Linear.Projection:inversePerspective from linear-1.19.1.3, C

Average Error: 15.3 → 0.0

Time: 2.7s

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	15.3
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 15.3

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

2. Simplified 8.1

   \[\leadsto \color{blue}{\frac{\frac{x + y}{x}}{y \cdot 2}} \]
   Proof
   (/.f64 (/.f64 (+.f64 x y) x) (*.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 x y) x) (Rewrite<= *-commutative_binary64 (*.f64 2 y))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 x y) (*.f64 x (*.f64 2 y)))): 81 points increase in error, 57 points decrease in error
   (/.f64 (+.f64 x y) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 2) y))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{0.5 \cdot \frac{1}{y} + 0.5 \cdot \frac{1}{x}} \]

4. Simplified 0.0

   \[\leadsto \color{blue}{\frac{0.5}{y} + \frac{0.5}{x}} \]
   Proof
   (+.f64 (/.f64 1/2 y) (/.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) y) (/.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 y))) (/.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 1/2 (/.f64 1 y)) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) x)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 1/2 (/.f64 1 y)) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	24.6
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq 1.4882338842088687 \cdot 10^{-103}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 1.0084416611267336 \cdot 10^{-14}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;y \leq 7.757367463636331 \cdot 10^{+49}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 2
Error	31.1
Cost	192

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

Reproduce

herbie shell --seed 2022291 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

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

  (/ (+ x y) (* (* x 2.0) y)))