Kahan p9 Example

Percentage Accurate: 68.0% → 100.0%

Time: 6.6s

Alternatives: 7

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: 68.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. add-sqr-sqrt 72.7%

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

   2. times-frac 73.5%

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

   3. hypot-define 73.5%

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

   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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. frac-times 100.0%

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

   3. *-commutative 100.0%

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

   4. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 100.0% 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 72.7%

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

   1. add-sqr-sqrt 72.7%

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

   2. times-frac 73.5%

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

   3. hypot-define 73.5%

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

   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. Add Preprocessing

Alternative 3: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.55 \cdot 10^{-168}:\\ \;\;\;\;1 - \frac{\frac{y\_m}{x} \cdot \left(y\_m + y\_m\right)}{x}\\ \mathbf{elif}\;y\_m \leq 1.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-168)
   (- 1.0 (/ (* (/ y_m x) (+ y_m y_m)) x))
   (if (<= y_m 1.1e-32)
     (/ (* (- 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 <= 1.55e-168) {
		tmp = 1.0 - (((y_m / x) * (y_m + y_m)) / x);
	} else if (y_m <= 1.1e-32) {
		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 <= 1.55d-168) then
        tmp = 1.0d0 - (((y_m / x) * (y_m + y_m)) / x)
    else if (y_m <= 1.1d-32) 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 <= 1.55e-168) {
		tmp = 1.0 - (((y_m / x) * (y_m + y_m)) / x);
	} else if (y_m <= 1.1e-32) {
		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 <= 1.55e-168:
		tmp = 1.0 - (((y_m / x) * (y_m + y_m)) / x)
	elif y_m <= 1.1e-32:
		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 <= 1.55e-168)
		tmp = Float64(1.0 - Float64(Float64(Float64(y_m / x) * Float64(y_m + y_m)) / x));
	elseif (y_m <= 1.1e-32)
		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 <= 1.55e-168)
		tmp = 1.0 - (((y_m / x) * (y_m + y_m)) / x);
	elseif (y_m <= 1.1e-32)
		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, 1.55e-168], N[(1.0 - N[(N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.1e-32], 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 < 1.55e-168

   1. Initial program 65.7%

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

      1. associate-/l* 66.5%

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

      2. +-commutative 66.5%

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

      3. fma-define 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in x around -inf 35.0%

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

      1. mul-1-neg 35.0%

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

      2. unsub-neg 35.0%

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

      3. div-sub 35.0%

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

   7. Simplified 35.2%

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

      1. unpow2 35.2%

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

      2. associate-*r/ 35.5%

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

      3. *-commutative 35.5%

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

   9. Applied egg-rr 35.5%

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

       1. fma-undefine 35.5%

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

       2. *-commutative 35.5%

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

       3. distribute-lft-out 35.5%

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

   11. Applied egg-rr 35.5%

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

   if 1.55e-168 < y < 1.1e-32

   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 1.1e-32 < 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* 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 100.0%

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

Alternative 4: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.4 \cdot 10^{-153}:\\ \;\;\;\;1 - \frac{\frac{y\_m}{x} \cdot \left(y\_m + y\_m\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 4.4e-153) (- 1.0 (/ (* (/ y_m x) (+ y_m y_m)) x)) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 4.4e-153) {
		tmp = 1.0 - (((y_m / x) * (y_m + y_m)) / x);
	} 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 <= 4.4d-153) then
        tmp = 1.0d0 - (((y_m / x) * (y_m + y_m)) / x)
    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 <= 4.4e-153) {
		tmp = 1.0 - (((y_m / x) * (y_m + y_m)) / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 4.4e-153:
		tmp = 1.0 - (((y_m / x) * (y_m + y_m)) / x)
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 4.4e-153)
		tmp = Float64(1.0 - Float64(Float64(Float64(y_m / x) * Float64(y_m + y_m)) / x));
	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 <= 4.4e-153)
		tmp = 1.0 - (((y_m / x) * (y_m + y_m)) / x);
	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, 4.4e-153], N[(1.0 - N[(N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.40000000000000001e-153

   1. Initial program 66.2%

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

      1. associate-/l* 66.9%

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

      2. +-commutative 66.9%

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

      3. fma-define 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in x around -inf 36.0%

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

      1. mul-1-neg 36.0%

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

      2. unsub-neg 36.0%

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

      3. div-sub 36.0%

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

   7. Simplified 36.2%

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

      1. unpow2 36.2%

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

      2. associate-*r/ 36.4%

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

      3. *-commutative 36.4%

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

   9. Applied egg-rr 36.4%

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

       1. fma-undefine 36.4%

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

       2. *-commutative 36.4%

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

       3. distribute-lft-out 36.4%

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

   11. Applied egg-rr 36.4%

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

   if 4.40000000000000001e-153 < 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* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 81.9%

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

Alternative 5: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.4 \cdot 10^{-153}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1 + \frac{y\_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 4.4e-153) (* (- x y_m) (/ (+ 1.0 (/ y_m x)) x)) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 4.4e-153) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	} 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 <= 4.4d-153) then
        tmp = (x - y_m) * ((1.0d0 + (y_m / x)) / x)
    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 <= 4.4e-153) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 4.4e-153:
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x)
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 4.4e-153)
		tmp = Float64(Float64(x - y_m) * Float64(Float64(1.0 + Float64(y_m / x)) / x));
	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 <= 4.4e-153)
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	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, 4.4e-153], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.40000000000000001e-153

   1. Initial program 66.2%

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

      1. associate-/l* 66.9%

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

      2. +-commutative 66.9%

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

      3. fma-define 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in x around inf 36.1%

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

   if 4.40000000000000001e-153 < 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* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 81.9%

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

Alternative 6: 83.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.3 \cdot 10^{-153}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 4.3e-153) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 4.3e-153) {
		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 <= 4.3d-153) 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 <= 4.3e-153) {
		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 <= 4.3e-153:
		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 <= 4.3e-153)
		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 <= 4.3e-153)
		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, 4.3e-153], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e-153

   1. Initial program 66.2%

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

      1. associate-/l* 66.9%

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

      2. +-commutative 66.9%

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

      3. fma-define 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in x around inf 34.8%

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

   if 4.3e-153 < 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* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 81.9%

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

Alternative 7: 66.8% 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 72.7%

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

   1. associate-/l* 73.2%

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

   2. +-commutative 73.2%

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

   3. fma-define 73.2%

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

3. Simplified 73.2%

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

5. Taylor expanded in x around 0 69.2%

   \[\leadsto \color{blue}{-1} \]
6. 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 2024108 
(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))))