Linear.Projection:inversePerspective from linear-1.19.1.3, C

Average Error: 15.6 → 0.0

Time: 3.8s

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	15.6
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 15.6

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

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

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

   [Start] 0.0	\[ 0.5 \cdot \frac{1}{y} + 0.5 \cdot \frac{1}{x} \]
   rational.json-simplify-2 [=>] 0.0	\[ 0.5 \cdot \frac{1}{y} + \color{blue}{\frac{1}{x} \cdot 0.5} \]
   rational.json-simplify-51 [=>] 0.0	\[ \color{blue}{0.5 \cdot \left(\frac{1}{x} + \frac{1}{y}\right)} \]
   rational.json-simplify-28 [<=] 15.6	\[ 0.5 \cdot \color{blue}{\frac{x + y}{x \cdot y}} \]
   rational.json-simplify-35 [=>] 15.7	\[ 0.5 \cdot \color{blue}{\frac{\left(x + y\right) + \left(x + y\right)}{x \cdot y + x \cdot y}} \]
   rational.json-simplify-7 [<=] 15.7	\[ 0.5 \cdot \frac{\color{blue}{\frac{\left(x + y\right) + \left(x + y\right)}{1}}}{x \cdot y + x \cdot y} \]
   rational.json-simplify-2 [=>] 15.7	\[ 0.5 \cdot \frac{\frac{\left(x + y\right) + \left(x + y\right)}{1}}{\color{blue}{y \cdot x} + x \cdot y} \]
   rational.json-simplify-51 [=>] 15.6	\[ 0.5 \cdot \frac{\frac{\left(x + y\right) + \left(x + y\right)}{1}}{\color{blue}{y \cdot \left(x + x\right)}} \]
   rational.json-simplify-7 [<=] 15.6	\[ 0.5 \cdot \frac{\frac{\left(x + y\right) + \left(x + y\right)}{1}}{\color{blue}{\frac{y \cdot \left(x + x\right)}{1}}} \]
   rational.json-simplify-55 [<=] 16.2	\[ 0.5 \cdot \color{blue}{\left(\frac{1}{y \cdot \left(x + x\right)} \cdot \frac{\left(x + y\right) + \left(x + y\right)}{1}\right)} \]
   rational.json-simplify-2 [=>] 16.2	\[ 0.5 \cdot \left(\frac{1}{\color{blue}{\left(x + x\right) \cdot y}} \cdot \frac{\left(x + y\right) + \left(x + y\right)}{1}\right) \]
   rational.json-simplify-47 [<=] 15.6	\[ 0.5 \cdot \left(\color{blue}{\frac{\frac{1}{x + x}}{y}} \cdot \frac{\left(x + y\right) + \left(x + y\right)}{1}\right) \]
   metadata-eval [<=] 15.6	\[ 0.5 \cdot \left(\frac{\frac{\color{blue}{0.5 + 0.5}}{x + x}}{y} \cdot \frac{\left(x + y\right) + \left(x + y\right)}{1}\right) \]
   rational.json-simplify-35 [<=] 15.6	\[ 0.5 \cdot \left(\frac{\color{blue}{\frac{0.5}{x}}}{y} \cdot \frac{\left(x + y\right) + \left(x + y\right)}{1}\right) \]
   rational.json-simplify-7 [=>] 15.6	\[ 0.5 \cdot \left(\frac{\frac{0.5}{x}}{y} \cdot \color{blue}{\left(\left(x + y\right) + \left(x + y\right)\right)}\right) \]
   rational.json-simplify-51 [<=] 15.6	\[ 0.5 \cdot \color{blue}{\left(\frac{\frac{0.5}{x}}{y} \cdot \left(x + y\right) + \left(x + y\right) \cdot \frac{\frac{0.5}{x}}{y}\right)} \]
   rational.json-simplify-2 [<=] 15.6	\[ 0.5 \cdot \left(\color{blue}{\left(x + y\right) \cdot \frac{\frac{0.5}{x}}{y}} + \left(x + y\right) \cdot \frac{\frac{0.5}{x}}{y}\right) \]
   rational.json-simplify-2 [=>] 15.6	\[ 0.5 \cdot \left(\left(x + y\right) \cdot \frac{\frac{0.5}{x}}{y} + \color{blue}{\frac{\frac{0.5}{x}}{y} \cdot \left(x + y\right)}\right) \]
   rational.json-simplify-51 [=>] 15.6	\[ 0.5 \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\frac{\frac{0.5}{x}}{y} + \frac{\frac{0.5}{x}}{y}\right)\right)} \]
   rational.json-simplify-7 [<=] 15.6	\[ 0.5 \cdot \left(\left(x + y\right) \cdot \left(\frac{\frac{0.5}{x}}{y} + \color{blue}{\frac{\frac{\frac{0.5}{x}}{y}}{1}}\right)\right) \]
   rational.json-simplify-30 [<=] 15.6	\[ 0.5 \cdot \left(\left(x + y\right) \cdot \color{blue}{\left(\left(1 + 1\right) \cdot \frac{\frac{\frac{0.5}{x}}{y}}{1}\right)}\right) \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	24.2
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -3850:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-78}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 2
Error	31.4
Cost	192

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

Reproduce

herbie shell --seed 2023064 
(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)))