Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 15.4 → 0.2

Time: 4.3s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1.6e+43)
   (/ (* x 2.0) (+ (/ x y) -1.0))
   (if (<= y 2e-30)
     (/ y (fma (/ y x) -0.5 0.5))
     (* 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 <= -1.6e+43) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else if (y <= 2e-30) {
		tmp = y / fma((y / x), -0.5, 0.5);
	} 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 <= -1.6e+43)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	elseif (y <= 2e-30)
		tmp = Float64(y / fma(Float64(y / x), -0.5, 0.5));
	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, -1.6e+43], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e-30], N[(y / N[(N[(y / x), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(-2.0 * N[(y / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	15.4
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if y < -1.60000000000000007e43

   1. Initial program 18.4

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

   2. Simplified 0.0

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

      [Start] 18.4	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-/l* [=>] 0.0	\[ \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]

   3. Taylor expanded in x around 0 0.0

      \[\leadsto \frac{x \cdot 2}{\color{blue}{\frac{x}{y} - 1}} \]

   if -1.60000000000000007e43 < y < 2e-30

   1. Initial program 14.4

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

   2. Simplified 0.3

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

      [Start] 14.4	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      *-commutative [=>] 14.4	\[ \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]
      associate-/l* [=>] 0.3	\[ \color{blue}{\frac{y}{\frac{x - y}{x \cdot 2}}} \]
      div-sub [=>] 0.3	\[ \frac{y}{\color{blue}{\frac{x}{x \cdot 2} - \frac{y}{x \cdot 2}}} \]
      sub-neg [=>] 0.3	\[ \frac{y}{\color{blue}{\frac{x}{x \cdot 2} + \left(-\frac{y}{x \cdot 2}\right)}} \]
      +-commutative [=>] 0.3	\[ \frac{y}{\color{blue}{\left(-\frac{y}{x \cdot 2}\right) + \frac{x}{x \cdot 2}}} \]
      distribute-neg-frac [=>] 0.3	\[ \frac{y}{\color{blue}{\frac{-y}{x \cdot 2}} + \frac{x}{x \cdot 2}} \]
      neg-mul-1 [=>] 0.3	\[ \frac{y}{\frac{\color{blue}{-1 \cdot y}}{x \cdot 2} + \frac{x}{x \cdot 2}} \]
      *-commutative [=>] 0.3	\[ \frac{y}{\frac{\color{blue}{y \cdot -1}}{x \cdot 2} + \frac{x}{x \cdot 2}} \]
      times-frac [=>] 0.3	\[ \frac{y}{\color{blue}{\frac{y}{x} \cdot \frac{-1}{2}} + \frac{x}{x \cdot 2}} \]
      metadata-eval [=>] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \color{blue}{-0.5} + \frac{x}{x \cdot 2}} \]
      metadata-eval [<=] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \color{blue}{\left(-0.5\right)} + \frac{x}{x \cdot 2}} \]
      metadata-eval [<=] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \left(-\color{blue}{\frac{1}{2}}\right) + \frac{x}{x \cdot 2}} \]
      *-inverses [<=] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{\color{blue}{\frac{y}{y}}}{2}\right) + \frac{x}{x \cdot 2}} \]
      associate-/r* [<=] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \left(-\color{blue}{\frac{y}{y \cdot 2}}\right) + \frac{x}{x \cdot 2}} \]
      *-commutative [<=] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{\color{blue}{2 \cdot y}}\right) + \frac{x}{x \cdot 2}} \]
      associate-/r* [=>] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \color{blue}{\frac{\frac{x}{x}}{2}}} \]
      *-inverses [=>] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \frac{\color{blue}{1}}{2}} \]
      *-inverses [<=] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \frac{\color{blue}{\frac{y}{y}}}{2}} \]
      associate-/r* [<=] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \color{blue}{\frac{y}{y \cdot 2}}} \]
      *-commutative [<=] 0.3	\[ \frac{y}{\frac{y}{x} \cdot \left(-\frac{y}{2 \cdot y}\right) + \frac{y}{\color{blue}{2 \cdot y}}} \]
      fma-def [=>] 0.3	\[ \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{y}{x}, -\frac{y}{2 \cdot y}, \frac{y}{2 \cdot y}\right)}} \]

   if 2e-30 < y

   1. Initial program 14.9

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

   2. Simplified 0.1

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

      [Start] 14.9	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      *-lft-identity [<=] 14.9	\[ \color{blue}{1 \cdot \frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]
      *-inverses [<=] 14.9	\[ \color{blue}{\frac{y}{y}} \cdot \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-/l* [=>] 0.3	\[ \frac{y}{y} \cdot \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]
      *-commutative [=>] 0.3	\[ \frac{y}{y} \cdot \frac{\color{blue}{2 \cdot x}}{\frac{x - y}{y}} \]
      associate-*l/ [<=] 0.3	\[ \frac{y}{y} \cdot \color{blue}{\left(\frac{2}{\frac{x - y}{y}} \cdot x\right)} \]
      associate-*r* [=>] 0.3	\[ \color{blue}{\left(\frac{y}{y} \cdot \frac{2}{\frac{x - y}{y}}\right) \cdot x} \]
      *-commutative [<=] 0.3	\[ \color{blue}{\left(\frac{2}{\frac{x - y}{y}} \cdot \frac{y}{y}\right)} \cdot x \]
      *-commutative [=>] 0.3	\[ \color{blue}{x \cdot \left(\frac{2}{\frac{x - y}{y}} \cdot \frac{y}{y}\right)} \]
      *-inverses [=>] 0.3	\[ x \cdot \left(\frac{2}{\frac{x - y}{y}} \cdot \color{blue}{1}\right) \]
      *-rgt-identity [=>] 0.3	\[ x \cdot \color{blue}{\frac{2}{\frac{x - y}{y}}} \]
      associate-/l* [<=] 0.2	\[ x \cdot \color{blue}{\frac{2 \cdot y}{x - y}} \]
      sub-neg [=>] 0.2	\[ x \cdot \frac{2 \cdot y}{\color{blue}{x + \left(-y\right)}} \]
      +-commutative [=>] 0.2	\[ x \cdot \frac{2 \cdot y}{\color{blue}{\left(-y\right) + x}} \]
      neg-sub0 [=>] 0.2	\[ x \cdot \frac{2 \cdot y}{\color{blue}{\left(0 - y\right)} + x} \]
      associate-+l- [=>] 0.2	\[ x \cdot \frac{2 \cdot y}{\color{blue}{0 - \left(y - x\right)}} \]
      sub0-neg [=>] 0.2	\[ x \cdot \frac{2 \cdot y}{\color{blue}{-\left(y - x\right)}} \]
      neg-mul-1 [=>] 0.2	\[ x \cdot \frac{2 \cdot y}{\color{blue}{-1 \cdot \left(y - x\right)}} \]
      times-frac [=>] 0.1	\[ x \cdot \color{blue}{\left(\frac{2}{-1} \cdot \frac{y}{y - x}\right)} \]
      metadata-eval [=>] 0.1	\[ x \cdot \left(\color{blue}{-2} \cdot \frac{y}{y - x}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-169} \lor \neg \left(y \leq 3 \cdot 10^{-101}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 2
Error	0.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-130} \lor \neg \left(y \leq 2 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \end{array} \]

Alternative 4
Error	15.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{-9}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+26}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 5
Error	31.5
Cost	192

\[y \cdot 2 \]

Reproduce

herbie shell --seed 2023010 
(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)))