Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 14.9 → 0.2

Time: 6.3s

Precision: binary64

Cost: 841

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-33} \lor \neg \left(y \leq 3.8 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6e-33) (not (<= y 3.8e-52)))
   (/ (* x 2.0) (+ (/ x y) -1.0))
   (* 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 <= -1.6e-33) || !(y <= 3.8e-52)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} 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 <= (-1.6d-33)) .or. (.not. (y <= 3.8d-52))) then
        tmp = (x * 2.0d0) / ((x / y) + (-1.0d0))
    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 <= -1.6e-33) || !(y <= 3.8e-52)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} 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 <= -1.6e-33) or not (y <= 3.8e-52):
		tmp = (x * 2.0) / ((x / y) + -1.0)
	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 <= -1.6e-33) || !(y <= 3.8e-52))
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	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 <= -1.6e-33) || ~((y <= 3.8e-52)))
		tmp = (x * 2.0) / ((x / y) + -1.0);
	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, -1.6e-33], N[Not[LessEqual[y, 3.8e-52]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	14.9
Target	0.4
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 2 regimes

2. if y < -1.59999999999999988e-33 or 3.8000000000000003e-52 < y

   1. Initial program 14.4

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

   2. Simplified 0.3

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

      [Start] 14.4	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-/l* [=>] 0.3	\[ \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}} \]

   3. Taylor expanded in x around 0 0.3

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

   if -1.59999999999999988e-33 < y < 3.8000000000000003e-52

   1. Initial program 15.7

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

   2. Simplified 0.0

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

      [Start] 15.7	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-*l/ [<=] 0.0	\[ \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-33} \lor \neg \left(y \leq 3.8 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-78}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-104}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \left(\frac{x}{y} \cdot -2\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-42}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 2
Error	0.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+42} \lor \neg \left(y \leq 2 \cdot 10^{-52}\right):\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternative 3
Error	3.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+173}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+181}:\\ \;\;\;\;x \cdot \left(-2 \cdot \frac{y}{y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 4
Error	17.1
Cost	722

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-77} \lor \neg \left(y \leq -8.2 \cdot 10^{-104}\right) \land \left(y \leq -2.45 \cdot 10^{-119} \lor \neg \left(y \leq 4.3 \cdot 10^{-42}\right)\right):\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 5
Error	31.8
Cost	192

\[y \cdot 2 \]

Reproduce

herbie shell --seed 2023060 
(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)))