Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H

Average Error: 0.1 → 0.1

Time: 2.6s

Precision: binary64

Cost: 448

Math

\[\frac{x \cdot x - 3}{6} \]

↓

\[\frac{x}{\frac{6}{x}} + -0.5 \]

FPCore

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))

↓

(FPCore (x) :precision binary64 (+ (/ x (/ 6.0 x)) -0.5))

C

double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

↓

double code(double x) {
	return (x / (6.0 / x)) + -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) - 3.0d0) / 6.0d0
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / (6.0d0 / x)) + (-0.5d0)
end function

Java

public static double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

↓

public static double code(double x) {
	return (x / (6.0 / x)) + -0.5;
}

Python

def code(x):
	return ((x * x) - 3.0) / 6.0

↓

def code(x):
	return (x / (6.0 / x)) + -0.5

Julia

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end

↓

function code(x)
	return Float64(Float64(x / Float64(6.0 / x)) + -0.5)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) - 3.0) / 6.0;
end

↓

function tmp = code(x)
	tmp = (x / (6.0 / x)) + -0.5;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]

↓

code[x_] := N[(N[(x / N[(6.0 / x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[\frac{x \cdot x - 3}{6} \]

2. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{x}{\frac{6}{x}} - 0.5} \]

3. Final simplification 0.1

   \[\leadsto \frac{x}{\frac{6}{x}} + -0.5 \]

Alternatives

Alternative 1
Error	1.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 2
Error	21.9
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))