Linear.Projection:inversePerspective from linear-1.19.1.3, C

Average Error: 15.8 → 0.0

Time: 4.2s

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	15.8
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 15.8

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

2. Simplified 15.8

   \[\leadsto \color{blue}{\frac{x + y}{x \cdot \left(y \cdot 2\right)}} \]
   Proof

   [Start] 15.8	\[ \frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   rational.json-simplify-2 [=>] 15.8	\[ \frac{x + y}{\color{blue}{y \cdot \left(x \cdot 2\right)}} \]
   rational.json-simplify-43 [=>] 15.8	\[ \frac{x + y}{\color{blue}{x \cdot \left(2 \cdot y\right)}} \]
   rational.json-simplify-2 [=>] 15.8	\[ \frac{x + y}{x \cdot \color{blue}{\left(y \cdot 2\right)}} \]

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

   \[\leadsto \color{blue}{\frac{0.5}{x} + \frac{0.5}{y}} \]
   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-29 [=>] 15.7	\[ 0.5 \cdot \color{blue}{\frac{x + y}{x \cdot y}} \]
   rational.json-simplify-35 [=>] 15.8	\[ 0.5 \cdot \color{blue}{\frac{\left(x + y\right) + \left(x + y\right)}{x \cdot y + x \cdot y}} \]
   rational.json-simplify-2 [<=] 15.8	\[ 0.5 \cdot \frac{\left(x + y\right) + \left(x + y\right)}{\color{blue}{y \cdot x} + x \cdot y} \]
   rational.json-simplify-51 [=>] 15.8	\[ 0.5 \cdot \frac{\left(x + y\right) + \left(x + y\right)}{\color{blue}{y \cdot \left(x + x\right)}} \]
   rational.json-simplify-6 [<=] 15.8	\[ 0.5 \cdot \frac{\color{blue}{1 \cdot \left(x + y\right)} + \left(x + y\right)}{y \cdot \left(x + x\right)} \]
   rational.json-simplify-2 [<=] 15.8	\[ 0.5 \cdot \frac{\color{blue}{\left(x + y\right) \cdot 1} + \left(x + y\right)}{y \cdot \left(x + x\right)} \]
   rational.json-simplify-6 [<=] 15.8	\[ 0.5 \cdot \frac{\left(x + y\right) \cdot 1 + \color{blue}{1 \cdot \left(x + y\right)}}{y \cdot \left(x + x\right)} \]
   rational.json-simplify-51 [=>] 15.8	\[ 0.5 \cdot \frac{\color{blue}{\left(x + y\right) \cdot \left(1 + 1\right)}}{y \cdot \left(x + x\right)} \]
   metadata-eval [=>] 15.8	\[ 0.5 \cdot \frac{\left(x + y\right) \cdot \color{blue}{2}}{y \cdot \left(x + x\right)} \]
   rational.json-simplify-49 [<=] 15.8	\[ \color{blue}{\frac{\left(\left(x + y\right) \cdot 2\right) \cdot 0.5}{y \cdot \left(x + x\right)}} \]
   rational.json-simplify-2 [<=] 15.8	\[ \frac{\color{blue}{0.5 \cdot \left(\left(x + y\right) \cdot 2\right)}}{y \cdot \left(x + x\right)} \]
   rational.json-simplify-43 [=>] 15.8	\[ \frac{\color{blue}{\left(x + y\right) \cdot \left(2 \cdot 0.5\right)}}{y \cdot \left(x + x\right)} \]
   metadata-eval [=>] 15.8	\[ \frac{\left(x + y\right) \cdot \color{blue}{1}}{y \cdot \left(x + x\right)} \]
   rational.json-simplify-2 [=>] 15.8	\[ \frac{\color{blue}{1 \cdot \left(x + y\right)}}{y \cdot \left(x + x\right)} \]
   rational.json-simplify-6 [=>] 15.8	\[ \frac{\color{blue}{x + y}}{y \cdot \left(x + x\right)} \]
   rational.json-simplify-35 [=>] 15.8	\[ \color{blue}{\frac{\left(x + y\right) + \left(x + y\right)}{y \cdot \left(x + x\right) + y \cdot \left(x + x\right)}} \]
   rational.json-simplify-2 [=>] 15.8	\[ \frac{\left(x + y\right) + \left(x + y\right)}{\color{blue}{\left(x + x\right) \cdot y} + y \cdot \left(x + x\right)} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	24.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-17}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 2
Error	31.4
Cost	192

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

Reproduce

herbie shell --seed 2023068 
(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)))