Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 23.45% → 0.3%

Time: 5.0s

Precision: binary64

Cost: 841

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+47} \lor \neg \left(y \leq 3 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e+47) (not (<= y 3e-49)))
   (* x (* -2.0 (/ y (- y x))))
   (* y (/ (* x 2.0) (- x y)))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y <= -8e+47) || !(y <= 3e-49)) {
		tmp = x * (-2.0 * (y / (y - x)));
	} else {
		tmp = y * ((x * 2.0) / (x - y));
	}
	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) :: tmp
    if ((y <= (-8d+47)) .or. (.not. (y <= 3d-49))) then
        tmp = x * ((-2.0d0) * (y / (y - x)))
    else
        tmp = y * ((x * 2.0d0) / (x - y))
    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 tmp;
	if ((y <= -8e+47) || !(y <= 3e-49)) {
		tmp = x * (-2.0 * (y / (y - x)));
	} else {
		tmp = y * ((x * 2.0) / (x - y));
	}
	return tmp;
}

Python

def code(x, y):
	return ((x * 2.0) * y) / (x - y)

↓

def code(x, y):
	tmp = 0
	if (y <= -8e+47) or not (y <= 3e-49):
		tmp = x * (-2.0 * (y / (y - x)))
	else:
		tmp = y * ((x * 2.0) / (x - y))
	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 <= -8e+47) || !(y <= 3e-49))
		tmp = Float64(x * Float64(-2.0 * Float64(y / Float64(y - x))));
	else
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) * y) / (x - y);
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8e+47) || ~((y <= 3e-49)))
		tmp = x * (-2.0 * (y / (y - x)));
	else
		tmp = y * ((x * 2.0) / (x - y));
	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_] := If[Or[LessEqual[y, -8e+47], N[Not[LessEqual[y, 3e-49]], $MachinePrecision]], N[(x * N[(-2.0 * N[(y / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	23.45%
Target	0.64%
Herbie	0.3%

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

Derivation

1. Split input into 2 regimes

2. if y < -8.0000000000000004e47 or 3e-49 < y

   1. Initial program 24.66

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

   2. Simplified 0.26

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

      [Start] 24.66	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      *-lft-identity [<=] 24.66	\[ \color{blue}{1 \cdot \frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]
      *-inverses [<=] 24.66	\[ \color{blue}{\frac{y}{y}} \cdot \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-/l* [=>] 0.5	\[ \frac{y}{y} \cdot \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]
      *-commutative [=>] 0.5	\[ \frac{y}{y} \cdot \frac{\color{blue}{2 \cdot x}}{\frac{x - y}{y}} \]
      associate-*l/ [<=] 0.46	\[ \frac{y}{y} \cdot \color{blue}{\left(\frac{2}{\frac{x - y}{y}} \cdot x\right)} \]
      associate-*r* [=>] 0.46	\[ \color{blue}{\left(\frac{y}{y} \cdot \frac{2}{\frac{x - y}{y}}\right) \cdot x} \]
      *-commutative [<=] 0.46	\[ \color{blue}{\left(\frac{2}{\frac{x - y}{y}} \cdot \frac{y}{y}\right)} \cdot x \]
      *-commutative [=>] 0.46	\[ \color{blue}{x \cdot \left(\frac{2}{\frac{x - y}{y}} \cdot \frac{y}{y}\right)} \]
      *-inverses [=>] 0.46	\[ x \cdot \left(\frac{2}{\frac{x - y}{y}} \cdot \color{blue}{1}\right) \]
      *-rgt-identity [=>] 0.46	\[ x \cdot \color{blue}{\frac{2}{\frac{x - y}{y}}} \]
      associate-/l* [<=] 0.35	\[ x \cdot \color{blue}{\frac{2 \cdot y}{x - y}} \]
      sub-neg [=>] 0.35	\[ x \cdot \frac{2 \cdot y}{\color{blue}{x + \left(-y\right)}} \]
      +-commutative [=>] 0.35	\[ x \cdot \frac{2 \cdot y}{\color{blue}{\left(-y\right) + x}} \]
      neg-sub0 [=>] 0.35	\[ x \cdot \frac{2 \cdot y}{\color{blue}{\left(0 - y\right)} + x} \]
      associate-+l- [=>] 0.35	\[ x \cdot \frac{2 \cdot y}{\color{blue}{0 - \left(y - x\right)}} \]
      sub0-neg [=>] 0.35	\[ x \cdot \frac{2 \cdot y}{\color{blue}{-\left(y - x\right)}} \]
      neg-mul-1 [=>] 0.35	\[ x \cdot \frac{2 \cdot y}{\color{blue}{-1 \cdot \left(y - x\right)}} \]
      times-frac [=>] 0.26	\[ x \cdot \color{blue}{\left(\frac{2}{-1} \cdot \frac{y}{y - x}\right)} \]
      metadata-eval [=>] 0.26	\[ x \cdot \left(\color{blue}{-2} \cdot \frac{y}{y - x}\right) \]

   if -8.0000000000000004e47 < y < 3e-49

   1. Initial program 22.25

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

   2. Simplified 0.33

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

      [Start] 22.25	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-*l/ [<=] 0.33	\[ \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+47} \lor \neg \left(y \leq 3 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.54%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-243} \lor \neg \left(y \leq 3.8 \cdot 10^{-177}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 2
Error	27.33%
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+42}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq -1.62 \cdot 10^{-93} \lor \neg \left(x \leq -9.8 \cdot 10^{-164}\right) \land x \leq 1.25 \cdot 10^{-25}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 3
Error	48.25%
Cost	192

\[y \cdot 2 \]

Reproduce

herbie shell --seed 2023121 
(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)))