Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, A

Average Error: 0.6 → 0.0

Time: 1.4s

Precision: binary64

Cost: 6528

Math

\[\frac{1}{x \cdot x} \]

↓

\[{x}^{-2} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* x x)))

↓

(FPCore (x) :precision binary64 (pow x -2.0))

C

double code(double x) {
	return 1.0 / (x * x);
}

↓

double code(double x) {
	return pow(x, -2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x * x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = x ** (-2.0d0)
end function

Java

public static double code(double x) {
	return 1.0 / (x * x);
}

↓

public static double code(double x) {
	return Math.pow(x, -2.0);
}

Python

def code(x):
	return 1.0 / (x * x)

↓

def code(x):
	return math.pow(x, -2.0)

Julia

function code(x)
	return Float64(1.0 / Float64(x * x))
end

↓

function code(x)
	return x ^ -2.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x * x);
end

↓

function tmp = code(x)
	tmp = x ^ -2.0;
end

Wolfram

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

↓

code[x_] := N[Power[x, -2.0], $MachinePrecision]

Target

Original	0.6
Target	0.2
Herbie	0.0

\[\frac{\frac{1}{x}}{x} \]

Derivation

1. Initial program 0.6

   \[\frac{1}{x \cdot x} \]

2. Taylor expanded in x around 0 0.6

   \[\leadsto \color{blue}{\frac{1}{{x}^{2}}} \]

3. Simplified 0.0

   \[\leadsto \color{blue}{{x}^{-2}} \]
   Proof

   [Start] 0.6	\[ \frac{1}{{x}^{2}} \]
   unpow2 [=>] 0.6	\[ \frac{1}{\color{blue}{x \cdot x}} \]
   associate-/r* [=>] 0.2	\[ \color{blue}{\frac{\frac{1}{x}}{x}} \]
   /-rgt-identity [<=] 0.2	\[ \frac{\frac{1}{x}}{\color{blue}{\frac{x}{1}}} \]
   associate-/l* [<=] 0.2	\[ \color{blue}{\frac{\frac{1}{x} \cdot 1}{x}} \]
   associate-*r/ [<=] 0.3	\[ \color{blue}{\frac{1}{x} \cdot \frac{1}{x}} \]
   unpow-1 [<=] 0.3	\[ \color{blue}{{x}^{-1}} \cdot \frac{1}{x} \]
   unpow-1 [<=] 0.3	\[ {x}^{-1} \cdot \color{blue}{{x}^{-1}} \]
   pow-sqr [=>] 0.0	\[ \color{blue}{{x}^{\left(2 \cdot -1\right)}} \]
   metadata-eval [=>] 0.0	\[ {x}^{\color{blue}{-2}} \]

4. Final simplification 0.0

   \[\leadsto {x}^{-2} \]

Alternatives

Alternative 1
Error	0.6
Cost	320

\[\frac{1}{x \cdot x} \]

Alternative 2
Error	0.2
Cost	320

\[\frac{\frac{1}{x}}{x} \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ 1.0 x) x)

  (/ 1.0 (* x x)))