Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 15.4 → 0.1

Time: 2.7s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\frac{\frac{x}{2}}{y} + -0.5}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ (/ (/ x 2.0) y) -0.5))))
   (if (<= y -2e+21) t_0 (if (<= y 2e-21) (* y (/ (* x 2.0) (- 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 / (((x / 2.0) / y) + -0.5);
	double tmp;
	if (y <= -2e+21) {
		tmp = t_0;
	} else if (y <= 2e-21) {
		tmp = y * ((x * 2.0) / (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 / (((x / 2.0d0) / y) + (-0.5d0))
    if (y <= (-2d+21)) then
        tmp = t_0
    else if (y <= 2d-21) then
        tmp = y * ((x * 2.0d0) / (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 / (((x / 2.0) / y) + -0.5);
	double tmp;
	if (y <= -2e+21) {
		tmp = t_0;
	} else if (y <= 2e-21) {
		tmp = y * ((x * 2.0) / (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 / (((x / 2.0) / y) + -0.5)
	tmp = 0
	if y <= -2e+21:
		tmp = t_0
	elif y <= 2e-21:
		tmp = y * ((x * 2.0) / (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(Float64(x / 2.0) / y) + -0.5))
	tmp = 0.0
	if (y <= -2e+21)
		tmp = t_0;
	elseif (y <= 2e-21)
		tmp = Float64(y * Float64(Float64(x * 2.0) / 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 / (((x / 2.0) / y) + -0.5);
	tmp = 0.0;
	if (y <= -2e+21)
		tmp = t_0;
	elseif (y <= 2e-21)
		tmp = y * ((x * 2.0) / (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[(N[(x / 2.0), $MachinePrecision] / y), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e+21], t$95$0, If[LessEqual[y, 2e-21], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	15.4
Target	0.3
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 < -2e21 or 1.99999999999999982e-21 < y

   1. Initial program 17.2

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{x}{2}}{y} + -0.5}} \]
      Proof
      (/.f64 x (+.f64 (/.f64 (/.f64 x 2) y) -1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 2 y))) -1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 (/.f64 x (*.f64 2 y)) (Rewrite<= metadata-eval (neg.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 (/.f64 x (*.f64 2 y)) (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 (/.f64 x (*.f64 2 y)) (neg.f64 (/.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 (/.f64 x (*.f64 2 y)) (neg.f64 (Rewrite<= associate-/r*_binary64 (/.f64 y (*.f64 y 2)))))): 1 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 (/.f64 x (*.f64 2 y)) (neg.f64 (/.f64 y (Rewrite<= *-commutative_binary64 (*.f64 2 y)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 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)))): 3 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 x (neg.f64 y))) (*.f64 2 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 y) x)) (*.f64 2 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 y)) x) (*.f64 2 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 y x))) (*.f64 2 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 y x))) (*.f64 2 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 y x))) (*.f64 2 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 y x) -1)) (*.f64 2 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (Rewrite=> associate-/l*_binary64 (/.f64 (-.f64 y x) (/.f64 (*.f64 2 y) -1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 x (-.f64 y x)) (/.f64 (*.f64 2 y) -1))): 46 points increase in error, 45 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x (*.f64 2 y)) (*.f64 (-.f64 y x) -1))): 87 points increase in error, 44 points decrease in error
      (/.f64 (*.f64 x (*.f64 2 y)) (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 y x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x (*.f64 2 y)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 y x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x (*.f64 2 y)) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 y x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x (*.f64 2 y)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 y) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x (*.f64 2 y)) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 y)) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x (*.f64 2 y)) (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x (*.f64 2 y)) (Rewrite<= sub-neg_binary64 (-.f64 x y))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 2) y)) (-.f64 x y)): 0 points increase in error, 0 points decrease in error

   if -2e21 < y < 1.99999999999999982e-21

   1. Initial program 13.7

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
      Proof
      (*.f64 (/.f64 (*.f64 x 2) (-.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))): 88 points increase in error, 44 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{\frac{\frac{x}{2}}{y} + -0.5}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\frac{x}{2}}{y} + -0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.3
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-25}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-89}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+17}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+113}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \]

Alternative 2
Error	3.9
Cost	840

\[\begin{array}{l} t_0 := y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-98}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	840

\[\begin{array}{l} t_0 := \frac{2}{\frac{\frac{x}{y} + -1}{x}}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 100000:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	31.7
Cost	192

\[x \cdot -2 \]

Reproduce

herbie shell --seed 2022329 
(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)))