Linear.Projection:inversePerspective from linear-1.19.1.3, B

Average Error: 15.5 → 0.0

Time: 2.2s

Precision: binary64

Cost: 576

Math

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

↓

\[0.5 \cdot \left(\frac{1}{y} - \frac{1}{x}\right) \]

FPCore

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

↓

(FPCore (x y) :precision binary64 (* 0.5 (- (/ 1.0 y) (/ 1.0 x))))

C

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

↓

double code(double x, double y) {
	return 0.5 * ((1.0 / y) - (1.0 / 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 * ((1.0d0 / y) - (1.0d0 / 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 * ((1.0 / y) - (1.0 / x));
}

Python

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

↓

def code(x, y):
	return 0.5 * ((1.0 / y) - (1.0 / x))

Julia

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

↓

function code(x, y)
	return Float64(0.5 * Float64(Float64(1.0 / y) - Float64(1.0 / x)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = 0.5 * ((1.0 / y) - (1.0 / x));
end

Wolfram

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

↓

code[x_, y_] := N[(0.5 * N[(N[(1.0 / y), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	15.5
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 15.5

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

2. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{y} - \frac{1}{x}\right)} \]
   Proof

   [Start] 0.0	\[ 0.5 \cdot \frac{1}{y} - 0.5 \cdot \frac{1}{x} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \color{blue}{\frac{1}{y} \cdot 0.5} - 0.5 \cdot \frac{1}{x} \]
   rational_best_oopsla_all_46_json_45_simplify-102 [=>] 0.0	\[ \color{blue}{0.5 \cdot \left(\frac{1}{y} - \frac{1}{x}\right)} \]

4. Final simplification 0.0

   \[\leadsto 0.5 \cdot \left(\frac{1}{y} - \frac{1}{x}\right) \]

Alternatives

Alternative 1
Error	16.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+36}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \]

Alternative 2
Error	32.4
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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)))