Kahan p9 Example

Percentage Accurate: 69.0% → 100.0%

Time: 6.3s

Alternatives: 9

Speedup: 0.5×

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: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.4%

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

   1. add-sqr-sqrt 66.4%

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

   2. times-frac 66.9%

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

   3. hypot-def 67.0%

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

   4. hypot-def 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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.4%

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

   1. add-sqr-sqrt 66.4%

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

   2. times-frac 66.9%

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

   3. hypot-def 67.0%

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

   4. hypot-def 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 3: 73.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x + y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-def 3.1%

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

      4. hypot-def 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 6.7%

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

      1. +-commutative 6.7%

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

   6. Simplified 6.7%

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

4. Final simplification 68.7%

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

Alternative 4: 92.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-def 3.1%

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

      4. hypot-def 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 6.7%

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

      1. +-commutative 6.7%

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

   6. Simplified 6.7%

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

   7. Taylor expanded in x around 0 85.6%

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

4. Final simplification 95.2%

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

Alternative 5: 42.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-206}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-151} \lor \neg \left(y \leq 1.7 \cdot 10^{-146}\right):\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.7e-206)
   (* (- 1.0 (/ y x)) (+ 1.0 (/ y x)))
   (if (or (<= y 6.5e-151) (not (<= y 1.7e-146)))
     (* (+ (/ x y) 1.0) (+ (/ x y) -1.0))
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.7e-206) {
		tmp = (1.0 - (y / x)) * (1.0 + (y / x));
	} else if ((y <= 6.5e-151) || !(y <= 1.7e-146)) {
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.7d-206) then
        tmp = (1.0d0 - (y / x)) * (1.0d0 + (y / x))
    else if ((y <= 6.5d-151) .or. (.not. (y <= 1.7d-146))) then
        tmp = ((x / y) + 1.0d0) * ((x / y) + (-1.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.7e-206) {
		tmp = (1.0 - (y / x)) * (1.0 + (y / x));
	} else if ((y <= 6.5e-151) || !(y <= 1.7e-146)) {
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.7e-206:
		tmp = (1.0 - (y / x)) * (1.0 + (y / x))
	elif (y <= 6.5e-151) or not (y <= 1.7e-146):
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.7e-206)
		tmp = Float64(Float64(1.0 - Float64(y / x)) * Float64(1.0 + Float64(y / x)));
	elseif ((y <= 6.5e-151) || !(y <= 1.7e-146))
		tmp = Float64(Float64(Float64(x / y) + 1.0) * Float64(Float64(x / y) + -1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.7e-206)
		tmp = (1.0 - (y / x)) * (1.0 + (y / x));
	elseif ((y <= 6.5e-151) || ~((y <= 1.7e-146)))
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.7e-206], N[(N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 6.5e-151], N[Not[LessEqual[y, 1.7e-146]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 3.69999999999999998e-206

   1. Initial program 61.0%

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

      1. add-sqr-sqrt 61.0%

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

      2. times-frac 61.7%

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

      3. hypot-def 61.7%

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

      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 inf 25.8%

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

      1. mul-1-neg 25.8%

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

      2. unsub-neg 25.8%

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

   6. Simplified 25.8%

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

   7. Taylor expanded in x around inf 25.2%

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

   if 3.69999999999999998e-206 < y < 6.4999999999999994e-151 or 1.7e-146 < y

   1. Initial program 92.3%

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

      1. add-sqr-sqrt 92.2%

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

      2. times-frac 92.3%

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

      3. hypot-def 92.3%

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

      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. Taylor expanded in x around 0 91.5%

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

      1. +-commutative 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in x around 0 91.4%

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

   if 6.4999999999999994e-151 < y < 1.7e-146

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-206}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-151} \lor \neg \left(y \leq 1.7 \cdot 10^{-146}\right):\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 41.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-205}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \left(1 - \frac{y}{x}\right) - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-151} \lor \neg \left(y \leq 1.14 \cdot 10^{-147}\right):\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.5e-205)
   (+ 1.0 (- (* (/ y x) (- 1.0 (/ y x))) (/ y x)))
   (if (or (<= y 7.8e-151) (not (<= y 1.14e-147)))
     (* (+ (/ x y) 1.0) (+ (/ x y) -1.0))
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-205) {
		tmp = 1.0 + (((y / x) * (1.0 - (y / x))) - (y / x));
	} else if ((y <= 7.8e-151) || !(y <= 1.14e-147)) {
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.5d-205) then
        tmp = 1.0d0 + (((y / x) * (1.0d0 - (y / x))) - (y / x))
    else if ((y <= 7.8d-151) .or. (.not. (y <= 1.14d-147))) then
        tmp = ((x / y) + 1.0d0) * ((x / y) + (-1.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-205) {
		tmp = 1.0 + (((y / x) * (1.0 - (y / x))) - (y / x));
	} else if ((y <= 7.8e-151) || !(y <= 1.14e-147)) {
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.5e-205:
		tmp = 1.0 + (((y / x) * (1.0 - (y / x))) - (y / x))
	elif (y <= 7.8e-151) or not (y <= 1.14e-147):
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-205)
		tmp = Float64(1.0 + Float64(Float64(Float64(y / x) * Float64(1.0 - Float64(y / x))) - Float64(y / x)));
	elseif ((y <= 7.8e-151) || !(y <= 1.14e-147))
		tmp = Float64(Float64(Float64(x / y) + 1.0) * Float64(Float64(x / y) + -1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.5e-205)
		tmp = 1.0 + (((y / x) * (1.0 - (y / x))) - (y / x));
	elseif ((y <= 7.8e-151) || ~((y <= 1.14e-147)))
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.5e-205], N[(1.0 + N[(N[(N[(y / x), $MachinePrecision] * N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 7.8e-151], N[Not[LessEqual[y, 1.14e-147]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 3.5e-205

   1. Initial program 61.0%

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

      1. add-sqr-sqrt 61.0%

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

      2. times-frac 61.7%

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

      3. hypot-def 61.7%

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

      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 inf 25.8%

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

      1. mul-1-neg 25.8%

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

      2. unsub-neg 25.8%

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

   6. Simplified 25.8%

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

   7. Taylor expanded in x around inf 25.2%

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

      1. distribute-rgt-in 24.5%

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

      2. *-un-lft-identity 24.5%

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

      3. associate-+l- 24.5%

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

   9. Applied egg-rr 24.5%

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

   if 3.5e-205 < y < 7.80000000000000013e-151 or 1.14e-147 < y

   1. Initial program 92.3%

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

      1. add-sqr-sqrt 92.2%

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

      2. times-frac 92.3%

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

      3. hypot-def 92.3%

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

      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. Taylor expanded in x around 0 91.5%

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

      1. +-commutative 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in x around 0 91.4%

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

   if 7.80000000000000013e-151 < y < 1.14e-147

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-205}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \left(1 - \frac{y}{x}\right) - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-151} \lor \neg \left(y \leq 1.14 \cdot 10^{-147}\right):\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 41.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-205}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-147}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-205)
   (* (- 1.0 (/ y x)) (+ 1.0 (/ y x)))
   (if (<= y 7.1e-152) -1.0 (if (<= y 1e-147) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-205) {
		tmp = (1.0 - (y / x)) * (1.0 + (y / x));
	} else if (y <= 7.1e-152) {
		tmp = -1.0;
	} else if (y <= 1e-147) {
		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 <= 3.3d-205) then
        tmp = (1.0d0 - (y / x)) * (1.0d0 + (y / x))
    else if (y <= 7.1d-152) then
        tmp = -1.0d0
    else if (y <= 1d-147) 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 <= 3.3e-205) {
		tmp = (1.0 - (y / x)) * (1.0 + (y / x));
	} else if (y <= 7.1e-152) {
		tmp = -1.0;
	} else if (y <= 1e-147) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e-205:
		tmp = (1.0 - (y / x)) * (1.0 + (y / x))
	elif y <= 7.1e-152:
		tmp = -1.0
	elif y <= 1e-147:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-205)
		tmp = Float64(Float64(1.0 - Float64(y / x)) * Float64(1.0 + Float64(y / x)));
	elseif (y <= 7.1e-152)
		tmp = -1.0;
	elseif (y <= 1e-147)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e-205)
		tmp = (1.0 - (y / x)) * (1.0 + (y / x));
	elseif (y <= 7.1e-152)
		tmp = -1.0;
	elseif (y <= 1e-147)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e-205], N[(N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.1e-152], -1.0, If[LessEqual[y, 1e-147], 1.0, -1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.2999999999999999e-205

   1. Initial program 61.0%

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

      1. add-sqr-sqrt 61.0%

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

      2. times-frac 61.7%

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

      3. hypot-def 61.7%

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

      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 inf 25.8%

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

      1. mul-1-neg 25.8%

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

      2. unsub-neg 25.8%

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

   6. Simplified 25.8%

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

   7. Taylor expanded in x around inf 25.2%

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

   if 3.2999999999999999e-205 < y < 7.10000000000000011e-152 or 9.9999999999999997e-148 < y

   1. Initial program 92.3%

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

   2. Taylor expanded in x around 0 91.0%

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

   if 7.10000000000000011e-152 < y < 9.9999999999999997e-148

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-205}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-147}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 40.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-146}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-205)
   1.0
   (if (<= y 2.2e-151) -1.0 (if (<= y 1.2e-146) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-205) {
		tmp = 1.0;
	} else if (y <= 2.2e-151) {
		tmp = -1.0;
	} else if (y <= 1.2e-146) {
		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 <= 3.3d-205) then
        tmp = 1.0d0
    else if (y <= 2.2d-151) then
        tmp = -1.0d0
    else if (y <= 1.2d-146) 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 <= 3.3e-205) {
		tmp = 1.0;
	} else if (y <= 2.2e-151) {
		tmp = -1.0;
	} else if (y <= 1.2e-146) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e-205:
		tmp = 1.0
	elif y <= 2.2e-151:
		tmp = -1.0
	elif y <= 1.2e-146:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-205)
		tmp = 1.0;
	elseif (y <= 2.2e-151)
		tmp = -1.0;
	elseif (y <= 1.2e-146)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e-205)
		tmp = 1.0;
	elseif (y <= 2.2e-151)
		tmp = -1.0;
	elseif (y <= 1.2e-146)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e-205], 1.0, If[LessEqual[y, 2.2e-151], -1.0, If[LessEqual[y, 1.2e-146], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2999999999999999e-205 or 2.1999999999999999e-151 < y < 1.2000000000000001e-146

   1. Initial program 61.8%

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

   2. Taylor expanded in x around inf 24.7%

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

   if 3.2999999999999999e-205 < y < 2.1999999999999999e-151 or 1.2000000000000001e-146 < y

   1. Initial program 92.3%

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

   2. Taylor expanded in x around 0 91.0%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-146}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9: 67.0% 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 66.4%

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

2. Taylor expanded in x around 0 78.6%

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

3. Final simplification 78.6%

   \[\leadsto -1 \]

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 2023322 
(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))))