Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 14.7 → 0.1

Time: 3.7s

Precision: binary64

Cost: 841

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+22} \lor \neg \left(x \leq 10^{-7}\right):\\ \;\;\;\;\frac{x}{x - y} \cdot \left(y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.9e+22) (not (<= x 1e-7)))
   (* (/ x (- x y)) (* y 2.0))
   (* 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 ((x <= -1.9e+22) || !(x <= 1e-7)) {
		tmp = (x / (x - y)) * (y * 2.0);
	} else {
		tmp = x * (-2.0 * (y / (y - x)));
	}
	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 ((x <= (-1.9d+22)) .or. (.not. (x <= 1d-7))) then
        tmp = (x / (x - y)) * (y * 2.0d0)
    else
        tmp = x * ((-2.0d0) * (y / (y - x)))
    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 ((x <= -1.9e+22) || !(x <= 1e-7)) {
		tmp = (x / (x - y)) * (y * 2.0);
	} else {
		tmp = x * (-2.0 * (y / (y - x)));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if (x <= -1.9e+22) or not (x <= 1e-7):
		tmp = (x / (x - y)) * (y * 2.0)
	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 ((x <= -1.9e+22) || !(x <= 1e-7))
		tmp = Float64(Float64(x / Float64(x - y)) * Float64(y * 2.0));
	else
		tmp = Float64(x * Float64(-2.0 * Float64(y / Float64(y - x))));
	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 ((x <= -1.9e+22) || ~((x <= 1e-7)))
		tmp = (x / (x - y)) * (y * 2.0);
	else
		tmp = x * (-2.0 * (y / (y - x)));
	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[x, -1.9e+22], N[Not[LessEqual[x, 1e-7]], $MachinePrecision]], N[(N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(-2.0 * N[(y / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	14.7
Target	0.4
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if x < -1.9000000000000002e22 or 9.9999999999999995e-8 < x

   1. Initial program 17.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{x - y} \cdot \left(2 \cdot y\right)} \]
      Proof
      (*.f64 (/.f64 x (-.f64 x y)) (*.f64 2 y)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x (*.f64 2 y)) (-.f64 x y))): 1 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 2) y)) (-.f64 x y)): 0 points increase in error, 1 points decrease in error

   if -1.9000000000000002e22 < x < 9.9999999999999995e-8

   1. Initial program 12.5

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{x \cdot \left(-2 \cdot \frac{y}{y - x}\right)} \]
      Proof
      (*.f64 (/.f64 x (-.f64 x y)) (*.f64 2 y)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x (*.f64 2 y)) (-.f64 x y))): 1 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 2) y)) (-.f64 x y)): 0 points increase in error, 1 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+22} \lor \neg \left(x \leq 10^{-7}\right):\\ \;\;\;\;\frac{x}{x - y} \cdot \left(y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+149}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+126}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 2
Error	16.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2200000000000:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 3
Error	32.3
Cost	192

\[y \cdot 2 \]

Reproduce

herbie shell --seed 2022343 
(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)))