Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 12.5s

Precision: binary64

Cost: 7232

Math

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

↓

\[\frac{\frac{x + -1}{x + \left(1 + 4 \cdot \sqrt{x}\right)}}{0.16666666666666666} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/ (/ (+ x -1.0) (+ x (+ 1.0 (* 4.0 (sqrt x))))) 0.16666666666666666))

C

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

↓

double code(double x) {
	return ((x + -1.0) / (x + (1.0 + (4.0 * sqrt(x))))) / 0.16666666666666666;
}

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 = ((x + (-1.0d0)) / (x + (1.0d0 + (4.0d0 * sqrt(x))))) / 0.16666666666666666d0
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 ((x + -1.0) / (x + (1.0 + (4.0 * Math.sqrt(x))))) / 0.16666666666666666;
}

Python

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

↓

def code(x):
	return ((x + -1.0) / (x + (1.0 + (4.0 * math.sqrt(x))))) / 0.16666666666666666

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(Float64(Float64(x + -1.0) / Float64(x + Float64(1.0 + Float64(4.0 * sqrt(x))))) / 0.16666666666666666)
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 = ((x + -1.0) / (x + (1.0 + (4.0 * sqrt(x))))) / 0.16666666666666666;
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[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.16666666666666666), $MachinePrecision]

Target

Original	0.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.1

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

   [Start] 0.2	\[ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{\left(x - 1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]
   rational.json-simplify-16 [=>] 0.1	\[ \color{blue}{\left(x + -1\right)} \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

3. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\frac{x + -1}{x + \left(1 + 4 \cdot \sqrt{x}\right)}}{0.16666666666666666}} \]

4. Final simplification 0.1

   \[\leadsto \frac{\frac{x + -1}{x + \left(1 + 4 \cdot \sqrt{x}\right)}}{0.16666666666666666} \]

Alternatives

Alternative 1
Error	0.0
Cost	7232

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

Alternative 2
Error	2.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\frac{x + -1}{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{4}{x} + -2\right) \cdot -3\\ \end{array} \]

Alternative 3
Error	2.8
Cost	576

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

Alternative 4
Error	2.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 5
Error	2.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\frac{x + -1}{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 6
Error	2.9
Cost	196

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

Alternative 7
Error	33.1
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023064 
(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)))))