Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 22.93% → 0.51%

Time: 4.9s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1e+83)
   (* x (* -2.0 (/ y (- y x))))
   (if (<= y 2.15e-60)
     (* y (/ (* x 2.0) (- x y)))
     (/ (* x 2.0) (+ (/ x y) -1.0)))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (y <= -1e+83) {
		tmp = x * (-2.0 * (y / (y - x)));
	} else if (y <= 2.15e-60) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	}
	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 <= (-1d+83)) then
        tmp = x * ((-2.0d0) * (y / (y - x)))
    else if (y <= 2.15d-60) then
        tmp = y * ((x * 2.0d0) / (x - y))
    else
        tmp = (x * 2.0d0) / ((x / y) + (-1.0d0))
    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 <= -1e+83) {
		tmp = x * (-2.0 * (y / (y - x)));
	} else if (y <= 2.15e-60) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if y <= -1e+83:
		tmp = x * (-2.0 * (y / (y - x)))
	elif y <= 2.15e-60:
		tmp = y * ((x * 2.0) / (x - y))
	else:
		tmp = (x * 2.0) / ((x / y) + -1.0)
	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 <= -1e+83)
		tmp = Float64(x * Float64(-2.0 * Float64(y / Float64(y - x))));
	elseif (y <= 2.15e-60)
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	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 <= -1e+83)
		tmp = x * (-2.0 * (y / (y - x)));
	elseif (y <= 2.15e-60)
		tmp = y * ((x * 2.0) / (x - y));
	else
		tmp = (x * 2.0) / ((x / y) + -1.0);
	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[LessEqual[y, -1e+83], N[(x * N[(-2.0 * N[(y / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e-60], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	22.93%
Target	0.55%
Herbie	0.51%

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

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000003e83

   1. Initial program 33.85

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

   2. Simplified 0.06

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

      [Start] 33.85	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      *-lft-identity [<=] 33.85	\[ \color{blue}{1 \cdot \frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]
      *-inverses [<=] 33.85	\[ \color{blue}{\frac{y}{y}} \cdot \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-/l* [=>] 0.04	\[ \frac{y}{y} \cdot \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]
      *-commutative [=>] 0.04	\[ \frac{y}{y} \cdot \frac{\color{blue}{2 \cdot x}}{\frac{x - y}{y}} \]
      associate-*l/ [<=] 0.11	\[ \frac{y}{y} \cdot \color{blue}{\left(\frac{2}{\frac{x - y}{y}} \cdot x\right)} \]
      associate-*r* [=>] 0.11	\[ \color{blue}{\left(\frac{y}{y} \cdot \frac{2}{\frac{x - y}{y}}\right) \cdot x} \]
      *-commutative [<=] 0.11	\[ \color{blue}{\left(\frac{2}{\frac{x - y}{y}} \cdot \frac{y}{y}\right)} \cdot x \]
      *-commutative [=>] 0.11	\[ \color{blue}{x \cdot \left(\frac{2}{\frac{x - y}{y}} \cdot \frac{y}{y}\right)} \]
      *-inverses [=>] 0.11	\[ x \cdot \left(\frac{2}{\frac{x - y}{y}} \cdot \color{blue}{1}\right) \]
      *-rgt-identity [=>] 0.11	\[ x \cdot \color{blue}{\frac{2}{\frac{x - y}{y}}} \]
      associate-/l* [<=] 0.13	\[ x \cdot \color{blue}{\frac{2 \cdot y}{x - y}} \]
      sub-neg [=>] 0.13	\[ x \cdot \frac{2 \cdot y}{\color{blue}{x + \left(-y\right)}} \]
      +-commutative [=>] 0.13	\[ x \cdot \frac{2 \cdot y}{\color{blue}{\left(-y\right) + x}} \]
      neg-sub0 [=>] 0.13	\[ x \cdot \frac{2 \cdot y}{\color{blue}{\left(0 - y\right)} + x} \]
      associate-+l- [=>] 0.13	\[ x \cdot \frac{2 \cdot y}{\color{blue}{0 - \left(y - x\right)}} \]
      sub0-neg [=>] 0.13	\[ x \cdot \frac{2 \cdot y}{\color{blue}{-\left(y - x\right)}} \]
      neg-mul-1 [=>] 0.13	\[ x \cdot \frac{2 \cdot y}{\color{blue}{-1 \cdot \left(y - x\right)}} \]
      times-frac [=>] 0.06	\[ x \cdot \color{blue}{\left(\frac{2}{-1} \cdot \frac{y}{y - x}\right)} \]
      metadata-eval [=>] 0.06	\[ x \cdot \left(\color{blue}{-2} \cdot \frac{y}{y - x}\right) \]

   if -1.00000000000000003e83 < y < 2.15e-60

   1. Initial program 20.1

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

   2. Simplified 0.61

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

      [Start] 20.1	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-*l/ [<=] 0.61	\[ \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]

   if 2.15e-60 < y

   1. Initial program 21.28

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

   2. Simplified 0.6

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

      [Start] 21.28	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-/l* [=>] 0.6	\[ \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]

   3. Taylor expanded in x around 0 0.59

      \[\leadsto \frac{x \cdot 2}{\color{blue}{\frac{x}{y} - 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.51

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.44%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+84} \lor \neg \left(y \leq 1.95 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternative 2
Error	6.25%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -4.85 \cdot 10^{+172}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 3
Error	26.07%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+83}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-45}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 4
Error	49.75%
Cost	192

\[y \cdot 2 \]

Reproduce

herbie shell --seed 2023088 
(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)))