Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.1% → 99.6%

Time: 7.1s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-50} \lor \neg \left(y \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.1e-50) (not (<= y 4e-32)))
   (/ (* x 2.0) (/ (- x y) y))
   (/ y (fma (/ y x) -0.5 0.5))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y <= -1.1e-50) || !(y <= 4e-32)) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = y / fma((y / x), -0.5, 0.5);
	}
	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 <= -1.1e-50) || !(y <= 4e-32))
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	else
		tmp = Float64(y / fma(Float64(y / x), -0.5, 0.5));
	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[Or[LessEqual[y, -1.1e-50], N[Not[LessEqual[y, 4e-32]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(y / x), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]

Target

Original	77.1%
Target	99.4%
Herbie	99.6%

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

Derivation

1. Split input into 2 regimes

2. if y < -1.0999999999999999e-50 or 4.00000000000000022e-32 < y

   1. Initial program 73.8%

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

   2. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]
      Step-by-step derivation

      [Start] 73.8%	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-/l* [=>] 100.0%	\[ \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]

   if -1.0999999999999999e-50 < y < 4.00000000000000022e-32

   1. Initial program 79.5%

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

   2. Simplified 100.0%

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

      [Start] 79.5%	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      *-commutative [=>] 79.5%	\[ \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]
      associate-/l* [=>] 100.0%	\[ \color{blue}{\frac{y}{\frac{x - y}{x \cdot 2}}} \]
      div-sub [=>] 100.0%	\[ \frac{y}{\color{blue}{\frac{x}{x \cdot 2} - \frac{y}{x \cdot 2}}} \]
      sub-neg [=>] 100.0%	\[ \frac{y}{\color{blue}{\frac{x}{x \cdot 2} + \left(-\frac{y}{x \cdot 2}\right)}} \]
      +-commutative [=>] 100.0%	\[ \frac{y}{\color{blue}{\left(-\frac{y}{x \cdot 2}\right) + \frac{x}{x \cdot 2}}} \]
      distribute-neg-frac [=>] 100.0%	\[ \frac{y}{\color{blue}{\frac{-y}{x \cdot 2}} + \frac{x}{x \cdot 2}} \]
      neg-mul-1 [=>] 100.0%	\[ \frac{y}{\frac{\color{blue}{-1 \cdot y}}{x \cdot 2} + \frac{x}{x \cdot 2}} \]
      *-commutative [=>] 100.0%	\[ \frac{y}{\frac{\color{blue}{y \cdot -1}}{x \cdot 2} + \frac{x}{x \cdot 2}} \]
      times-frac [=>] 100.0%	\[ \frac{y}{\color{blue}{\frac{y}{x} \cdot \frac{-1}{2}} + \frac{x}{x \cdot 2}} \]
      metadata-eval [=>] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \color{blue}{-0.5} + \frac{x}{x \cdot 2}} \]
      metadata-eval [<=] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \color{blue}{\left(-0.5\right)} + \frac{x}{x \cdot 2}} \]
      metadata-eval [<=] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \left(-\color{blue}{\frac{1}{2}}\right) + \frac{x}{x \cdot 2}} \]
      *-inverses [<=] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{\color{blue}{\frac{y}{y}}}{2}\right) + \frac{x}{x \cdot 2}} \]
      associate-/r* [<=] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \left(-\color{blue}{\frac{y}{y \cdot 2}}\right) + \frac{x}{x \cdot 2}} \]
      *-commutative [<=] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{\color{blue}{2 \cdot y}}\right) + \frac{x}{x \cdot 2}} \]
      associate-/r* [=>] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \color{blue}{\frac{\frac{x}{x}}{2}}} \]
      *-inverses [=>] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \frac{\color{blue}{1}}{2}} \]
      *-inverses [<=] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \frac{\color{blue}{\frac{y}{y}}}{2}} \]
      associate-/r* [<=] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \color{blue}{\frac{y}{y \cdot 2}}} \]
      *-commutative [<=] 100.0%	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \frac{y}{\color{blue}{2 \cdot y}}} \]
      fma-def [=>] 100.0%	\[ \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{y}{x}, -\frac{y}{2 \cdot y}, \frac{y}{2 \cdot y}\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-50} \lor \neg \left(y \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-50} \lor \neg \left(y \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \]

Alternative 2
Accuracy	93.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-144} \lor \neg \left(y \leq 3.5 \cdot 10^{-107}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1000000 \lor \neg \left(x \leq 1.3 \cdot 10^{-13}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{y}{x - y}\\ \end{array} \]

Alternative 4
Accuracy	75.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 5
Accuracy	50.0%
Cost	192

\[x \cdot -2 \]

Reproduce

herbie shell --seed 2023272 
(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)))