Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.38% → 0.08%

Time: 10.0s

Precision: binary64

Cost: 7232

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = (1.0d0 - x) / ((1.0d0 + (x + (4.0d0 * sqrt(x)))) / (-6.0d0))
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 (1.0 - x) / ((1.0 + (x + (4.0 * Math.sqrt(x)))) / -6.0);
}

Python

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

↓

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

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(Float64(1.0 - x) / Float64(Float64(1.0 + Float64(x + Float64(4.0 * sqrt(x)))) / -6.0))
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 = (1.0 - x) / ((1.0 + (x + (4.0 * sqrt(x)))) / -6.0);
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[(N[(1.0 - x), $MachinePrecision] / N[(N[(1.0 + N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -6.0), $MachinePrecision]), $MachinePrecision]

Target

Original	0.38%
Target	0.08%
Herbie	0.08%

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

Derivation

1. Initial program 0.38

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

2. Simplified 0.08

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

   [Start] 0.38	\[ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   associate-*l/ [<=] 0.14	\[ \color{blue}{\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot \left(x - 1\right)} \]
   sub-neg [=>] 0.14	\[ \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot \color{blue}{\left(x + \left(-1\right)\right)} \]
   +-commutative [=>] 0.14	\[ \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot \color{blue}{\left(\left(-1\right) + x\right)} \]
   distribute-rgt-in [=>] 0.14	\[ \color{blue}{\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} + x \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]
   *-commutative [=>] 0.14	\[ \left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} + \color{blue}{\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot x} \]
   cancel-sign-sub [<=] 0.14	\[ \color{blue}{\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} - \left(-\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \cdot x} \]
   mul-1-neg [<=] 0.14	\[ \left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} - \color{blue}{\left(-1 \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \cdot x \]
   metadata-eval [<=] 0.14	\[ \left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} - \left(\color{blue}{\left(-1\right)} \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \cdot x \]
   *-commutative [=>] 0.14	\[ \left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} - \color{blue}{x \cdot \left(\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]
   cancel-sign-sub-inv [=>] 0.14	\[ \color{blue}{\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} + \left(-x\right) \cdot \left(\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]
   distribute-rgt1-in [=>] 0.14	\[ \color{blue}{\left(\left(-x\right) + 1\right) \cdot \left(\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]

3. Applied egg-rr 0.41

   \[\leadsto \color{blue}{\frac{6}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \cdot \left(1 - x\right)} \]

4. Simplified 0.36

   \[\leadsto \color{blue}{-6 \cdot \frac{-1 + x}{\left(-1 - x\right) - \sqrt{x \cdot 16}}} \]
   Proof

   [Start] 0.41	\[ \frac{6}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \cdot \left(1 - x\right) \]
   metadata-eval [<=] 0.41	\[ \frac{\color{blue}{\frac{-6}{-1}}}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \cdot \left(1 - x\right) \]
   associate-/r* [<=] 0.41	\[ \color{blue}{\frac{-6}{-1 \cdot \left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)}} \cdot \left(1 - x\right) \]
   neg-mul-1 [<=] 0.41	\[ \frac{-6}{\color{blue}{-\left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)}} \cdot \left(1 - x\right) \]
   associate-*l/ [=>] 0.47	\[ \color{blue}{\frac{-6 \cdot \left(1 - x\right)}{-\left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)}} \]
   neg-mul-1 [=>] 0.47	\[ \frac{-6 \cdot \left(1 - x\right)}{\color{blue}{-1 \cdot \left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)}} \]
   metadata-eval [<=] 0.47	\[ \frac{\color{blue}{\left(6 \cdot -1\right)} \cdot \left(1 - x\right)}{-1 \cdot \left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)} \]
   associate-*r* [<=] 0.47	\[ \frac{\color{blue}{6 \cdot \left(-1 \cdot \left(1 - x\right)\right)}}{-1 \cdot \left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)} \]
   neg-mul-1 [<=] 0.47	\[ \frac{6 \cdot \color{blue}{\left(-\left(1 - x\right)\right)}}{-1 \cdot \left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)} \]
   neg-sub0 [=>] 0.47	\[ \frac{6 \cdot \color{blue}{\left(0 - \left(1 - x\right)\right)}}{-1 \cdot \left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)} \]
   associate--r- [=>] 0.47	\[ \frac{6 \cdot \color{blue}{\left(\left(0 - 1\right) + x\right)}}{-1 \cdot \left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)} \]
   metadata-eval [=>] 0.47	\[ \frac{6 \cdot \left(\color{blue}{-1} + x\right)}{-1 \cdot \left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)} \]
   +-commutative [<=] 0.47	\[ \frac{6 \cdot \color{blue}{\left(x + -1\right)}}{-1 \cdot \left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)} \]
   times-frac [=>] 0.35	\[ \color{blue}{\frac{6}{-1} \cdot \frac{x + -1}{-1 - \left(x + \sqrt{x \cdot 16}\right)}} \]
   metadata-eval [=>] 0.35	\[ \color{blue}{-6} \cdot \frac{x + -1}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \]
   +-commutative [=>] 0.35	\[ -6 \cdot \frac{\color{blue}{-1 + x}}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \]
   associate--r+ [=>] 0.36	\[ -6 \cdot \frac{-1 + x}{\color{blue}{\left(-1 - x\right) - \sqrt{x \cdot 16}}} \]

5. Applied egg-rr 0.08

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

6. Final simplification 0.08

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

Alternatives

Alternative 1
Error	3.54%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;-6 + x \cdot 12\\ \mathbf{elif}\;x \leq 10^{+33}:\\ \;\;\;\;\frac{x \cdot 6}{4 \cdot \sqrt{x} + \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 2
Error	2.54%
Cost	7236

\[\begin{array}{l} t_0 := 4 \cdot \sqrt{x} + \left(1 + x\right)\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{t_0}\\ \end{array} \]

Alternative 3
Error	0.36%
Cost	7232

\[-6 \cdot \frac{x + -1}{\left(-1 - x\right) - \sqrt{x \cdot 16}} \]

Alternative 4
Error	0.08%
Cost	7232

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

Alternative 5
Error	4.71%
Cost	576

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

Alternative 6
Error	4.71%
Cost	576

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

Alternative 7
Error	4.72%
Cost	452

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

Alternative 8
Error	4.72%
Cost	452

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

Alternative 9
Error	4.72%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;-6 + x \cdot 12\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-12}{x}\\ \end{array} \]

Alternative 10
Error	4.72%
Cost	196

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

Alternative 11
Error	50.79%
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023090 
(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)))))