Linear.Projection:perspective from linear-1.19.1.3, B

Average Accuracy: 76.1% → 99.5%

Time: 7.4s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+80}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-35)
   (/ x (fma 0.5 (/ x y) -0.5))
   (if (<= y 2.6e+80)
     (* y (/ (* x 2.0) (- x y)))
     (* x (* -2.0 (/ y (- y x)))))))

C

double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

↓

double code(double x, double y) {
	double tmp;
	if (y <= -5e-35) {
		tmp = x / fma(0.5, (x / y), -0.5);
	} else if (y <= 2.6e+80) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = x * (-2.0 * (y / (y - x)));
	}
	return tmp;
}

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end

↓

function code(x, y)
	tmp = 0.0
	if (y <= -5e-35)
		tmp = Float64(x / fma(0.5, Float64(x / y), -0.5));
	elseif (y <= 2.6e+80)
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(x * Float64(-2.0 * Float64(y / Float64(y - x))));
	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_] := If[LessEqual[y, -5e-35], N[(x / N[(0.5 * N[(x / y), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+80], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(-2.0 * N[(y / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	76.1%
Target	99.4%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if y < -4.99999999999999964e-35

   1. Initial program 75.1%

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

   2. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}} \]
      Proof

      [Start] 75.1	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-*l* [=>] 75.0	\[ \frac{\color{blue}{x \cdot \left(2 \cdot y\right)}}{x - y} \]
      associate-/l* [=>] 99.6	\[ \color{blue}{\frac{x}{\frac{x - y}{2 \cdot y}}} \]
      associate-/r* [=>] 99.7	\[ \frac{x}{\color{blue}{\frac{\frac{x - y}{2}}{y}}} \]
      associate-/r* [<=] 99.6	\[ \frac{x}{\color{blue}{\frac{x - y}{2 \cdot y}}} \]
      div-sub [=>] 99.6	\[ \frac{x}{\color{blue}{\frac{x}{2 \cdot y} - \frac{y}{2 \cdot y}}} \]
      *-lft-identity [<=] 99.6	\[ \frac{x}{\frac{\color{blue}{1 \cdot x}}{2 \cdot y} - \frac{y}{2 \cdot y}} \]
      *-inverses [<=] 99.6	\[ \frac{x}{\frac{\color{blue}{\frac{y}{y}} \cdot x}{2 \cdot y} - \frac{y}{2 \cdot y}} \]
      times-frac [=>] 99.6	\[ \frac{x}{\color{blue}{\frac{\frac{y}{y}}{2} \cdot \frac{x}{y}} - \frac{y}{2 \cdot y}} \]
      associate-/r* [<=] 99.6	\[ \frac{x}{\color{blue}{\frac{y}{y \cdot 2}} \cdot \frac{x}{y} - \frac{y}{2 \cdot y}} \]
      *-commutative [<=] 99.6	\[ \frac{x}{\frac{y}{\color{blue}{2 \cdot y}} \cdot \frac{x}{y} - \frac{y}{2 \cdot y}} \]
      fma-neg [=>] 99.6	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{y}{2 \cdot y}, \frac{x}{y}, -\frac{y}{2 \cdot y}\right)}} \]
      *-commutative [=>] 99.6	\[ \frac{x}{\mathsf{fma}\left(\frac{y}{\color{blue}{y \cdot 2}}, \frac{x}{y}, -\frac{y}{2 \cdot y}\right)} \]
      associate-/r* [=>] 99.6	\[ \frac{x}{\mathsf{fma}\left(\color{blue}{\frac{\frac{y}{y}}{2}}, \frac{x}{y}, -\frac{y}{2 \cdot y}\right)} \]
      *-inverses [=>] 99.6	\[ \frac{x}{\mathsf{fma}\left(\frac{\color{blue}{1}}{2}, \frac{x}{y}, -\frac{y}{2 \cdot y}\right)} \]
      metadata-eval [=>] 99.6	\[ \frac{x}{\mathsf{fma}\left(\color{blue}{0.5}, \frac{x}{y}, -\frac{y}{2 \cdot y}\right)} \]
      *-commutative [=>] 99.6	\[ \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\frac{y}{\color{blue}{y \cdot 2}}\right)} \]
      associate-/r* [=>] 99.7	\[ \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\color{blue}{\frac{\frac{y}{y}}{2}}\right)} \]
      *-inverses [=>] 99.7	\[ \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\frac{\color{blue}{1}}{2}\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -\color{blue}{0.5}\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, \color{blue}{-0.5}\right)} \]

   if -4.99999999999999964e-35 < y < 2.59999999999999982e80

   1. Initial program 79.5%

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

   2. Simplified 99.3%

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

      [Start] 79.5	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-*l/ [<=] 99.3	\[ \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]

   if 2.59999999999999982e80 < y

   1. Initial program 68.2%

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

   2. Simplified 99.9%

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

      [Start] 68.2	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      *-lft-identity [<=] 68.2	\[ \color{blue}{1 \cdot \frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]
      *-inverses [<=] 68.2	\[ \color{blue}{\frac{y}{y}} \cdot \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-/l* [=>] 99.9	\[ \frac{y}{y} \cdot \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]
      *-commutative [=>] 99.9	\[ \frac{y}{y} \cdot \frac{\color{blue}{2 \cdot x}}{\frac{x - y}{y}} \]
      associate-*l/ [<=] 99.9	\[ \frac{y}{y} \cdot \color{blue}{\left(\frac{2}{\frac{x - y}{y}} \cdot x\right)} \]
      associate-*r* [=>] 99.9	\[ \color{blue}{\left(\frac{y}{y} \cdot \frac{2}{\frac{x - y}{y}}\right) \cdot x} \]
      *-commutative [<=] 99.9	\[ \color{blue}{\left(\frac{2}{\frac{x - y}{y}} \cdot \frac{y}{y}\right)} \cdot x \]
      *-commutative [=>] 99.9	\[ \color{blue}{x \cdot \left(\frac{2}{\frac{x - y}{y}} \cdot \frac{y}{y}\right)} \]
      *-inverses [=>] 99.9	\[ x \cdot \left(\frac{2}{\frac{x - y}{y}} \cdot \color{blue}{1}\right) \]
      *-rgt-identity [=>] 99.9	\[ x \cdot \color{blue}{\frac{2}{\frac{x - y}{y}}} \]
      associate-/l* [<=] 99.9	\[ x \cdot \color{blue}{\frac{2 \cdot y}{x - y}} \]
      sub-neg [=>] 99.9	\[ x \cdot \frac{2 \cdot y}{\color{blue}{x + \left(-y\right)}} \]
      +-commutative [=>] 99.9	\[ x \cdot \frac{2 \cdot y}{\color{blue}{\left(-y\right) + x}} \]
      neg-sub0 [=>] 99.9	\[ x \cdot \frac{2 \cdot y}{\color{blue}{\left(0 - y\right)} + x} \]
      associate-+l- [=>] 99.9	\[ x \cdot \frac{2 \cdot y}{\color{blue}{0 - \left(y - x\right)}} \]
      sub0-neg [=>] 99.9	\[ x \cdot \frac{2 \cdot y}{\color{blue}{-\left(y - x\right)}} \]
      neg-mul-1 [=>] 99.9	\[ x \cdot \frac{2 \cdot y}{\color{blue}{-1 \cdot \left(y - x\right)}} \]
      times-frac [=>] 99.9	\[ x \cdot \color{blue}{\left(\frac{2}{-1} \cdot \frac{y}{y - x}\right)} \]
      metadata-eval [=>] 99.9	\[ x \cdot \left(\color{blue}{-2} \cdot \frac{y}{y - x}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+80}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-35} \lor \neg \left(y \leq 5.5 \cdot 10^{+79}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternative 2
Accuracy	73.8%
Cost	721

\[\begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-146} \lor \neg \left(y \leq 4.8 \cdot 10^{-129}\right) \land y \leq 5.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-0.5}\\ \end{array} \]

Alternative 3
Accuracy	91.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{y}{0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \end{array} \]

Alternative 4
Accuracy	51.3%
Cost	192

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

Reproduce

herbie shell --seed 2023137 
(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)))