Average Error: 14.7 → 0.1
Time: 4.6s
Precision: binary64
Cost: 840
\[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\ \mathbf{if}\;y \leq -3.2168284734468214 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.0637076401127374 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{x + x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ (* 0.5 (/ x y)) -0.5))))
   (if (<= y -3.2168284734468214e-6)
     t_0
     (if (<= y 2.0637076401127374e+52) (* y (/ (+ x x) (- x y))) t_0))))
double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}
↓
double code(double x, double y) {
	double t_0 = x / ((0.5 * (x / y)) + -0.5);
	double tmp;
	if (y <= -3.2168284734468214e-6) {
		tmp = t_0;
	} else if (y <= 2.0637076401127374e+52) {
		tmp = y * ((x + x) / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) * y) / (x - y)
end function
↓
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / ((0.5d0 * (x / y)) + (-0.5d0))
    if (y <= (-3.2168284734468214d-6)) then
        tmp = t_0
    else if (y <= 2.0637076401127374d+52) then
        tmp = y * ((x + x) / (x - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

public static double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}
↓
public static double code(double x, double y) {
	double t_0 = x / ((0.5 * (x / y)) + -0.5);
	double tmp;
	if (y <= -3.2168284734468214e-6) {
		tmp = t_0;
	} else if (y <= 2.0637076401127374e+52) {
		tmp = y * ((x + x) / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

def code(x, y):
	return ((x * 2.0) * y) / (x - y)
↓
def code(x, y):
	t_0 = x / ((0.5 * (x / y)) + -0.5)
	tmp = 0
	if y <= -3.2168284734468214e-6:
		tmp = t_0
	elif y <= 2.0637076401127374e+52:
		tmp = y * ((x + x) / (x - y))
	else:
		tmp = t_0
	return tmp

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end
↓
function code(x, y)
	t_0 = Float64(x / Float64(Float64(0.5 * Float64(x / y)) + -0.5))
	tmp = 0.0
	if (y <= -3.2168284734468214e-6)
		tmp = t_0;
	elseif (y <= 2.0637076401127374e+52)
		tmp = Float64(y * Float64(Float64(x + x) / Float64(x - y)));
	else
		tmp = t_0;
	end
	return tmp
end

function tmp = code(x, y)
	tmp = ((x * 2.0) * y) / (x - y);
end
↓
function tmp_2 = code(x, y)
	t_0 = x / ((0.5 * (x / y)) + -0.5);
	tmp = 0.0;
	if (y <= -3.2168284734468214e-6)
		tmp = t_0;
	elseif (y <= 2.0637076401127374e+52)
		tmp = y * ((x + x) / (x - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := Block[{t$95$0 = N[(x / N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2168284734468214e-6], t$95$0, If[LessEqual[y, 2.0637076401127374e+52], N[(y * N[(N[(x + x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

\frac{\left(x \cdot 2\right) \cdot y}{x - y}
↓
\begin{array}{l}
t_0 := \frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\
\mathbf{if}\;y \leq -3.2168284734468214 \cdot 10^{-6}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;y \leq 2.0637076401127374 \cdot 10^{+52}:\\
\;\;\;\;y \cdot \frac{x + x}{x - y}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.3
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.2168284734468214e-6 or 2.0637076401127374e52 < y

    1. Initial program 16.9

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}} \]
      Proof
      (/.f64 x (fma.f64 1/2 (/.f64 x y) -1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (fma.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (/.f64 x y) -1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (fma.f64 (/.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) 2) (/.f64 x y) -1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (fma.f64 (Rewrite<= associate-/r*_binary64 (/.f64 y (*.f64 y 2))) (/.f64 x y) -1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (fma.f64 (/.f64 y (Rewrite<= *-commutative_binary64 (*.f64 2 y))) (/.f64 x y) -1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (fma.f64 (/.f64 y (*.f64 2 y)) (/.f64 x y) (Rewrite<= metadata-eval (neg.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (fma.f64 (/.f64 y (*.f64 2 y)) (/.f64 x y) (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (fma.f64 (/.f64 y (*.f64 2 y)) (/.f64 x y) (neg.f64 (/.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (fma.f64 (/.f64 y (*.f64 2 y)) (/.f64 x y) (neg.f64 (Rewrite<= associate-/r*_binary64 (/.f64 y (*.f64 y 2)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (fma.f64 (/.f64 y (*.f64 2 y)) (/.f64 x y) (neg.f64 (/.f64 y (Rewrite<= *-commutative_binary64 (*.f64 2 y)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 y (*.f64 2 y)) (/.f64 x y)) (/.f64 y (*.f64 2 y))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (-.f64 (*.f64 (/.f64 y (Rewrite=> *-commutative_binary64 (*.f64 y 2))) (/.f64 x y)) (/.f64 y (*.f64 2 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (-.f64 (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 y y) 2)) (/.f64 x y)) (/.f64 y (*.f64 2 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (-.f64 (*.f64 (/.f64 (Rewrite=> *-inverses_binary64 1) 2) (/.f64 x y)) (/.f64 y (*.f64 2 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (-.f64 (*.f64 (/.f64 (Rewrite<= *-inverses_binary64 (/.f64 x x)) 2) (/.f64 x y)) (/.f64 y (*.f64 2 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 x x) x) (*.f64 2 y))) (/.f64 y (*.f64 2 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (-.f64 (/.f64 (*.f64 (Rewrite=> *-inverses_binary64 1) x) (*.f64 2 y)) (/.f64 y (*.f64 2 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (-.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 x) (*.f64 2 y)) (/.f64 y (*.f64 2 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (-.f64 (/.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 x 0)) (*.f64 2 y)) (/.f64 y (*.f64 2 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (-.f64 (/.f64 (Rewrite=> +-rgt-identity_binary64 x) (*.f64 2 y)) (/.f64 y (*.f64 2 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 x y) (*.f64 2 y)))): 1 points increase in error, 1 points decrease in error
      (/.f64 x (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (-.f64 x y) y) 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x 2) (/.f64 (-.f64 x y) y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y))): 72 points increase in error, 42 points decrease in error

    3. Taylor expanded in x around 0 0.0

      \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \frac{x}{y} - 0.5}} \]

    if -3.2168284734468214e-6 < y < 2.0637076401127374e52

    1. Initial program 12.7

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

    2. Applied egg-rr0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2168284734468214 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\ \mathbf{elif}\;y \leq 2.0637076401127374 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{x + x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error3.6
Cost840
\[\begin{array}{l} t_0 := \frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\ \mathbf{if}\;y \leq -1.7868132413280266 \cdot 10^{-168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.530445046190567 \cdot 10^{-129}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error17.0
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -6.3978617110119816 \cdot 10^{-80}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 1.8961062316395846 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
Alternative 3
Error31.3
Cost192
\[y \cdot 2 \]

Error

Reproduce

herbie shell --seed 2022306 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))