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

Average Error: 0.1 → 0.1

Time: 2.4s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = (-0.5d0) - (x * (x / (-6.0d0)))
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

code[x_] := N[(-0.5 - N[(x * N[(x / -6.0), $MachinePrecision]), $MachinePrecision]), $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. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{x}{-6} \cdot \left(-x\right)} - 0.5 \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.0
Cost	584

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;x \leq -1.72:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.72:\\ \;\;\;\;\frac{x}{\frac{6}{x}}\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]

Alternative 3
Error	1.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 3:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]

Alternative 4
Error	1.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]

Alternative 5
Error	0.1
Cost	448

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

Alternative 6
Error	21.3
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))