Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 76.6% → 100.0%

Time: 6.4s

Precision: binary64

Cost: 448

• Unsound rule application detected in e-graph. Results may not be sound.

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	76.6%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 83.6%

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

2. Taylor expanded in x around 0 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{0.5}{y} + \frac{0.5}{x}} \]
   Step-by-step derivation

   [Start] 100.0%	\[ 0.5 \cdot \frac{1}{y} + 0.5 \cdot \frac{1}{x} \]
   associate-*r/ [=>] 100.0%	\[ \color{blue}{\frac{0.5 \cdot 1}{y}} + 0.5 \cdot \frac{1}{x} \]
   metadata-eval [=>] 100.0%	\[ \frac{\color{blue}{0.5}}{y} + 0.5 \cdot \frac{1}{x} \]
   associate-*r/ [=>] 100.0%	\[ \frac{0.5}{y} + \color{blue}{\frac{0.5 \cdot 1}{x}} \]
   metadata-eval [=>] 100.0%	\[ \frac{0.5}{y} + \frac{\color{blue}{0.5}}{x} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	448

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

Alternative 2
Accuracy	61.6%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 3
Accuracy	50.6%
Cost	192

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

Alternative 4
Accuracy	3.4%
Cost	64

\[-4 \]

Alternative 5
Accuracy	3.3%
Cost	64

\[-0.08333333333333333 \]

Alternative 6
Accuracy	3.3%
Cost	64

\[0.08333333333333333 \]

Alternative 7
Accuracy	3.3%
Cost	64

\[0.25 \]

Reproduce

herbie shell --seed 2023263 
(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)))