Linear.Projection:inversePerspective from linear-1.19.1.3, B

Average Error: 14.4 → 0.0

Time: 2.8s

Precision: binary64

Cost: 448

Math

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

↓

\[\frac{0.5}{y} + \frac{-0.5}{x} \]

FPCore

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

↓

(FPCore (x y) :precision binary64 (+ (/ 0.5 y) (/ -0.5 x)))

C

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

↓

double code(double x, double y) {
	return (0.5 / y) + (-0.5 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / ((x * 2.0d0) * y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.5d0 / y) + ((-0.5d0) / x)
end function

Java

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

↓

public static double code(double x, double y) {
	return (0.5 / y) + (-0.5 / x);
}

Python

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

↓

def code(x, y):
	return (0.5 / y) + (-0.5 / x)

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(Float64(x * 2.0) * y))
end

↓

function code(x, y)
	return Float64(Float64(0.5 / y) + Float64(-0.5 / x))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (0.5 / y) + (-0.5 / x);
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(0.5 / y), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]

Target

Original	14.4
Target	0.0
Herbie	0.0

\[\frac{0.5}{y} - \frac{0.5}{x} \]

Derivation

1. Initial program 14.4

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{0.5}{y} - \frac{0.5}{x}} \]
   Proof

   [Start] 14.4	\[ \frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   div-sub [=>] 14.4	\[ \color{blue}{\frac{x}{\left(x \cdot 2\right) \cdot y} - \frac{y}{\left(x \cdot 2\right) \cdot y}} \]
   associate-/r* [=>] 11.0	\[ \color{blue}{\frac{\frac{x}{x \cdot 2}}{y}} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]
   associate-/r* [=>] 10.9	\[ \frac{\color{blue}{\frac{\frac{x}{x}}{2}}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]
   *-inverses [=>] 10.9	\[ \frac{\frac{\color{blue}{1}}{2}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]
   metadata-eval [=>] 10.9	\[ \frac{\color{blue}{0.5}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]
   associate-/l/ [<=] 0.0	\[ \frac{0.5}{y} - \color{blue}{\frac{\frac{y}{y}}{x \cdot 2}} \]
   *-commutative [=>] 0.0	\[ \frac{0.5}{y} - \frac{\frac{y}{y}}{\color{blue}{2 \cdot x}} \]
   associate-/r* [=>] 0.0	\[ \frac{0.5}{y} - \color{blue}{\frac{\frac{\frac{y}{y}}{2}}{x}} \]
   *-inverses [=>] 0.0	\[ \frac{0.5}{y} - \frac{\frac{\color{blue}{1}}{2}}{x} \]
   metadata-eval [=>] 0.0	\[ \frac{0.5}{y} - \frac{\color{blue}{0.5}}{x} \]

3. Final simplification 0.0

   \[\leadsto \frac{0.5}{y} + \frac{-0.5}{x} \]

Alternatives

Alternative 1
Error	15.9
Cost	722

\[\begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+31} \lor \neg \left(x \leq -4.2 \cdot 10^{-79}\right) \land \left(x \leq -7 \cdot 10^{-90} \lor \neg \left(x \leq 1.75 \cdot 10^{+14}\right)\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]

Alternative 2
Error	31.3
Cost	192

\[\frac{-0.5}{x} \]

Reproduce

herbie shell --seed 2023054 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (- (/ 0.5 y) (/ 0.5 x))

  (/ (- x y) (* (* x 2.0) y)))