Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 14.8 → 0.1

Time: 5.3s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \frac{x + x}{x - y}\\ \mathbf{if}\;x \leq -2.952044422514514 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.155516129976019 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (/ (+ x x) (- x y)))))
   (if (<= x -2.952044422514514e-8)
     t_0
     (if (<= x 4.155516129976019e-22) (/ x (fma 0.5 (/ x y) -0.5)) 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 = y * ((x + x) / (x - y));
	double tmp;
	if (x <= -2.952044422514514e-8) {
		tmp = t_0;
	} else if (x <= 4.155516129976019e-22) {
		tmp = x / fma(0.5, (x / y), -0.5);
	} 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(y * Float64(Float64(x + x) / Float64(x - y)))
	tmp = 0.0
	if (x <= -2.952044422514514e-8)
		tmp = t_0;
	elseif (x <= 4.155516129976019e-22)
		tmp = Float64(x / fma(0.5, Float64(x / y), -0.5));
	else
		tmp = t_0;
	end
	return 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[(y * N[(N[(x + x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.952044422514514e-8], t$95$0, If[LessEqual[x, 4.155516129976019e-22], N[(x / N[(0.5 * N[(x / y), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	14.8
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 x < -2.9520444225145141e-8 or 4.155516129976019e-22 < x

   1. Initial program 15.2

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

   2. Applied egg-rr 0.1

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

   if -2.9520444225145141e-8 < x < 4.155516129976019e-22

   1. Initial program 14.4

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

   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)))): 1 points increase in error, 0 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))): 81 points increase in error, 29 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.952044422514514 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{x + x}{x - y}\\ \mathbf{elif}\;x \leq 4.155516129976019 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x + x}{x - y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	3152

\[\begin{array}{l} t_0 := \frac{y \cdot \left(x \cdot 2\right)}{x - y}\\ t_1 := y \cdot \frac{x + x}{x - y}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	3.9
Cost	840

\[\begin{array}{l} t_0 := y \cdot \frac{x + x}{x - y}\\ \mathbf{if}\;x \leq -1.0494655353882887 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4456416384429955 \cdot 10^{-133}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	840

\[\begin{array}{l} t_0 := \left(x + x\right) \cdot \frac{y}{x - y}\\ \mathbf{if}\;y \leq -2.4476923778725644 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9475749879832593 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \frac{x + x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	17.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2219178185025503 \cdot 10^{+78}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 4.155516129976019 \cdot 10^{-22}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;x \leq 60.01595918475139:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 4.1406460568066846 \cdot 10^{+77}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 5
Error	31.4
Cost	192

\[y \cdot 2 \]

Reproduce

herbie shell --seed 2022317 
(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)))