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

Average Error: 0.6 → 0.2

Time: 7.5s

Precision: binary64

Cost: 320

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = (1.0d0 / x) / x
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	0.6
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.6

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\frac{\frac{1}{x}}{x}} \]
   Proof

   [Start] 0.6	\[ \frac{1}{x \cdot x} \]
   rational.json-simplify-46 [=>] 0.2	\[ \color{blue}{\frac{\frac{1}{x}}{x}} \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.6
Cost	320

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

Reproduce

herbie shell --seed 2023074 
(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)))