Kahan p9 Example

Percentage Accurate: 68.5% → 99.9%

Time: 7.4s

Alternatives: 10

Speedup: 2.9×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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 71.9%

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

   1. add-sqr-sqrt 71.9%

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

   2. times-frac 71.2%

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

   3. hypot-def 71.3%

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

   4. hypot-def 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. associate-*r/ 70.9%

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

   2. +-commutative 70.9%

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

   3. fma-def 71.0%

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

3. Simplified 71.0%

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

   1. fma-udef 70.9%

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

   2. +-commutative 70.9%

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

   3. *-un-lft-identity 70.9%

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

   4. add-sqr-sqrt 71.0%

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

   5. times-frac 71.1%

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

   6. hypot-def 71.2%

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

   7. hypot-def 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

   3. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 92.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-def 3.1%

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

      4. hypot-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 54.2%

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

      1. fma-neg 54.2%

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

      2. unpow2 54.2%

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

      3. unpow2 54.2%

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

      4. times-frac 77.8%

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

      5. metadata-eval 77.8%

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

   6. Simplified 77.8%

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

      1. fma-udef 77.8%

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

      2. *-commutative 77.8%

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

      3. pow2 77.8%

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

   8. Applied egg-rr 77.8%

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

      1. unpow2 77.8%

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

      2. clear-num 77.8%

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

      3. un-div-inv 77.8%

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

   10. Applied egg-rr 77.8%

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

4. Final simplification 93.7%

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

Alternative 4: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-127} \lor \neg \left(y \leq -1.52 \cdot 10^{-149} \lor \neg \left(y \leq -1.22 \cdot 10^{-182}\right) \land y \leq 3.6 \cdot 10^{-109}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.5e-127)
         (not
          (or (<= y -1.52e-149)
              (and (not (<= y -1.22e-182)) (<= y 3.6e-109)))))
   (+ -1.0 (* (/ x y) (/ x y)))
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.5e-127) || !((y <= -1.52e-149) || (!(y <= -1.22e-182) && (y <= 3.6e-109)))) {
		tmp = -1.0 + ((x / y) * (x / 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 <= (-7.5d-127)) .or. (.not. (y <= (-1.52d-149)) .or. (.not. (y <= (-1.22d-182))) .and. (y <= 3.6d-109))) then
        tmp = (-1.0d0) + ((x / y) * (x / y))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.5e-127) || !((y <= -1.52e-149) || (!(y <= -1.22e-182) && (y <= 3.6e-109)))) {
		tmp = -1.0 + ((x / y) * (x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.5e-127) or not ((y <= -1.52e-149) or (not (y <= -1.22e-182) and (y <= 3.6e-109))):
		tmp = -1.0 + ((x / y) * (x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.5e-127) || !((y <= -1.52e-149) || (!(y <= -1.22e-182) && (y <= 3.6e-109))))
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.5e-127) || ~(((y <= -1.52e-149) || (~((y <= -1.22e-182)) && (y <= 3.6e-109)))))
		tmp = -1.0 + ((x / y) * (x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.5e-127], N[Not[Or[LessEqual[y, -1.52e-149], And[N[Not[LessEqual[y, -1.22e-182]], $MachinePrecision], LessEqual[y, 3.6e-109]]]], $MachinePrecision]], N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -7.5000000000000004e-127 or -1.5199999999999999e-149 < y < -1.22e-182 or 3.6000000000000001e-109 < y

   1. Initial program 72.8%

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

      1. associate-*r/ 72.6%

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

      2. +-commutative 72.6%

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

      3. fma-def 72.6%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in y around inf 63.6%

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

      1. unpow2 63.6%

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

   6. Simplified 63.6%

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

   7. Taylor expanded in x around 0 88.5%

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

      1. sub-neg 88.5%

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

      2. metadata-eval 88.5%

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

      3. unpow2 88.5%

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

      4. unpow2 88.5%

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

      5. times-frac 91.1%

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

   9. Simplified 91.1%

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

   if -7.5000000000000004e-127 < y < -1.5199999999999999e-149 or -1.22e-182 < y < 3.6000000000000001e-109

   1. Initial program 70.4%

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

      1. associate-*r/ 68.3%

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

      2. +-commutative 68.3%

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

      3. fma-def 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in x around inf 80.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-127} \lor \neg \left(y \leq -1.52 \cdot 10^{-149} \lor \neg \left(y \leq -1.22 \cdot 10^{-182}\right) \land y \leq 3.6 \cdot 10^{-109}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{if}\;y \leq -7.6 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-152}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ (/ x y) 1.0) (+ (/ x y) -1.0))))
   (if (<= y -7.6e-126)
     t_0
     (if (<= y -1.45e-152)
       1.0
       (if (<= y -5e-184)
         t_0
         (if (<= y 3.6e-109) 1.0 (+ -1.0 (* (/ x y) (/ x y)))))))))

C

double code(double x, double y) {
	double t_0 = ((x / y) + 1.0) * ((x / y) + -1.0);
	double tmp;
	if (y <= -7.6e-126) {
		tmp = t_0;
	} else if (y <= -1.45e-152) {
		tmp = 1.0;
	} else if (y <= -5e-184) {
		tmp = t_0;
	} else if (y <= 3.6e-109) {
		tmp = 1.0;
	} else {
		tmp = -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 = ((x / y) + 1.0d0) * ((x / y) + (-1.0d0))
    if (y <= (-7.6d-126)) then
        tmp = t_0
    else if (y <= (-1.45d-152)) then
        tmp = 1.0d0
    else if (y <= (-5d-184)) then
        tmp = t_0
    else if (y <= 3.6d-109) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + ((x / y) * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x / y) + 1.0) * ((x / y) + -1.0);
	double tmp;
	if (y <= -7.6e-126) {
		tmp = t_0;
	} else if (y <= -1.45e-152) {
		tmp = 1.0;
	} else if (y <= -5e-184) {
		tmp = t_0;
	} else if (y <= 3.6e-109) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((x / y) * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x / y) + 1.0) * ((x / y) + -1.0)
	tmp = 0
	if y <= -7.6e-126:
		tmp = t_0
	elif y <= -1.45e-152:
		tmp = 1.0
	elif y <= -5e-184:
		tmp = t_0
	elif y <= 3.6e-109:
		tmp = 1.0
	else:
		tmp = -1.0 + ((x / y) * (x / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x / y) + 1.0) * Float64(Float64(x / y) + -1.0))
	tmp = 0.0
	if (y <= -7.6e-126)
		tmp = t_0;
	elseif (y <= -1.45e-152)
		tmp = 1.0;
	elseif (y <= -5e-184)
		tmp = t_0;
	elseif (y <= 3.6e-109)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x / y) + 1.0) * ((x / y) + -1.0);
	tmp = 0.0;
	if (y <= -7.6e-126)
		tmp = t_0;
	elseif (y <= -1.45e-152)
		tmp = 1.0;
	elseif (y <= -5e-184)
		tmp = t_0;
	elseif (y <= 3.6e-109)
		tmp = 1.0;
	else
		tmp = -1.0 + ((x / y) * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.6e-126], t$95$0, If[LessEqual[y, -1.45e-152], 1.0, If[LessEqual[y, -5e-184], t$95$0, If[LessEqual[y, 3.6e-109], 1.0, N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5999999999999997e-126 or -1.4500000000000001e-152 < y < -5.00000000000000003e-184

   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-*r/ 65.5%

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

      2. +-commutative 65.5%

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

      3. fma-def 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in y around inf 57.7%

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

      1. unpow2 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in x around 0 89.0%

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

      1. sub-neg 89.0%

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

      2. metadata-eval 89.0%

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

      3. unpow2 89.0%

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

      4. unpow2 89.0%

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

      5. times-frac 92.3%

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

   9. Simplified 92.3%

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

       1. difference-of-sqr--1 92.3%

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

       2. sub-neg 92.3%

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

       3. metadata-eval 92.3%

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

   11. Applied egg-rr 92.3%

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

   if -7.5999999999999997e-126 < y < -1.4500000000000001e-152 or -5.00000000000000003e-184 < y < 3.6000000000000001e-109

   1. Initial program 70.4%

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

      1. associate-*r/ 68.3%

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

      2. +-commutative 68.3%

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

      3. fma-def 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in x around inf 80.7%

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

   if 3.6000000000000001e-109 < 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-*r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 86.1%

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

      1. unpow2 86.1%

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

   6. Simplified 86.1%

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

   7. Taylor expanded in x around 0 86.3%

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

      1. sub-neg 86.3%

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

      2. metadata-eval 86.3%

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

      3. unpow2 86.3%

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

      4. unpow2 86.3%

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

      5. times-frac 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-126}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-152}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 6: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-127} \lor \neg \left(y \leq 3.6 \cdot 10^{-109}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{\frac{x}{\frac{y}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.5e-127) (not (<= y 3.6e-109)))
   (+ (* 2.0 (/ (/ x y) (/ y x))) -1.0)
   (+ 1.0 (* -2.0 (/ y (/ x (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.5e-127) || !(y <= 3.6e-109)) {
		tmp = (2.0 * ((x / y) / (y / x))) + -1.0;
	} else {
		tmp = 1.0 + (-2.0 * (y / (x / (y / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.5d-127)) .or. (.not. (y <= 3.6d-109))) then
        tmp = (2.0d0 * ((x / y) / (y / x))) + (-1.0d0)
    else
        tmp = 1.0d0 + ((-2.0d0) * (y / (x / (y / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.5e-127) || !(y <= 3.6e-109)) {
		tmp = (2.0 * ((x / y) / (y / x))) + -1.0;
	} else {
		tmp = 1.0 + (-2.0 * (y / (x / (y / x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.5e-127) or not (y <= 3.6e-109):
		tmp = (2.0 * ((x / y) / (y / x))) + -1.0
	else:
		tmp = 1.0 + (-2.0 * (y / (x / (y / x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.5e-127) || !(y <= 3.6e-109))
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y) / Float64(y / x))) + -1.0);
	else
		tmp = Float64(1.0 + Float64(-2.0 * Float64(y / Float64(x / Float64(y / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.5e-127) || ~((y <= 3.6e-109)))
		tmp = (2.0 * ((x / y) / (y / x))) + -1.0;
	else
		tmp = 1.0 + (-2.0 * (y / (x / (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.5e-127], N[Not[LessEqual[y, 3.6e-109]], $MachinePrecision]], N[(N[(2.0 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-2.0 * N[(y / N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5000000000000004e-127 or 3.6000000000000001e-109 < y

   1. Initial program 73.3%

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

      1. add-sqr-sqrt 73.3%

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

      2. times-frac 74.1%

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

      3. hypot-def 74.1%

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

      4. hypot-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 93.1%

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

      1. fma-neg 93.1%

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

      2. unpow2 93.1%

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

      3. unpow2 93.1%

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

      4. times-frac 93.1%

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

      5. metadata-eval 93.1%

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

   6. Simplified 93.1%

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

      1. fma-udef 93.1%

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

      2. *-commutative 93.1%

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

      3. pow2 93.1%

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

   8. Applied egg-rr 93.1%

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

      1. unpow2 93.1%

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

      2. clear-num 93.1%

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

      3. un-div-inv 93.1%

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

   10. Applied egg-rr 93.1%

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

   if -7.5000000000000004e-127 < y < 3.6000000000000001e-109

   1. Initial program 70.0%

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

      1. associate-*r/ 67.1%

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

      2. +-commutative 67.1%

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

      3. fma-def 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in y around 0 60.5%

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

      1. unpow2 60.5%

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

      2. unpow2 60.5%

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

   6. Simplified 60.5%

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

      1. associate-/l* 61.0%

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

      2. div-inv 61.0%

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

   8. Applied egg-rr 61.0%

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

      1. associate-*r/ 61.0%

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

      2. *-rgt-identity 61.0%

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

      3. associate-/l* 76.8%

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

   10. Simplified 76.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-127} \lor \neg \left(y \leq 3.6 \cdot 10^{-109}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{\frac{x}{\frac{y}{x}}}\\ \end{array} \]

Alternative 7: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-127}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-109}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{\frac{x}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e-127)
   (* (+ (/ x y) 1.0) (+ (/ x y) -1.0))
   (if (<= y 3.7e-109)
     (+ 1.0 (* -2.0 (/ y (/ x (/ y x)))))
     (+ -1.0 (* (/ x y) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e-127) {
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	} else if (y <= 3.7e-109) {
		tmp = 1.0 + (-2.0 * (y / (x / (y / x))));
	} else {
		tmp = -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) :: tmp
    if (y <= (-9d-127)) then
        tmp = ((x / y) + 1.0d0) * ((x / y) + (-1.0d0))
    else if (y <= 3.7d-109) then
        tmp = 1.0d0 + ((-2.0d0) * (y / (x / (y / x))))
    else
        tmp = (-1.0d0) + ((x / y) * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9e-127) {
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	} else if (y <= 3.7e-109) {
		tmp = 1.0 + (-2.0 * (y / (x / (y / x))));
	} else {
		tmp = -1.0 + ((x / y) * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e-127:
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0)
	elif y <= 3.7e-109:
		tmp = 1.0 + (-2.0 * (y / (x / (y / x))))
	else:
		tmp = -1.0 + ((x / y) * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e-127)
		tmp = Float64(Float64(Float64(x / y) + 1.0) * Float64(Float64(x / y) + -1.0));
	elseif (y <= 3.7e-109)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(y / Float64(x / Float64(y / x)))));
	else
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9e-127)
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	elseif (y <= 3.7e-109)
		tmp = 1.0 + (-2.0 * (y / (x / (y / x))));
	else
		tmp = -1.0 + ((x / y) * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e-127], N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e-109], N[(1.0 + N[(-2.0 * N[(y / N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999998e-127

   1. Initial program 65.5%

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

      1. associate-*r/ 66.3%

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

      2. +-commutative 66.3%

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

      3. fma-def 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in y around inf 61.1%

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

      1. unpow2 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in x around 0 94.8%

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

      1. sub-neg 94.8%

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

      2. metadata-eval 94.8%

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

      3. unpow2 94.8%

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

      4. unpow2 94.8%

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

      5. times-frac 94.8%

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

   9. Simplified 94.8%

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

       1. difference-of-sqr--1 94.8%

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

       2. sub-neg 94.8%

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

       3. metadata-eval 94.8%

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

   11. Applied egg-rr 94.8%

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

   if -8.9999999999999998e-127 < y < 3.69999999999999981e-109

   1. Initial program 70.0%

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

      1. associate-*r/ 67.1%

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

      2. +-commutative 67.1%

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

      3. fma-def 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in y around 0 60.5%

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

      1. unpow2 60.5%

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

      2. unpow2 60.5%

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

   6. Simplified 60.5%

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

      1. associate-/l* 61.0%

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

      2. div-inv 61.0%

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

   8. Applied egg-rr 61.0%

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

      1. associate-*r/ 61.0%

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

      2. *-rgt-identity 61.0%

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

      3. associate-/l* 76.8%

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

   10. Simplified 76.8%

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

   if 3.69999999999999981e-109 < 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-*r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 86.1%

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

      1. unpow2 86.1%

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

   6. Simplified 86.1%

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

   7. Taylor expanded in x around 0 86.3%

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

      1. sub-neg 86.3%

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

      2. metadata-eval 86.3%

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

      3. unpow2 86.3%

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

      4. unpow2 86.3%

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

      5. times-frac 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-127}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-109}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{\frac{x}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 8: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-126}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.6e-126) -1.0 (if (<= y 3.6e-109) 1.0 (/ (- x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.6e-126) {
		tmp = -1.0;
	} else if (y <= 3.6e-109) {
		tmp = 1.0;
	} else {
		tmp = (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 <= (-7.6d-126)) then
        tmp = -1.0d0
    else if (y <= 3.6d-109) then
        tmp = 1.0d0
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.6e-126) {
		tmp = -1.0;
	} else if (y <= 3.6e-109) {
		tmp = 1.0;
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.6e-126:
		tmp = -1.0
	elif y <= 3.6e-109:
		tmp = 1.0
	else:
		tmp = (x - y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.6e-126)
		tmp = -1.0;
	elseif (y <= 3.6e-109)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.6e-126)
		tmp = -1.0;
	elseif (y <= 3.6e-109)
		tmp = 1.0;
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.6e-126], -1.0, If[LessEqual[y, 3.6e-109], 1.0, N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5999999999999997e-126

   1. Initial program 65.5%

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

      1. associate-*r/ 66.3%

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

      2. +-commutative 66.3%

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

      3. fma-def 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in x around 0 94.6%

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

   if -7.5999999999999997e-126 < y < 3.6000000000000001e-109

   1. Initial program 70.0%

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

      1. associate-*r/ 67.1%

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

      2. +-commutative 67.1%

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

      3. fma-def 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in x around inf 75.7%

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

   if 3.6000000000000001e-109 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 85.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-126}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 9: 82.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-126}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.4e-126) -1.0 (if (<= y 5e-109) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.4e-126) {
		tmp = -1.0;
	} else if (y <= 5e-109) {
		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 <= (-4.4d-126)) then
        tmp = -1.0d0
    else if (y <= 5d-109) 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 <= -4.4e-126) {
		tmp = -1.0;
	} else if (y <= 5e-109) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.4e-126:
		tmp = -1.0
	elif y <= 5e-109:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.4e-126)
		tmp = -1.0;
	elseif (y <= 5e-109)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.4e-126)
		tmp = -1.0;
	elseif (y <= 5e-109)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.4e-126], -1.0, If[LessEqual[y, 5e-109], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.40000000000000029e-126 or 5.0000000000000002e-109 < y

   1. Initial program 73.3%

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

      1. associate-*r/ 73.8%

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

      2. +-commutative 73.8%

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

      3. fma-def 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in x around 0 92.5%

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

   if -4.40000000000000029e-126 < y < 5.0000000000000002e-109

   1. Initial program 70.0%

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

      1. associate-*r/ 67.1%

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

      2. +-commutative 67.1%

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

      3. fma-def 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in x around inf 75.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-126}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 66.4% 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 71.9%

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

   1. associate-*r/ 70.9%

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

   2. +-commutative 70.9%

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

   3. fma-def 71.0%

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

3. Simplified 71.0%

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

4. Taylor expanded in x around 0 63.3%

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

5. Final simplification 63.3%

   \[\leadsto -1 \]

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023192 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

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

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