Kahan p9 Example

Percentage Accurate: 69.1% → 99.9%

Time: 8.7s

Alternatives: 9

Speedup: 0.4×

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

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

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 67.2%

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

   1. add-sqr-sqrt 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 67.0%

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

   3. hypot-define 67.1%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 92.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (+ (* 2.0 (* (/ x y) (/ 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 = (2.0 * ((x / y) * (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 = (2.0d0 * ((x / y) * (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 = (2.0 * ((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 = (2.0 * ((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(2.0 * Float64(Float64(x / y) * 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 = (2.0 * ((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[(2.0 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   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%

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

      2. +-commutative 3.1%

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

      3. fma-define 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 52.4%

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

      1. unpow2 52.4%

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

      2. unpow2 52.4%

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

      3. frac-times 78.8%

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

   7. Applied egg-rr 78.8%

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

4. Final simplification 93.0%

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

Alternative 3: 41.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-148} \lor \neg \left(y \leq 1.1 \cdot 10^{-116}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.8e-178)
   1.0
   (if (or (<= y 1.05e-148) (not (<= y 1.1e-116)))
     (* (- x y) (/ (+ (/ x y) 1.0) y))
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.8e-178) {
		tmp = 1.0;
	} else if ((y <= 1.05e-148) || !(y <= 1.1e-116)) {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.8d-178) then
        tmp = 1.0d0
    else if ((y <= 1.05d-148) .or. (.not. (y <= 1.1d-116))) then
        tmp = (x - y) * (((x / y) + 1.0d0) / y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.8e-178) {
		tmp = 1.0;
	} else if ((y <= 1.05e-148) || !(y <= 1.1e-116)) {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.8e-178:
		tmp = 1.0
	elif (y <= 1.05e-148) or not (y <= 1.1e-116):
		tmp = (x - y) * (((x / y) + 1.0) / y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.8e-178)
		tmp = 1.0;
	elseif ((y <= 1.05e-148) || !(y <= 1.1e-116))
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.8e-178)
		tmp = 1.0;
	elseif ((y <= 1.05e-148) || ~((y <= 1.1e-116)))
		tmp = (x - y) * (((x / y) + 1.0) / y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.8e-178], 1.0, If[Or[LessEqual[y, 1.05e-148], N[Not[LessEqual[y, 1.1e-116]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 9.80000000000000041e-178 or 1.05e-148 < y < 1.10000000000000005e-116

   1. Initial program 63.9%

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

      1. associate-/l* 63.7%

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

      2. +-commutative 63.7%

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

      3. fma-define 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in x around inf 41.6%

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

   if 9.80000000000000041e-178 < y < 1.05e-148 or 1.10000000000000005e-116 < y

   1. Initial program 88.2%

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

      1. associate-/l* 86.9%

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

      2. +-commutative 86.9%

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

      3. fma-define 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in y around inf 86.3%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-148} \lor \neg \left(y \leq 1.1 \cdot 10^{-116}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-177} \lor \neg \left(y \leq 6.5 \cdot 10^{-152}\right) \land y \leq 1.2 \cdot 10^{-118}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 1.05e-177) (and (not (<= y 6.5e-152)) (<= y 1.2e-118)))
   (* (- x y) (/ (+ 1.0 (/ y x)) x))
   (* (- x y) (/ (+ (/ x y) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 1.05e-177) || (!(y <= 6.5e-152) && (y <= 1.2e-118))) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 1.05d-177) .or. (.not. (y <= 6.5d-152)) .and. (y <= 1.2d-118)) then
        tmp = (x - y) * ((1.0d0 + (y / x)) / x)
    else
        tmp = (x - y) * (((x / y) + 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 1.05e-177) || (!(y <= 6.5e-152) && (y <= 1.2e-118))) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 1.05e-177) or (not (y <= 6.5e-152) and (y <= 1.2e-118)):
		tmp = (x - y) * ((1.0 + (y / x)) / x)
	else:
		tmp = (x - y) * (((x / y) + 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 1.05e-177) || (!(y <= 6.5e-152) && (y <= 1.2e-118)))
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(y / x)) / x));
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 1.05e-177) || (~((y <= 6.5e-152)) && (y <= 1.2e-118)))
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	else
		tmp = (x - y) * (((x / y) + 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 1.05e-177], And[N[Not[LessEqual[y, 6.5e-152]], $MachinePrecision], LessEqual[y, 1.2e-118]]], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e-177 or 6.5000000000000001e-152 < y < 1.2000000000000001e-118

   1. Initial program 64.1%

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

      1. associate-/l* 63.8%

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

      2. +-commutative 63.8%

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

      3. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around inf 42.8%

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

   if 1.05e-177 < y < 6.5000000000000001e-152 or 1.2000000000000001e-118 < y

   1. Initial program 87.9%

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

      1. associate-/l* 86.5%

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

      2. +-commutative 86.5%

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

      3. fma-define 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in y around inf 85.9%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-177} \lor \neg \left(y \leq 6.5 \cdot 10^{-152}\right) \land y \leq 1.2 \cdot 10^{-118}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-178} \lor \neg \left(y \leq 1.55 \cdot 10^{-154}\right) \land y \leq 3.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{x - y}{x} \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 8.5e-178) (and (not (<= y 1.55e-154)) (<= y 3.4e-119)))
   (* (/ (- x y) x) (+ 1.0 (/ y x)))
   (* (- x y) (/ (+ (/ x y) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 8.5e-178) || (!(y <= 1.55e-154) && (y <= 3.4e-119))) {
		tmp = ((x - y) / x) * (1.0 + (y / x));
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 8.5d-178) .or. (.not. (y <= 1.55d-154)) .and. (y <= 3.4d-119)) then
        tmp = ((x - y) / x) * (1.0d0 + (y / x))
    else
        tmp = (x - y) * (((x / y) + 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 8.5e-178) || (!(y <= 1.55e-154) && (y <= 3.4e-119))) {
		tmp = ((x - y) / x) * (1.0 + (y / x));
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 8.5e-178) or (not (y <= 1.55e-154) and (y <= 3.4e-119)):
		tmp = ((x - y) / x) * (1.0 + (y / x))
	else:
		tmp = (x - y) * (((x / y) + 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 8.5e-178) || (!(y <= 1.55e-154) && (y <= 3.4e-119)))
		tmp = Float64(Float64(Float64(x - y) / x) * Float64(1.0 + Float64(y / x)));
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 8.5e-178) || (~((y <= 1.55e-154)) && (y <= 3.4e-119)))
		tmp = ((x - y) / x) * (1.0 + (y / x));
	else
		tmp = (x - y) * (((x / y) + 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 8.5e-178], And[N[Not[LessEqual[y, 1.55e-154]], $MachinePrecision], LessEqual[y, 3.4e-119]]], N[(N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000001e-178 or 1.54999999999999991e-154 < y < 3.40000000000000024e-119

   1. Initial program 64.1%

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

      1. associate-/l* 63.8%

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

      2. +-commutative 63.8%

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

      3. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around inf 42.8%

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

      1. *-commutative 42.8%

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

      2. sub-neg 42.8%

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

      3. distribute-lft-in 42.7%

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

   7. Applied egg-rr 42.7%

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

      1. distribute-lft-out 42.8%

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

      2. sub-neg 42.8%

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

      3. /-rgt-identity 42.8%

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

      4. times-frac 42.9%

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

      5. *-commutative 42.9%

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

      6. times-frac 42.9%

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

      7. /-rgt-identity 42.9%

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

   9. Simplified 42.9%

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

   if 8.5000000000000001e-178 < y < 1.54999999999999991e-154 or 3.40000000000000024e-119 < y

   1. Initial program 87.9%

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

      1. associate-/l* 86.5%

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

      2. +-commutative 86.5%

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

      3. fma-define 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in y around inf 85.9%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-178} \lor \neg \left(y \leq 1.55 \cdot 10^{-154}\right) \land y \leq 3.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{x - y}{x} \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-178} \lor \neg \left(y \leq 1.36 \cdot 10^{-148}\right) \land y \leq 3.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{x - y}{x} \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 9.5e-178) (and (not (<= y 1.36e-148)) (<= y 3.8e-119)))
   (* (/ (- x y) x) (+ 1.0 (/ y x)))
   (+ (* 2.0 (* (/ x y) (/ x y))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 9.5e-178) || (!(y <= 1.36e-148) && (y <= 3.8e-119))) {
		tmp = ((x - y) / x) * (1.0 + (y / x));
	} else {
		tmp = (2.0 * ((x / y) * (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) :: tmp
    if ((y <= 9.5d-178) .or. (.not. (y <= 1.36d-148)) .and. (y <= 3.8d-119)) then
        tmp = ((x - y) / x) * (1.0d0 + (y / x))
    else
        tmp = (2.0d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 9.5e-178) || (!(y <= 1.36e-148) && (y <= 3.8e-119))) {
		tmp = ((x - y) / x) * (1.0 + (y / x));
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 9.5e-178) or (not (y <= 1.36e-148) and (y <= 3.8e-119)):
		tmp = ((x - y) / x) * (1.0 + (y / x))
	else:
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 9.5e-178) || (!(y <= 1.36e-148) && (y <= 3.8e-119)))
		tmp = Float64(Float64(Float64(x - y) / x) * Float64(1.0 + Float64(y / x)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 9.5e-178) || (~((y <= 1.36e-148)) && (y <= 3.8e-119)))
		tmp = ((x - y) / x) * (1.0 + (y / x));
	else
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 9.5e-178], And[N[Not[LessEqual[y, 1.36e-148]], $MachinePrecision], LessEqual[y, 3.8e-119]]], N[(N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.50000000000000009e-178 or 1.36e-148 < y < 3.79999999999999975e-119

   1. Initial program 63.9%

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

      1. associate-/l* 63.7%

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

      2. +-commutative 63.7%

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

      3. fma-define 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in x around inf 42.9%

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

      1. *-commutative 42.9%

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

      2. sub-neg 42.9%

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

      3. distribute-lft-in 42.8%

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

   7. Applied egg-rr 42.8%

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

      1. distribute-lft-out 42.9%

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

      2. sub-neg 42.9%

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

      3. /-rgt-identity 42.9%

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

      4. times-frac 43.1%

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

      5. *-commutative 43.1%

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

      6. times-frac 43.1%

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

      7. /-rgt-identity 43.1%

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

   9. Simplified 43.1%

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

   if 9.50000000000000009e-178 < y < 1.36e-148 or 3.79999999999999975e-119 < y

   1. Initial program 88.2%

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

      1. associate-/l* 86.9%

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

      2. +-commutative 86.9%

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

      3. fma-define 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in x around 0 77.3%

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

      1. unpow2 77.3%

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

      2. unpow2 77.3%

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

      3. frac-times 86.5%

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

   7. Applied egg-rr 86.5%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-178} \lor \neg \left(y \leq 1.36 \cdot 10^{-148}\right) \land y \leq 3.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{x - y}{x} \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-178} \lor \neg \left(y \leq 1.52 \cdot 10^{-148}\right) \land y \leq 3.6 \cdot 10^{-118}:\\ \;\;\;\;1 - \frac{\frac{y}{x} \cdot \left(y + y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 9.8e-178) (and (not (<= y 1.52e-148)) (<= y 3.6e-118)))
   (- 1.0 (/ (* (/ y x) (+ y y)) x))
   (+ (* 2.0 (* (/ x y) (/ x y))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 9.8e-178) || (!(y <= 1.52e-148) && (y <= 3.6e-118))) {
		tmp = 1.0 - (((y / x) * (y + y)) / x);
	} else {
		tmp = (2.0 * ((x / y) * (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) :: tmp
    if ((y <= 9.8d-178) .or. (.not. (y <= 1.52d-148)) .and. (y <= 3.6d-118)) then
        tmp = 1.0d0 - (((y / x) * (y + y)) / x)
    else
        tmp = (2.0d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 9.8e-178) || (!(y <= 1.52e-148) && (y <= 3.6e-118))) {
		tmp = 1.0 - (((y / x) * (y + y)) / x);
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 9.8e-178) or (not (y <= 1.52e-148) and (y <= 3.6e-118)):
		tmp = 1.0 - (((y / x) * (y + y)) / x)
	else:
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 9.8e-178) || (!(y <= 1.52e-148) && (y <= 3.6e-118)))
		tmp = Float64(1.0 - Float64(Float64(Float64(y / x) * Float64(y + y)) / x));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 9.8e-178) || (~((y <= 1.52e-148)) && (y <= 3.6e-118)))
		tmp = 1.0 - (((y / x) * (y + y)) / x);
	else
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 9.8e-178], And[N[Not[LessEqual[y, 1.52e-148]], $MachinePrecision], LessEqual[y, 3.6e-118]]], N[(1.0 - N[(N[(N[(y / x), $MachinePrecision] * N[(y + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.80000000000000041e-178 or 1.52000000000000002e-148 < y < 3.6000000000000002e-118

   1. Initial program 63.9%

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

      1. associate-/l* 63.7%

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

      2. +-commutative 63.7%

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

      3. fma-define 63.7%

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

   3. Simplified 63.7%

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

      1. *-commutative 63.7%

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

      2. sub-neg 63.7%

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

      3. distribute-lft-in 63.4%

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

      4. fma-undefine 63.4%

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

      5. +-commutative 63.4%

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

      6. add-sqr-sqrt 63.4%

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

      7. pow2 63.4%

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

      8. hypot-define 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. distribute-lft-out 63.6%

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

      2. sub-neg 63.6%

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

      3. associate-*l/ 63.9%

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

      4. +-commutative 63.9%

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

   8. Simplified 63.9%

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

   9. Taylor expanded in x around -inf 43.0%

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

       1. mul-1-neg 43.0%

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

       2. unsub-neg 43.0%

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

   11. Simplified 43.5%

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

   if 9.80000000000000041e-178 < y < 1.52000000000000002e-148 or 3.6000000000000002e-118 < y

   1. Initial program 88.2%

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

      1. associate-/l* 86.9%

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

      2. +-commutative 86.9%

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

      3. fma-define 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in x around 0 77.3%

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

      1. unpow2 77.3%

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

      2. unpow2 77.3%

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

      3. frac-times 86.5%

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

   7. Applied egg-rr 86.5%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-178} \lor \neg \left(y \leq 1.52 \cdot 10^{-148}\right) \land y \leq 3.6 \cdot 10^{-118}:\\ \;\;\;\;1 - \frac{\frac{y}{x} \cdot \left(y + y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-177}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e-177) 1.0 (if (<= y 2e-151) -1.0 (if (<= y 3.3e-103) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e-177) {
		tmp = 1.0;
	} else if (y <= 2e-151) {
		tmp = -1.0;
	} else if (y <= 3.3e-103) {
		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 <= 1d-177) then
        tmp = 1.0d0
    else if (y <= 2d-151) then
        tmp = -1.0d0
    else if (y <= 3.3d-103) 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 <= 1e-177) {
		tmp = 1.0;
	} else if (y <= 2e-151) {
		tmp = -1.0;
	} else if (y <= 3.3e-103) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e-177:
		tmp = 1.0
	elif y <= 2e-151:
		tmp = -1.0
	elif y <= 3.3e-103:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e-177)
		tmp = 1.0;
	elseif (y <= 2e-151)
		tmp = -1.0;
	elseif (y <= 3.3e-103)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e-177)
		tmp = 1.0;
	elseif (y <= 2e-151)
		tmp = -1.0;
	elseif (y <= 3.3e-103)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e-177], 1.0, If[LessEqual[y, 2e-151], -1.0, If[LessEqual[y, 3.3e-103], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999952e-178 or 1.9999999999999999e-151 < y < 3.2999999999999999e-103

   1. Initial program 64.4%

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

      1. associate-/l* 64.1%

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

      2. +-commutative 64.1%

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

      3. fma-define 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around inf 41.5%

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

   if 9.99999999999999952e-178 < y < 1.9999999999999999e-151 or 3.2999999999999999e-103 < y

   1. Initial program 87.1%

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

      1. associate-/l* 85.7%

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

      2. +-commutative 85.7%

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

      3. fma-define 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in x around 0 87.3%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-177}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.9% 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 67.2%

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

   1. associate-/l* 66.8%

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

   2. +-commutative 66.8%

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

   3. fma-define 66.8%

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

3. Simplified 66.8%

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

5. Taylor expanded in x around 0 62.3%

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

6. Final simplification 62.3%

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

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024079 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

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

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