Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 14.7 → 0.1

Time: 6.4s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\ \mathbf{if}\;y \leq -10710713085907712:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.969338444072506 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{x + x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(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 -10710713085907712.0)
     t_0
     (if (<= y 3.969338444072506e+21) (* y (/ (+ x x) (- x y))) t_0))))

C

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 <= -10710713085907712.0) {
		tmp = t_0;
	} else if (y <= 3.969338444072506e+21) {
		tmp = y * ((x + x) / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 <= (-10710713085907712.0d0)) then
        tmp = t_0
    else if (y <= 3.969338444072506d+21) then
        tmp = y * ((x + x) / (x - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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 <= -10710713085907712.0) {
		tmp = t_0;
	} else if (y <= 3.969338444072506e+21) {
		tmp = y * ((x + x) / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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 <= -10710713085907712.0:
		tmp = t_0
	elif y <= 3.969338444072506e+21:
		tmp = y * ((x + x) / (x - y))
	else:
		tmp = t_0
	return tmp

Julia

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 <= -10710713085907712.0)
		tmp = t_0;
	elseif (y <= 3.969338444072506e+21)
		tmp = Float64(y * Float64(Float64(x + x) / Float64(x - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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 <= -10710713085907712.0)
		tmp = t_0;
	elseif (y <= 3.969338444072506e+21)
		tmp = y * ((x + x) / (x - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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, -10710713085907712.0], t$95$0, If[LessEqual[y, 3.969338444072506e+21], N[(y * N[(N[(x + x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	14.7
Target	0.4
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -10710713085907710 or 3.9693384440725058e21 < y

   1. Initial program 16.5

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}} \]
      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)))): 2 points increase in error, 2 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))): 85 points increase in error, 36 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 -10710713085907710 < y < 3.9693384440725058e21

   1. Initial program 13.1

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

   2. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10710713085907712:\\ \;\;\;\;\frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\ \mathbf{elif}\;y \leq 3.969338444072506 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{x + x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7365041612758146 \cdot 10^{+40}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 2.649406543447422 \cdot 10^{+20}:\\ \;\;\;\;-2 \cdot \left(x + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 2
Error	15.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7365041612758146 \cdot 10^{+40}:\\ \;\;\;\;2 \cdot \left(y + y \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 2.649406543447422 \cdot 10^{+20}:\\ \;\;\;\;-2 \cdot \left(x + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 3
Error	3.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -8.835618387144482 \cdot 10^{+174}:\\ \;\;\;\;2 \cdot \left(y + y \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 3.221516957003474 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 4
Error	15.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7365041612758146 \cdot 10^{+40}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 2.649406543447422 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 5
Error	32.4
Cost	192

\[\frac{x}{-0.5} \]

Reproduce

herbie shell --seed 2022316 
(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)))