Kahan p9 Example

Percentage Accurate: 68.2% → 92.8%

Time: 7.8s

Alternatives: 5

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: 68.2% 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: 92.8% accurate, 0.4× speedup

Math

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

FPCore

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

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 = -1.0 + (((x * 2.0) * (x / y)) / 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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (-1.0d0) + (((x * 2.0d0) * (x / y)) / y)
    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 = -1.0 + (((x * 2.0) * (x / y)) / y);
	}
	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 = -1.0 + (((x * 2.0) * (x / y)) / y)
	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(-1.0 + Float64(Float64(Float64(x * 2.0) * Float64(x / y)) / y));
	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 = -1.0 + (((x * 2.0) * (x / y)) / y);
	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[(-1.0 + N[(N[(N[(x * 2.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $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%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

         \[\leadsto {x}^{2} \cdot \frac{2 \cdot 1}{{y}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto {x}^{2} \cdot \left(2 \cdot \frac{1}{{y}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({x}^{2} \cdot \left(2 \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2 \cdot 1}{\color{blue}{{y}^{2}}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2}{{\color{blue}{y}}^{2}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} \cdot 2}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{2 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\left(1 + 1\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + 1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + \left(\mathsf{neg}\left(-1\right)\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y \cdot \color{blue}{y}}\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}}{\color{blue}{y}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}\right), \color{blue}{y}\right)\right) \]

   5. Simplified 76.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(2 \cdot \left(x \cdot \frac{x}{y}\right)\right), y\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{x}{y}\right), y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot x\right), \left(\frac{x}{y}\right)\right), y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, x\right), \left(\frac{x}{y}\right)\right), y\right)\right) \]

      5. /-lowering-/.f64 77.8%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, x\right), \mathsf{/.f64}\left(x, y\right)\right), y\right)\right) \]

   7. Applied egg-rr 77.8%

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

4. Final simplification 93.2%

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

Alternative 2: 43.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.75 \cdot 10^{-160}:\\ \;\;\;\;1 + \frac{y \cdot \frac{y \cdot -2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2 \cdot \frac{x \cdot x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.75e-160)
   (+ 1.0 (/ (* y (/ (* y -2.0) x)) x))
   (+ -1.0 (/ (* 2.0 (/ (* x x) y)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.75e-160) {
		tmp = 1.0 + ((y * ((y * -2.0) / x)) / x);
	} else {
		tmp = -1.0 + ((2.0 * ((x * x) / y)) / 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 <= 2.75d-160) then
        tmp = 1.0d0 + ((y * ((y * (-2.0d0)) / x)) / x)
    else
        tmp = (-1.0d0) + ((2.0d0 * ((x * x) / y)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.75e-160) {
		tmp = 1.0 + ((y * ((y * -2.0) / x)) / x);
	} else {
		tmp = -1.0 + ((2.0 * ((x * x) / y)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.75e-160:
		tmp = 1.0 + ((y * ((y * -2.0) / x)) / x)
	else:
		tmp = -1.0 + ((2.0 * ((x * x) / y)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.75e-160)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(Float64(y * -2.0) / x)) / x));
	else
		tmp = Float64(-1.0 + Float64(Float64(2.0 * Float64(Float64(x * x) / y)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.75e-160)
		tmp = 1.0 + ((y * ((y * -2.0) / x)) / x);
	else
		tmp = -1.0 + ((2.0 * ((x * x) / y)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.75e-160], N[(1.0 + N[(N[(y * N[(N[(y * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(2.0 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.75e-160

   1. Initial program 64.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 37.5%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot \left(y \cdot -2\right)}{x}\right), x\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{y \cdot -2}{x}\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{y \cdot -2}{x}\right)\right), x\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(y \cdot -2\right), x\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 38.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, -2\right), x\right)\right), x\right)\right) \]

   7. Applied egg-rr 38.0%

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

   if 2.75e-160 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

         \[\leadsto {x}^{2} \cdot \frac{2 \cdot 1}{{y}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto {x}^{2} \cdot \left(2 \cdot \frac{1}{{y}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({x}^{2} \cdot \left(2 \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2 \cdot 1}{\color{blue}{{y}^{2}}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2}{{\color{blue}{y}}^{2}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} \cdot 2}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{2 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\left(1 + 1\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + 1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + \left(\mathsf{neg}\left(-1\right)\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y \cdot \color{blue}{y}}\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}}{\color{blue}{y}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}\right), \color{blue}{y}\right)\right) \]

   5. Simplified 83.2%

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

Alternative 3: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-161}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2 \cdot \frac{x \cdot x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-161) 1.0 (+ -1.0 (/ (* 2.0 (/ (* x x) y)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-161) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((2.0 * ((x * x) / y)) / 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 <= 2.1d-161) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + ((2.0d0 * ((x * x) / y)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-161) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((2.0 * ((x * x) / y)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-161:
		tmp = 1.0
	else:
		tmp = -1.0 + ((2.0 * ((x * x) / y)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-161)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(2.0 * Float64(Float64(x * x) / y)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-161)
		tmp = 1.0;
	else
		tmp = -1.0 + ((2.0 * ((x * x) / y)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-161], 1.0, N[(-1.0 + N[(N[(2.0 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e-161

   1. Initial program 63.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 35.8%

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

   if 2.1e-161 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

         \[\leadsto {x}^{2} \cdot \frac{2 \cdot 1}{{y}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto {x}^{2} \cdot \left(2 \cdot \frac{1}{{y}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({x}^{2} \cdot \left(2 \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2 \cdot 1}{\color{blue}{{y}^{2}}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2}{{\color{blue}{y}}^{2}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} \cdot 2}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{2 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\left(1 + 1\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + 1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + \left(\mathsf{neg}\left(-1\right)\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y \cdot \color{blue}{y}}\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}}{\color{blue}{y}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}\right), \color{blue}{y}\right)\right) \]

   5. Simplified 81.2%

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

Alternative 4: 41.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.75 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.75e-160) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.75e-160) {
		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 <= 2.75d-160) 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 <= 2.75e-160) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.75e-160:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.75e-160)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.75e-160)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.75e-160], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.75e-160

   1. Initial program 64.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 36.1%

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

   if 2.75e-160 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 82.3%

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

Alternative 5: 67.1% 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 69.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 66.7%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Developer Target 1: 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 2024155 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (! :herbie-platform default (if (< 1/2 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y)))))))

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