Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.0

Time: 7.9s

Precision: binary64

Cost: 7232

Math

\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

↓

\[6 \cdot \frac{x + -1}{\left(x + 4 \cdot \sqrt{x}\right) + 1} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

↓

(FPCore (x)
 :precision binary64
 (* 6.0 (/ (+ x -1.0) (+ (+ x (* 4.0 (sqrt x))) 1.0))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

↓

double code(double x) {
	return 6.0 * ((x + -1.0) / ((x + (4.0 * sqrt(x))) + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 * ((x + (-1.0d0)) / ((x + (4.0d0 * sqrt(x))) + 1.0d0))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

↓

public static double code(double x) {
	return 6.0 * ((x + -1.0) / ((x + (4.0 * Math.sqrt(x))) + 1.0));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

↓

def code(x):
	return 6.0 * ((x + -1.0) / ((x + (4.0 * math.sqrt(x))) + 1.0))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

↓

function code(x)
	return Float64(6.0 * Float64(Float64(x + -1.0) / Float64(Float64(x + Float64(4.0 * sqrt(x))) + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

↓

function tmp = code(x)
	tmp = 6.0 * ((x + -1.0) / ((x + (4.0 * sqrt(x))) + 1.0));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.0
Herbie	0.0

\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}} \]

Derivation

1. Initial program 0.2

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

2. Applied egg-rr 0.0

   \[\leadsto \color{blue}{6 \cdot \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}} \]

3. Applied egg-rr 0.0

   \[\leadsto 6 \cdot \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right)} + 1} \]

4. Final simplification 0.0

   \[\leadsto 6 \cdot \frac{x + -1}{\left(x + 4 \cdot \sqrt{x}\right) + 1} \]

Alternatives

Alternative 1
Error	0.0
Cost	7232

\[\frac{6}{\frac{\left(x + 4 \cdot \sqrt{x}\right) + 1}{x + -1}} \]

Alternative 2
Error	2.8
Cost	576

\[\frac{6}{\frac{x + 1}{x + -1}} \]

Alternative 3
Error	3.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9528963858121105 \cdot 10^{-8}:\\ \;\;\;\;x \cdot 12 + -6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 4
Error	2.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9528963858121105 \cdot 10^{-8}:\\ \;\;\;\;x \cdot 12 + -6\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-6}{x}\\ \end{array} \]

Alternative 5
Error	2.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1.521228129001963:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-12}{x}\\ \end{array} \]

Alternative 6
Error	3.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9528963858121105 \cdot 10^{-8}:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 7
Error	32.9
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))