Kahan p9 Example

Percentage Accurate: 68.8% → 100.0%

Time: 15.2s

Alternatives: 9

Speedup: 2.5×

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: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ (/ (- x y_m) (/ (hypot x y_m) (+ x y_m))) (hypot x y_m)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x - y$95$m), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

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

   1. add-sqr-sqrt 66.8%

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

   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 66.9%

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

   4. hypot-def 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

6. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. un-div-inv 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* (/ (+ x y_m) (hypot x y_m)) (/ (- x y_m) (hypot x y_m))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

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

   1. add-sqr-sqrt 66.8%

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

   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 66.9%

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

   4. hypot-def 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ (* (- x y_m) (/ (+ x y_m) (hypot x y_m))) (hypot x y_m)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

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

   1. add-sqr-sqrt 66.8%

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

   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 66.9%

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

   4. hypot-def 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 4: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x - y_m}{y_m}\\ \mathbf{if}\;y_m \leq 5.6 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;y_m \leq 2.4 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;y_m \leq 8 \cdot 10^{-93}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (- x y_m) y_m)))
   (if (<= y_m 5.6e-147)
     1.0
     (if (<= y_m 2.4e-106)
       t_0
       (if (<= y_m 1e-99) 1.0 (if (<= y_m 8e-93) -1.0 t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (x - y_m) / y_m;
	double tmp;
	if (y_m <= 5.6e-147) {
		tmp = 1.0;
	} else if (y_m <= 2.4e-106) {
		tmp = t_0;
	} else if (y_m <= 1e-99) {
		tmp = 1.0;
	} else if (y_m <= 8e-93) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y_m) / y_m
    if (y_m <= 5.6d-147) then
        tmp = 1.0d0
    else if (y_m <= 2.4d-106) then
        tmp = t_0
    else if (y_m <= 1d-99) then
        tmp = 1.0d0
    else if (y_m <= 8d-93) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = (x - y_m) / y_m;
	double tmp;
	if (y_m <= 5.6e-147) {
		tmp = 1.0;
	} else if (y_m <= 2.4e-106) {
		tmp = t_0;
	} else if (y_m <= 1e-99) {
		tmp = 1.0;
	} else if (y_m <= 8e-93) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = (x - y_m) / y_m
	tmp = 0
	if y_m <= 5.6e-147:
		tmp = 1.0
	elif y_m <= 2.4e-106:
		tmp = t_0
	elif y_m <= 1e-99:
		tmp = 1.0
	elif y_m <= 8e-93:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(x - y_m) / y_m)
	tmp = 0.0
	if (y_m <= 5.6e-147)
		tmp = 1.0;
	elseif (y_m <= 2.4e-106)
		tmp = t_0;
	elseif (y_m <= 1e-99)
		tmp = 1.0;
	elseif (y_m <= 8e-93)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = (x - y_m) / y_m;
	tmp = 0.0;
	if (y_m <= 5.6e-147)
		tmp = 1.0;
	elseif (y_m <= 2.4e-106)
		tmp = t_0;
	elseif (y_m <= 1e-99)
		tmp = 1.0;
	elseif (y_m <= 8e-93)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(x - y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[y$95$m, 5.6e-147], 1.0, If[LessEqual[y$95$m, 2.4e-106], t$95$0, If[LessEqual[y$95$m, 1e-99], 1.0, If[LessEqual[y$95$m, 8e-93], -1.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.6000000000000001e-147 or 2.3999999999999998e-106 < y < 1e-99

   1. Initial program 59.9%

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

   3. Taylor expanded in x around inf 37.5%

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

   if 5.6000000000000001e-147 < y < 2.3999999999999998e-106 or 7.9999999999999992e-93 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. remove-double-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. remove-double-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-def 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 75.5%

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

   if 1e-99 < y < 7.9999999999999992e-93

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 58.3%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-93}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.15 \cdot 10^{-165}:\\ \;\;\;\;1\\ \mathbf{elif}\;y_m \leq 0.01:\\ \;\;\;\;\frac{\left(x - y_m\right) \cdot \left(x + y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.15e-165)
   1.0
   (if (<= y_m 0.01)
     (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.15e-165) {
		tmp = 1.0;
	} else if (y_m <= 0.01) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.15d-165) then
        tmp = 1.0d0
    else if (y_m <= 0.01d0) then
        tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.15e-165) {
		tmp = 1.0;
	} else if (y_m <= 0.01) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.15e-165:
		tmp = 1.0
	elif y_m <= 0.01:
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.15e-165)
		tmp = 1.0;
	elseif (y_m <= 0.01)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.15e-165)
		tmp = 1.0;
	elseif (y_m <= 0.01)
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.15e-165], 1.0, If[LessEqual[y$95$m, 0.01], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15e-165

   1. Initial program 58.9%

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

   3. Taylor expanded in x around inf 36.3%

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

   if 1.15e-165 < y < 0.0100000000000000002

   1. Initial program 99.9%

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

   if 0.0100000000000000002 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-165}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.01:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.15 \cdot 10^{-165}:\\ \;\;\;\;\frac{x - y_m}{\left(x - y_m\right) + 2 \cdot \frac{y_m}{\frac{x}{y_m}}}\\ \mathbf{elif}\;y_m \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(x - y_m\right) \cdot \left(x + y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.15e-165)
   (/ (- x y_m) (+ (- x y_m) (* 2.0 (/ y_m (/ x y_m)))))
   (if (<= y_m 2e-9)
     (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.15e-165) {
		tmp = (x - y_m) / ((x - y_m) + (2.0 * (y_m / (x / y_m))));
	} else if (y_m <= 2e-9) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.15d-165) then
        tmp = (x - y_m) / ((x - y_m) + (2.0d0 * (y_m / (x / y_m))))
    else if (y_m <= 2d-9) then
        tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.15e-165) {
		tmp = (x - y_m) / ((x - y_m) + (2.0 * (y_m / (x / y_m))));
	} else if (y_m <= 2e-9) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.15e-165:
		tmp = (x - y_m) / ((x - y_m) + (2.0 * (y_m / (x / y_m))))
	elif y_m <= 2e-9:
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.15e-165)
		tmp = Float64(Float64(x - y_m) / Float64(Float64(x - y_m) + Float64(2.0 * Float64(y_m / Float64(x / y_m)))));
	elseif (y_m <= 2e-9)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.15e-165)
		tmp = (x - y_m) / ((x - y_m) + (2.0 * (y_m / (x / y_m))));
	elseif (y_m <= 2e-9)
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.15e-165], N[(N[(x - y$95$m), $MachinePrecision] / N[(N[(x - y$95$m), $MachinePrecision] + N[(2.0 * N[(y$95$m / N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 2e-9], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15e-165

   1. Initial program 58.9%

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

      1. associate-/l* 59.0%

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

      2. remove-double-neg 59.0%

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

      3. sub-neg 59.0%

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

      4. sub-neg 59.0%

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

      5. remove-double-neg 59.0%

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

      6. +-commutative 59.0%

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

      7. fma-def 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in y around 0 37.0%

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

      1. associate-+r+ 37.0%

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

      2. mul-1-neg 37.0%

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

      3. sub-neg 37.0%

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

   7. Simplified 37.0%

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

      1. unpow2 37.0%

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

      2. *-un-lft-identity 37.0%

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

      3. times-frac 37.6%

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

   9. Applied egg-rr 37.6%

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

       1. clear-num 37.6%

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

       2. /-rgt-identity 37.6%

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

       3. un-div-inv 37.6%

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

   11. Applied egg-rr 37.6%

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

   if 1.15e-165 < y < 2.00000000000000012e-9

   1. Initial program 99.9%

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

   if 2.00000000000000012e-9 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-165}:\\ \;\;\;\;\frac{x - y}{\left(x - y\right) + 2 \cdot \frac{y}{\frac{x}{y}}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.15 \cdot 10^{-165}:\\ \;\;\;\;\frac{x - y_m}{x + \left(2 \cdot \frac{y_m}{\frac{x}{y_m}} - y_m\right)}\\ \mathbf{elif}\;y_m \leq 10^{-5}:\\ \;\;\;\;\frac{\left(x - y_m\right) \cdot \left(x + y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.15e-165)
   (/ (- x y_m) (+ x (- (* 2.0 (/ y_m (/ x y_m))) y_m)))
   (if (<= y_m 1e-5)
     (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.15e-165) {
		tmp = (x - y_m) / (x + ((2.0 * (y_m / (x / y_m))) - y_m));
	} else if (y_m <= 1e-5) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.15d-165) then
        tmp = (x - y_m) / (x + ((2.0d0 * (y_m / (x / y_m))) - y_m))
    else if (y_m <= 1d-5) then
        tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.15e-165) {
		tmp = (x - y_m) / (x + ((2.0 * (y_m / (x / y_m))) - y_m));
	} else if (y_m <= 1e-5) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.15e-165:
		tmp = (x - y_m) / (x + ((2.0 * (y_m / (x / y_m))) - y_m))
	elif y_m <= 1e-5:
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.15e-165)
		tmp = Float64(Float64(x - y_m) / Float64(x + Float64(Float64(2.0 * Float64(y_m / Float64(x / y_m))) - y_m)));
	elseif (y_m <= 1e-5)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.15e-165)
		tmp = (x - y_m) / (x + ((2.0 * (y_m / (x / y_m))) - y_m));
	elseif (y_m <= 1e-5)
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.15e-165], N[(N[(x - y$95$m), $MachinePrecision] / N[(x + N[(N[(2.0 * N[(y$95$m / N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1e-5], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15e-165

   1. Initial program 58.9%

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

      1. associate-/l* 59.0%

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

      2. remove-double-neg 59.0%

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

      3. sub-neg 59.0%

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

      4. sub-neg 59.0%

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

      5. remove-double-neg 59.0%

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

      6. +-commutative 59.0%

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

      7. fma-def 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in y around 0 37.0%

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

      1. associate-+r+ 37.0%

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

      2. mul-1-neg 37.0%

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

      3. sub-neg 37.0%

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

   7. Simplified 37.0%

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

      1. unpow2 37.0%

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

      2. *-un-lft-identity 37.0%

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

      3. times-frac 37.6%

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

   9. Applied egg-rr 37.6%

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

       1. clear-num 37.6%

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

       2. /-rgt-identity 37.6%

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

       3. un-div-inv 37.6%

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

   11. Applied egg-rr 37.6%

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

       1. associate-+l- 37.6%

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

   13. Applied egg-rr 37.6%

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

   if 1.15e-165 < y < 1.00000000000000008e-5

   1. Initial program 99.9%

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

   if 1.00000000000000008e-5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-165}:\\ \;\;\;\;\frac{x - y}{x + \left(2 \cdot \frac{y}{\frac{x}{y}} - y\right)}\\ \mathbf{elif}\;y \leq 10^{-5}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.2 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 1.2e-147) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.2e-147) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.2d-147) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.2e-147) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.2e-147:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.2e-147)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.2e-147)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.2e-147], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.19999999999999999e-147

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf 36.6%

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

   if 1.19999999999999999e-147 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 68.0%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.7% accurate, 15.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 66.8%

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

3. Taylor expanded in x around 0 63.4%

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

4. Final simplification 63.4%

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

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024024 
(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))))