Linear.Projection:inversePerspective from linear-1.19.1.3, B

Average Error: 15.3 → 0.0

Time: 3.3s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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) / x) + (0.5d0 / y)
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 / x) + (0.5 / y);
}

Python

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

↓

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

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 / x) + Float64(0.5 / y))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (-0.5 / x) + (0.5 / y);
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 / x), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision]

Target

Original	15.3
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 15.3

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{-0.5}{x} - \frac{-0.5}{y}} \]
   Proof
   (-.f64 (/.f64 -1/2 x) (/.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) x) (/.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 x))) (/.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 -1 (Rewrite<= *-commutative_binary64 (*.f64 x 2))) (/.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 -1 (Rewrite<= /-rgt-identity_binary64 (/.f64 (*.f64 x 2) 1))) (/.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 -1 (/.f64 (*.f64 x 2) (Rewrite<= *-inverses_binary64 (/.f64 y y)))) (/.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 x 2) y) y))) (/.f64 -1/2 y)): 47 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 y) (*.f64 (*.f64 x 2) y))) (/.f64 -1/2 y)): 19 points increase in error, 3 points decrease in error
   (-.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 y)) (*.f64 (*.f64 x 2) y)) (/.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (neg.f64 y) (*.f64 (*.f64 x 2) y)) (/.f64 (Rewrite<= metadata-eval (neg.f64 1/2)) y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (neg.f64 y) (*.f64 (*.f64 x 2) y)) (/.f64 (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2))) y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (neg.f64 y) (*.f64 (*.f64 x 2) y)) (/.f64 (neg.f64 (/.f64 (Rewrite<= *-inverses_binary64 (/.f64 x x)) 2)) y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (neg.f64 y) (*.f64 (*.f64 x 2) y)) (/.f64 (neg.f64 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 x 2)))) y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (neg.f64 y) (*.f64 (*.f64 x 2) y)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (/.f64 x (*.f64 x 2)) y)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (neg.f64 y) (*.f64 (*.f64 x 2) y)) (neg.f64 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 (*.f64 x 2) y))))): 44 points increase in error, 4 points decrease in error
   (-.f64 (/.f64 (neg.f64 y) (*.f64 (*.f64 x 2) y)) (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 x) (*.f64 (*.f64 x 2) y)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (neg.f64 y) (neg.f64 x)) (*.f64 (*.f64 x 2) y))): 1 points increase in error, 2 points decrease in error
   (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 y) (neg.f64 (neg.f64 x)))) (*.f64 (*.f64 x 2) y)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (neg.f64 y) (Rewrite=> remove-double-neg_binary64 x)) (*.f64 (*.f64 x 2) y)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 y))) (*.f64 (*.f64 x 2) y)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x y)) (*.f64 (*.f64 x 2) y)): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	16.5
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -8.410025894581865 \cdot 10^{-19}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;y \leq 1.1760454271792877 \cdot 10^{-88}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 29596512.912223004:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;y \leq 7.757367463636331 \cdot 10^{+49}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]

Alternative 2
Error	32.1
Cost	192

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

Reproduce

herbie shell --seed 2022291 
(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)))