePrecision]], N[(N[(N[(x - y), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.99999999999999912e-205

   1. Initial program 58.4%

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

      1. associate-/l* 58.3%

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

      2. fma-define 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around inf 36.6%

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

   6. Taylor expanded in x around 0 36.6%

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

      1. +-commutative 36.6%

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

   8. Simplified 36.6%

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

   if 8.99999999999999912e-205 < y < 2.75000000000000027e-175 or 4.3999999999999996e-168 < y

   1. Initial program 89.6%

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

      1. associate-/l* 88.9%

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

      2. fma-define 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 71.8%

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

      1. associate-*r/ 71.9%

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

   7. Applied egg-rr 71.9%

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

   if 2.75000000000000027e-175 < y < 4.3999999999999996e-168

   1. Initial program 66.7%

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

      1. associate-/l* 67.2%

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

      2. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in x around inf 69.3%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-175} \lor \neg \left(y \leq 4.4 \cdot 10^{-168}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-176}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-168}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) (/ (+ 1.0 (/ y x)) x))))
   (if (<= y 9e-205)
     t_0
     (if (<= y 8.6e-176)
       (* (- x y) (/ (+ 1.0 (/ x y)) y))
       (if (<= y 5.2e-168) t_0 (* (- x y) (/ (/ (+ x y) y) y)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) * ((1.0 + (y / x)) / x);
	double tmp;
	if (y <= 9e-205) {
		tmp = t_0;
	} else if (y <= 8.6e-176) {
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	} else if (y <= 5.2e-168) {
		tmp = t_0;
	} else {
		tmp = (x - y) * (((x + y) / 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) * ((1.0d0 + (y / x)) / x)
    if (y <= 9d-205) then
        tmp = t_0
    else if (y <= 8.6d-176) then
        tmp = (x - y) * ((1.0d0 + (x / y)) / y)
    else if (y <= 5.2d-168) then
        tmp = t_0
    else
        tmp = (x - y) * (((x + y) / y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) * ((1.0 + (y / x)) / x);
	double tmp;
	if (y <= 9e-205) {
		tmp = t_0;
	} else if (y <= 8.6e-176) {
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	} else if (y <= 5.2e-168) {
		tmp = t_0;
	} else {
		tmp = (x - y) * (((x + y) / y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) * ((1.0 + (y / x)) / x)
	tmp = 0
	if y <= 9e-205:
		tmp = t_0
	elif y <= 8.6e-176:
		tmp = (x - y) * ((1.0 + (x / y)) / y)
	elif y <= 5.2e-168:
		tmp = t_0
	else:
		tmp = (x - y) * (((x + y) / y) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(y / x)) / x))
	tmp = 0.0
	if (y <= 9e-205)
		tmp = t_0;
	elseif (y <= 8.6e-176)
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(x / y)) / y));
	elseif (y <= 5.2e-168)
		tmp = t_0;
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x + y) / y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) * ((1.0 + (y / x)) / x);
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = t_0;
	elseif (y <= 8.6e-176)
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	elseif (y <= 5.2e-168)
		tmp = t_0;
	else
		tmp = (x - y) * (((x + y) / y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9e-205], t$95$0, If[LessEqual[y, 8.6e-176], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-168], t$95$0, N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.99999999999999912e-205 or 8.60000000000000025e-176 < y < 5.2000000000000002e-168

   1. Initial program 58.6%

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

      1. associate-/l* 58.4%

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

      2. fma-define 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in x around inf 37.1%

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

   if 8.99999999999999912e-205 < y < 8.60000000000000025e-176

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in y around inf 83.5%

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

   if 5.2000000000000002e-168 < 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* 98.8%

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

      2. fma-define 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around inf 70.2%

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

   6. Taylor expanded in y around 0 70.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-176}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-168}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-175}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-168}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e-205)
   (* (- x y) (/ (/ (+ x y) x) x))
   (if (<= y 1.4e-175)
     (* (- x y) (/ (+ 1.0 (/ x y)) y))
     (if (<= y 5.2e-168)
       (* (- x y) (/ (+ 1.0 (/ y x)) x))
       (* (- x y) (/ (/ (+ x y) y) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = (x - y) * (((x + y) / x) / x);
	} else if (y <= 1.4e-175) {
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	} else if (y <= 5.2e-168) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = (x - y) * (((x + y) / y) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9d-205) then
        tmp = (x - y) * (((x + y) / x) / x)
    else if (y <= 1.4d-175) then
        tmp = (x - y) * ((1.0d0 + (x / y)) / y)
    else if (y <= 5.2d-168) then
        tmp = (x - y) * ((1.0d0 + (y / x)) / x)
    else
        tmp = (x - y) * (((x + y) / y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = (x - y) * (((x + y) / x) / x);
	} else if (y <= 1.4e-175) {
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	} else if (y <= 5.2e-168) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = (x - y) * (((x + y) / y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e-205:
		tmp = (x - y) * (((x + y) / x) / x)
	elif y <= 1.4e-175:
		tmp = (x - y) * ((1.0 + (x / y)) / y)
	elif y <= 5.2e-168:
		tmp = (x - y) * ((1.0 + (y / x)) / x)
	else:
		tmp = (x - y) * (((x + y) / y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e-205)
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x + y) / x) / x));
	elseif (y <= 1.4e-175)
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(x / y)) / y));
	elseif (y <= 5.2e-168)
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(y / x)) / x));
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x + y) / y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = (x - y) * (((x + y) / x) / x);
	elseif (y <= 1.4e-175)
		tmp = (x - y) * ((1.0 + (x / y)) / y);
	elseif (y <= 5.2e-168)
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	else
		tmp = (x - y) * (((x + y) / y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e-205], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e-175], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-168], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.99999999999999912e-205

   1. Initial program 58.4%

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

      1. associate-/l* 58.3%

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

      2. fma-define 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around inf 36.6%

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

   6. Taylor expanded in x around 0 36.6%

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

      1. +-commutative 36.6%

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

   8. Simplified 36.6%

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

   if 8.99999999999999912e-205 < y < 1.4e-175

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in y around inf 83.5%

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

   if 1.4e-175 < y < 5.2000000000000002e-168

   1. Initial program 66.7%

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

      1. associate-/l* 67.2%

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

      2. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in x around inf 69.3%

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

   if 5.2000000000000002e-168 < 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* 98.8%

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

      2. fma-define 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around inf 70.2%

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

   6. Taylor expanded in y around 0 70.2%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-175}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-168}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{x}}{x}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-168}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \frac{x + y}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e-205)
   (* (- x y) (/ (/ (+ x y) x) x))
   (if (<= y 8.8e-176)
     (/ (* (- x y) (+ 1.0 (/ x y))) y)
     (if (<= y 5.2e-168)
       (* (- x y) (/ (+ 1.0 (/ y x)) x))
       (/ (* (- x y) (/ (+ x y) y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = (x - y) * (((x + y) / x) / x);
	} else if (y <= 8.8e-176) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else if (y <= 5.2e-168) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = ((x - y) * ((x + y) / y)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9d-205) then
        tmp = (x - y) * (((x + y) / x) / x)
    else if (y <= 8.8d-176) then
        tmp = ((x - y) * (1.0d0 + (x / y))) / y
    else if (y <= 5.2d-168) then
        tmp = (x - y) * ((1.0d0 + (y / x)) / x)
    else
        tmp = ((x - y) * ((x + y) / y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-205) {
		tmp = (x - y) * (((x + y) / x) / x);
	} else if (y <= 8.8e-176) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else if (y <= 5.2e-168) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = ((x - y) * ((x + y) / y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e-205:
		tmp = (x - y) * (((x + y) / x) / x)
	elif y <= 8.8e-176:
		tmp = ((x - y) * (1.0 + (x / y))) / y
	elif y <= 5.2e-168:
		tmp = (x - y) * ((1.0 + (y / x)) / x)
	else:
		tmp = ((x - y) * ((x + y) / y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e-205)
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x + y) / x) / x));
	elseif (y <= 8.8e-176)
		tmp = Float64(Float64(Float64(x - y) * Float64(1.0 + Float64(x / y))) / y);
	elseif (y <= 5.2e-168)
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(y / x)) / x));
	else
		tmp = Float64(Float64(Float64(x - y) * Float64(Float64(x + y) / y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = (x - y) * (((x + y) / x) / x);
	elseif (y <= 8.8e-176)
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	elseif (y <= 5.2e-168)
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	else
		tmp = ((x - y) * ((x + y) / y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e-205], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.8e-176], N[(N[(N[(x - y), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 5.2e-168], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.99999999999999912e-205

   1. Initial program 58.4%

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

      1. associate-/l* 58.3%

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

      2. fma-define 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around inf 36.6%

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

   6. Taylor expanded in x around 0 36.6%

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

      1. +-commutative 36.6%

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

   8. Simplified 36.6%

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

   if 8.99999999999999912e-205 < y < 8.7999999999999994e-176

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in y around inf 83.5%

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

      1. associate-*r/ 84.0%

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

   7. Applied egg-rr 84.0%

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

   if 8.7999999999999994e-176 < y < 5.2000000000000002e-168

   1. Initial program 66.7%

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

      1. associate-/l* 67.2%

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

      2. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in x around inf 69.3%

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

   if 5.2000000000000002e-168 < 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* 98.8%

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

      2. fma-define 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around inf 70.2%

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

      1. associate-*r/ 70.2%

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

   7. Applied egg-rr 70.2%

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

   8. Taylor expanded in y around 0 70.2%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{x}}{x}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-168}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \frac{x + y}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{\frac{x}{1 + \frac{y}{x}}}\\ \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-168}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \frac{x + y}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (/ x (+ 1.0 (/ y x))))))
   (if (<= y 9e-205)
     t_0
     (if (<= y 8.8e-176)
       (/ (* (- x y) (+ 1.0 (/ x y))) y)
       (if (<= y 4.4e-168) t_0 (/ (* (- x y) (/ (+ x y) y)) y))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (x / (1.0 + (y / x)));
	double tmp;
	if (y <= 9e-205) {
		tmp = t_0;
	} else if (y <= 8.8e-176) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else if (y <= 4.4e-168) {
		tmp = t_0;
	} else {
		tmp = ((x - y) * ((x + y) / 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 / (1.0d0 + (y / x)))
    if (y <= 9d-205) then
        tmp = t_0
    else if (y <= 8.8d-176) then
        tmp = ((x - y) * (1.0d0 + (x / y))) / y
    else if (y <= 4.4d-168) then
        tmp = t_0
    else
        tmp = ((x - y) * ((x + y) / y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (x / (1.0 + (y / x)));
	double tmp;
	if (y <= 9e-205) {
		tmp = t_0;
	} else if (y <= 8.8e-176) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else if (y <= 4.4e-168) {
		tmp = t_0;
	} else {
		tmp = ((x - y) * ((x + y) / y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (x / (1.0 + (y / x)))
	tmp = 0
	if y <= 9e-205:
		tmp = t_0
	elif y <= 8.8e-176:
		tmp = ((x - y) * (1.0 + (x / y))) / y
	elif y <= 4.4e-168:
		tmp = t_0
	else:
		tmp = ((x - y) * ((x + y) / y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(x / Float64(1.0 + Float64(y / x))))
	tmp = 0.0
	if (y <= 9e-205)
		tmp = t_0;
	elseif (y <= 8.8e-176)
		tmp = Float64(Float64(Float64(x - y) * Float64(1.0 + Float64(x / y))) / y);
	elseif (y <= 4.4e-168)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(x - y) * Float64(Float64(x + y) / y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (x / (1.0 + (y / x)));
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = t_0;
	elseif (y <= 8.8e-176)
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	elseif (y <= 4.4e-168)
		tmp = t_0;
	else
		tmp = ((x - y) * ((x + y) / y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(x / N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9e-205], t$95$0, If[LessEqual[y, 8.8e-176], N[(N[(N[(x - y), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 4.4e-168], t$95$0, N[(N[(N[(x - y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.99999999999999912e-205 or 8.7999999999999994e-176 < y < 4.3999999999999996e-168

   1. Initial program 58.6%

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

      1. associate-/l* 58.4%

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

      2. fma-define 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in x around inf 37.1%

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

      1. clear-num 37.1%

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

      2. un-div-inv 37.2%

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

   7. Applied egg-rr 37.2%

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

   if 8.99999999999999912e-205 < y < 8.7999999999999994e-176

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in y around inf 83.5%

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

      1. associate-*r/ 84.0%

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

   7. Applied egg-rr 84.0%

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

   if 4.3999999999999996e-168 < 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* 98.8%

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

      2. fma-define 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around inf 70.2%

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

      1. associate-*r/ 70.2%

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

   7. Applied egg-rr 70.2%

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

   8. Taylor expanded in y around 0 70.2%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\frac{x - y}{\frac{x}{1 + \frac{y}{x}}}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-168}:\\ \;\;\;\;\frac{x - y}{\frac{x}{1 + \frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \frac{x + y}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{y}{x}\\ \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot t\_0}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-168}:\\ \;\;\;\;\frac{x - y}{\frac{x}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \frac{x + y}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ y x))))
   (if (<= y 9e-205)
     (/ 1.0 (/ x (* (- x y) t_0)))
     (if (<= y 8.2e-176)
       (/ (* (- x y) (+ 1.0 (/ x y))) y)
       (if (<= y 4.7e-168)
         (/ (- x y) (/ x t_0))
         (/ (* (- x y) (/ (+ x y) y)) y))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y / x);
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 / (x / ((x - y) * t_0));
	} else if (y <= 8.2e-176) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else if (y <= 4.7e-168) {
		tmp = (x - y) / (x / t_0);
	} else {
		tmp = ((x - y) * ((x + y) / 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 = 1.0d0 + (y / x)
    if (y <= 9d-205) then
        tmp = 1.0d0 / (x / ((x - y) * t_0))
    else if (y <= 8.2d-176) then
        tmp = ((x - y) * (1.0d0 + (x / y))) / y
    else if (y <= 4.7d-168) then
        tmp = (x - y) / (x / t_0)
    else
        tmp = ((x - y) * ((x + y) / y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y / x);
	double tmp;
	if (y <= 9e-205) {
		tmp = 1.0 / (x / ((x - y) * t_0));
	} else if (y <= 8.2e-176) {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	} else if (y <= 4.7e-168) {
		tmp = (x - y) / (x / t_0);
	} else {
		tmp = ((x - y) * ((x + y) / y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y / x)
	tmp = 0
	if y <= 9e-205:
		tmp = 1.0 / (x / ((x - y) * t_0))
	elif y <= 8.2e-176:
		tmp = ((x - y) * (1.0 + (x / y))) / y
	elif y <= 4.7e-168:
		tmp = (x - y) / (x / t_0)
	else:
		tmp = ((x - y) * ((x + y) / y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y / x))
	tmp = 0.0
	if (y <= 9e-205)
		tmp = Float64(1.0 / Float64(x / Float64(Float64(x - y) * t_0)));
	elseif (y <= 8.2e-176)
		tmp = Float64(Float64(Float64(x - y) * Float64(1.0 + Float64(x / y))) / y);
	elseif (y <= 4.7e-168)
		tmp = Float64(Float64(x - y) / Float64(x / t_0));
	else
		tmp = Float64(Float64(Float64(x - y) * Float64(Float64(x + y) / y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y / x);
	tmp = 0.0;
	if (y <= 9e-205)
		tmp = 1.0 / (x / ((x - y) * t_0));
	elseif (y <= 8.2e-176)
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	elseif (y <= 4.7e-168)
		tmp = (x - y) / (x / t_0);
	else
		tmp = ((x - y) * ((x + y) / y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9e-205], N[(1.0 / N[(x / N[(N[(x - y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-176], N[(N[(N[(x - y), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 4.7e-168], N[(N[(x - y), $MachinePrecision] / N[(x / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.99999999999999912e-205

   1. Initial program 58.4%

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

      1. associate-/l* 58.3%

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

      2. fma-define 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around inf 36.6%

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

      1. associate-*r/ 36.8%

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

      2. clear-num 36.8%

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

   7. Applied egg-rr 36.8%

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

   if 8.99999999999999912e-205 < y < 8.2000000000000005e-176

   1. Initial program 14.3%

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

      1. associate-/l* 17.0%

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

      2. fma-define 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in y around inf 83.5%

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

      1. associate-*r/ 84.0%

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

   7. Applied egg-rr 84.0%

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

   if 8.2000000000000005e-176 < y < 4.70000000000000026e-168

   1. Initial program 66.7%

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

      1. associate-/l* 67.2%

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

      2. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in x around inf 69.3%

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

      1. clear-num 69.3%

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

      2. un-div-inv 69.8%

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

   7. Applied egg-rr 69.8%

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

   if 4.70000000000000026e-168 < 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* 98.8%

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

      2. fma-define 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around inf 70.2%

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

      1. associate-*r/ 70.2%

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

   7. Applied egg-rr 70.2%

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

   8. Taylor expanded in y around 0 70.2%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(1 + \frac{y}{x}\right)}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-168}:\\ \;\;\;\;\frac{x - y}{\frac{x}{1 + \frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \frac{x + y}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-175}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-205)
   1.0
   (if (<= y 1.5e-175) -1.0 (if (<= y 5e-168) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-205) {
		tmp = 1.0;
	} else if (y <= 1.5e-175) {
		tmp = -1.0;
	} else if (y <= 5e-168) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.5d-205) then
        tmp = 1.0d0
    else if (y <= 1.5d-175) then
        tmp = -1.0d0
    else if (y <= 5d-168) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-205) {
		tmp = 1.0;
	} else if (y <= 1.5e-175) {
		tmp = -1.0;
	} else if (y <= 5e-168) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-205:
		tmp = 1.0
	elif y <= 1.5e-175:
		tmp = -1.0
	elif y <= 5e-168:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-205)
		tmp = 1.0;
	elseif (y <= 1.5e-175)
		tmp = -1.0;
	elseif (y <= 5e-168)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-205)
		tmp = 1.0;
	elseif (y <= 1.5e-175)
		tmp = -1.0;
	elseif (y <= 5e-168)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-205], 1.0, If[LessEqual[y, 1.5e-175], -1.0, If[LessEqual[y, 5e-168], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000005e-205 or 1.5e-175 < y < 5.00000000000000001e-168

   1. Initial program 58.6%

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

      1. associate-/l* 58.4%

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

      2. fma-define 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in x around inf 35.5%

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

   if 8.5000000000000005e-205 < y < 1.5e-175 or 5.00000000000000001e-168 < y

   1. Initial program 89.6%

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

      1. associate-/l* 88.9%

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

      2. fma-define 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in x around 0 70.2%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-175}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.8% 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 65.6%

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

   1. associate-/l* 65.4%

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

   2. fma-define 65.4%

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

3. Simplified 65.4%

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

5. Taylor expanded in x around 0 66.2%

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

6. Final simplification 66.2%

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

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if ((0.5d0 < t_0) .and. (t_0 < 2.0d0)) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = 1.0d0 - (2.0d0 / (1.0d0 + ((x / y) * (x / y))))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	t_0 = math.fabs((x / y))
	tmp = 0
	if (0.5 < t_0) and (t_0 < 2.0):
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs((x / y));
	tmp = 0.0;
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[And[Less[0.5, t$95$0], Less[t$95$0, 2.0]], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / N[(1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024055 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (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))))