Kahan p9 Example

Percentage Accurate: 69.1% → 99.9%

Time: 7.4s

Alternatives: 8

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (/ (+ x y) (* (hypot x y) (/ (hypot x y) (- x y)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.2%

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

   2. times-frac 67.3%

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

   3. hypot-def 67.3%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   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. frac-times 99.9%

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

   4. metadata-eval 99.9%

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

   5. div-inv 99.9%

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

   6. /-rgt-identity 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (* (/ (- x y) (hypot x y)) (/ (+ x y) (hypot x y))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.2%

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

   2. times-frac 67.3%

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

   3. hypot-def 67.3%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ t_1 := \frac{x}{\frac{y}{\frac{x}{y}}}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(t_1 + -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ x y) (- x y)) (+ (* x x) (* y y))))
        (t_1 (/ x (/ y (/ x y)))))
   (if (<= t_0 2.0) t_0 (+ t_1 (+ t_1 -1.0)))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = ((x + y) * (x - y)) / ((x * x) + (y * y));
	double t_1 = x / (y / (x / y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (t_1 + -1.0);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + y) * (x - y)) / ((x * x) + (y * y))
    t_1 = x / (y / (x / y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = t_1 + (t_1 + (-1.0d0))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = ((x + y) * (x - y)) / ((x * x) + (y * y));
	double t_1 = x / (y / (x / y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (t_1 + -1.0);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	t_0 = ((x + y) * (x - y)) / ((x * x) + (y * y))
	t_1 = x / (y / (x / y))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = t_1 + (t_1 + -1.0)
	return tmp

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(Float64(Float64(x + y) * Float64(x - y)) / Float64(Float64(x * x) + Float64(y * y)))
	t_1 = Float64(x / Float64(y / Float64(x / y)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(t_1 + -1.0));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	t_0 = ((x + y) * (x - y)) / ((x * x) + (y * y));
	t_1 = x / (y / (x / y));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = t_1 + (t_1 + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
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]}, Block[{t$95$1 = N[(x / N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(t$95$1 + N[(t$95$1 + -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 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 y around inf 46.4%

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

      1. associate--r+ 46.4%

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

      2. sub-neg 46.4%

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

      3. sub-neg 46.4%

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

      4. associate-+r+ 46.4%

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

      5. distribute-lft1-in 46.4%

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

      6. metadata-eval 46.4%

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

      7. mul0-lft 46.4%

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

      8. unpow2 46.4%

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

      9. unpow2 46.4%

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

      10. associate-/l* 46.4%

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

      11. associate-*r/ 46.4%

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

      12. metadata-eval 46.4%

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

      13. mul-1-neg 46.4%

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

      14. remove-double-neg 46.4%

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

      15. unpow2 46.4%

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

      16. unpow2 46.4%

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

   6. Simplified 73.3%

      \[\leadsto \color{blue}{\left(\left(0 + \frac{x}{y \cdot \frac{y}{x}}\right) + -1\right) + \frac{x}{y \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 73.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(0 + \frac{x}{y \cdot \frac{y}{x}}\right) + -1\right) + \frac{x}{y \cdot \frac{y}{x}}\right)\right)} \]

      2. expm1-udef 73.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(0 + \frac{x}{y \cdot \frac{y}{x}}\right) + -1\right) + \frac{x}{y \cdot \frac{y}{x}}\right)} - 1} \]

      3. +-commutative 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{y \cdot \frac{y}{x}} + \left(\left(0 + \frac{x}{y \cdot \frac{y}{x}}\right) + -1\right)}\right)} - 1 \]

      4. +-lft-identity 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{y \cdot \frac{y}{x}} + \left(\color{blue}{\frac{x}{y \cdot \frac{y}{x}}} + -1\right)\right)} - 1 \]

   8. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y \cdot \frac{y}{x}} + \left(\frac{x}{y \cdot \frac{y}{x}} + -1\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 73.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot \frac{y}{x}} + \left(\frac{x}{y \cdot \frac{y}{x}} + -1\right)\right)\right)} \]

      2. expm1-log1p 73.3%

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

      3. associate-*r/ 47.5%

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

      4. associate-/l* 73.3%

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

      5. +-commutative 73.3%

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

      6. associate-*r/ 47.5%

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

      7. associate-/l* 73.3%

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

   10. Simplified 73.3%

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

4. Final simplification 91.2%

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

Alternative 4: 92.7% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(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) y))))

C

y = abs(y);
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) / y;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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) / y
    end if
    code = tmp
end function

Java

y = Math.abs(y);
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) / y;
	}
	return tmp;
}

Python

y = abs(y)
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) / y
	return tmp

Julia

y = abs(y)
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(x - y) / y);
	end
	return tmp
end

MATLAB

y = abs(y)
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) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
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[(x - y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-/l* 3.1%

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

      2. +-commutative 3.1%

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

      3. remove-double-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. +-commutative 3.1%

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

      6. fma-def 3.1%

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

      7. sub-neg 3.1%

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

      8. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 71.2%

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

4. Final simplification 90.5%

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

Alternative 5: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-189} \lor \neg \left(y \leq 1.2 \cdot 10^{-135}\right) \land y \leq 3.5 \cdot 10^{-122}:\\ \;\;\;\;1 + \frac{y}{\frac{x}{y}} \cdot \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (or (<= y 6e-189) (and (not (<= y 1.2e-135)) (<= y 3.5e-122)))
   (+ 1.0 (* (/ y (/ x y)) (/ -2.0 x)))
   (/ (- x y) y)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((y <= 6e-189) || (!(y <= 1.2e-135) && (y <= 3.5e-122))) {
		tmp = 1.0 + ((y / (x / y)) * (-2.0 / x));
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 6d-189) .or. (.not. (y <= 1.2d-135)) .and. (y <= 3.5d-122)) then
        tmp = 1.0d0 + ((y / (x / y)) * ((-2.0d0) / x))
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((y <= 6e-189) || (!(y <= 1.2e-135) && (y <= 3.5e-122))) {
		tmp = 1.0 + ((y / (x / y)) * (-2.0 / x));
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if (y <= 6e-189) or (not (y <= 1.2e-135) and (y <= 3.5e-122)):
		tmp = 1.0 + ((y / (x / y)) * (-2.0 / x))
	else:
		tmp = (x - y) / y
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if ((y <= 6e-189) || (!(y <= 1.2e-135) && (y <= 3.5e-122)))
		tmp = Float64(1.0 + Float64(Float64(y / Float64(x / y)) * Float64(-2.0 / x)));
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 6e-189) || (~((y <= 1.2e-135)) && (y <= 3.5e-122)))
		tmp = 1.0 + ((y / (x / y)) * (-2.0 / x));
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[Or[LessEqual[y, 6e-189], And[N[Not[LessEqual[y, 1.2e-135]], $MachinePrecision], LessEqual[y, 3.5e-122]]], N[(1.0 + N[(N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision] * N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6e-189 or 1.1999999999999999e-135 < y < 3.5000000000000001e-122

   1. Initial program 62.5%

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

      1. add-sqr-sqrt 62.5%

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

      2. times-frac 62.9%

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

      3. hypot-def 62.9%

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

      4. hypot-def 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      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. frac-times 100.0%

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

      4. metadata-eval 100.0%

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

      5. div-inv 100.0%

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

      6. /-rgt-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 28.7%

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

      1. associate-*r/ 28.7%

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

      2. *-commutative 28.7%

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

      3. unpow2 28.7%

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

      4. times-frac 39.7%

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

      5. unpow2 39.7%

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

      6. associate-/l* 40.3%

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

   8. Simplified 40.3%

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

   if 6e-189 < y < 1.1999999999999999e-135 or 3.5000000000000001e-122 < y

   1. Initial program 87.5%

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

      1. associate-/l* 86.5%

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

      2. +-commutative 86.5%

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

      3. remove-double-neg 86.5%

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

      4. sub-neg 86.5%

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

      5. +-commutative 86.5%

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

      6. fma-def 86.5%

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

      7. sub-neg 86.5%

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

      8. remove-double-neg 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in x around 0 68.8%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-189} \lor \neg \left(y \leq 1.2 \cdot 10^{-135}\right) \land y \leq 3.5 \cdot 10^{-122}:\\ \;\;\;\;1 + \frac{y}{\frac{x}{y}} \cdot \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 6: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-134} \lor \neg \left(y \leq 1.75 \cdot 10^{-122}\right):\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-190)
   1.0
   (if (or (<= y 1.66e-134) (not (<= y 1.75e-122))) (/ (- x y) y) 1.0)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-190) {
		tmp = 1.0;
	} else if ((y <= 1.66e-134) || !(y <= 1.75e-122)) {
		tmp = (x - y) / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d-190) then
        tmp = 1.0d0
    else if ((y <= 1.66d-134) .or. (.not. (y <= 1.75d-122))) then
        tmp = (x - y) / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-190) {
		tmp = 1.0;
	} else if ((y <= 1.66e-134) || !(y <= 1.75e-122)) {
		tmp = (x - y) / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.5e-190:
		tmp = 1.0
	elif (y <= 1.66e-134) or not (y <= 1.75e-122):
		tmp = (x - y) / y
	else:
		tmp = 1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-190)
		tmp = 1.0;
	elseif ((y <= 1.66e-134) || !(y <= 1.75e-122))
		tmp = Float64(Float64(x - y) / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e-190)
		tmp = 1.0;
	elseif ((y <= 1.66e-134) || ~((y <= 1.75e-122)))
		tmp = (x - y) / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.5e-190], 1.0, If[Or[LessEqual[y, 1.66e-134], N[Not[LessEqual[y, 1.75e-122]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4999999999999999e-190 or 1.6599999999999999e-134 < y < 1.7500000000000001e-122

   1. Initial program 62.3%

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

   2. Taylor expanded in x around inf 38.1%

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

   if 1.4999999999999999e-190 < y < 1.6599999999999999e-134 or 1.7500000000000001e-122 < y

   1. Initial program 87.8%

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

      1. associate-/l* 86.8%

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

      2. +-commutative 86.8%

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

      3. remove-double-neg 86.8%

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

      4. sub-neg 86.8%

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

      5. +-commutative 86.8%

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

      6. fma-def 86.8%

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

      7. sub-neg 86.8%

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

      8. remove-double-neg 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in x around 0 67.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-134} \lor \neg \left(y \leq 1.75 \cdot 10^{-122}\right):\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 81.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e-189) 1.0 (if (<= y 1e-156) -1.0 (if (<= y 8e-121) 1.0 -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-189) {
		tmp = 1.0;
	} else if (y <= 1e-156) {
		tmp = -1.0;
	} else if (y <= 8e-121) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.2d-189) then
        tmp = 1.0d0
    else if (y <= 1d-156) then
        tmp = -1.0d0
    else if (y <= 8d-121) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-189) {
		tmp = 1.0;
	} else if (y <= 1e-156) {
		tmp = -1.0;
	} else if (y <= 8e-121) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 2.2e-189:
		tmp = 1.0
	elif y <= 1e-156:
		tmp = -1.0
	elif y <= 8e-121:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-189)
		tmp = 1.0;
	elseif (y <= 1e-156)
		tmp = -1.0;
	elseif (y <= 8e-121)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-189)
		tmp = 1.0;
	elseif (y <= 1e-156)
		tmp = -1.0;
	elseif (y <= 8e-121)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 2.2e-189], 1.0, If[LessEqual[y, 1e-156], -1.0, If[LessEqual[y, 8e-121], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.20000000000000019e-189 or 1.00000000000000004e-156 < y < 7.9999999999999998e-121

   1. Initial program 63.4%

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

   2. Taylor expanded in x around inf 39.4%

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

   if 2.20000000000000019e-189 < y < 1.00000000000000004e-156 or 7.9999999999999998e-121 < y

   1. Initial program 86.0%

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

   2. Taylor expanded in x around 0 72.5%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 66.3% accurate, 15.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

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

Python

y = abs(y)
def code(x, y):
	return -1.0

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := -1.0

Derivation

1. Initial program 67.2%

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

2. Taylor expanded in x around 0 62.7%

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

3. Final simplification 62.7%

   \[\leadsto -1 \]

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 2023297 
(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))))