Linear.Projection:perspective from linear-1.19.1.3, B

Average Accuracy: 77.3% → 99.9%

Time: 5.1s

Precision: binary64

Cost: 841

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+15} \lor \neg \left(y \leq 2 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+15) (not (<= y 2e-20)))
   (/ (* x 2.0) (/ (- x y) y))
   (* 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 <= -5e+15) || !(y <= 2e-20)) {
		tmp = (x * 2.0) / ((x - y) / y);
	} 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 <= (-5d+15)) .or. (.not. (y <= 2d-20))) then
        tmp = (x * 2.0d0) / ((x - y) / y)
    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 <= -5e+15) || !(y <= 2e-20)) {
		tmp = (x * 2.0) / ((x - y) / y);
	} 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 <= -5e+15) or not (y <= 2e-20):
		tmp = (x * 2.0) / ((x - y) / y)
	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 <= -5e+15) || !(y <= 2e-20))
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	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 <= -5e+15) || ~((y <= 2e-20)))
		tmp = (x * 2.0) / ((x - y) / y);
	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, -5e+15], N[Not[LessEqual[y, 2e-20]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	77.3%
Target	99.5%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if y < -5e15 or 1.99999999999999989e-20 < y

   1. Initial program 75.1%

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

   2. Simplified 99.9%

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

      [Start] 75.1	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-/l* [=>] 99.9	\[ \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]

   if -5e15 < y < 1.99999999999999989e-20

   1. Initial program 79.6%

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

   2. Simplified 99.9%

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

      [Start] 79.6	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-*l/ [<=] 99.9	\[ \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+15} \lor \neg \left(y \leq 2 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	93.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-208} \lor \neg \left(y \leq 5.8 \cdot 10^{-229}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+18} \lor \neg \left(y \leq 10^{-79}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternative 3
Accuracy	74.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+17}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+55}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 4
Accuracy	50.9%
Cost	192

\[x \cdot -2 \]

Reproduce

herbie shell --seed 2023135 
(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)))