Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.0

Time: 8.8s

Precision: binary64

Cost: 7360

Math

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

↓

\[\frac{x + -1}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{24}} \cdot 0.25 \]

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))) 24.0)) 0.25))

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))) / 24.0)) * 0.25;
}

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))) / 24.0d0)) * 0.25d0
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))) / 24.0)) * 0.25;
}

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))) / 24.0)) * 0.25

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(Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))) / 24.0)) * 0.25)
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))) / 24.0)) * 0.25;
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[(N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 24.0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]

Target

Original	0.2
Target	0.1
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. 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

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{24}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x + -1}}}{8}} \]

4. Simplified 0.0

   \[\leadsto \color{blue}{\frac{x + -1}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{24}} \cdot 0.25} \]
   Proof

   [Start] 0.1	\[ \frac{2 \cdot \frac{24}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x + -1}}}{8} \]
   rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{\frac{24}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x + -1}} \cdot \frac{2}{8}} \]
   rational.json-simplify-61 [=>] 0.0	\[ \color{blue}{\frac{x + -1}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{24}}} \cdot \frac{2}{8} \]
   metadata-eval [=>] 0.0	\[ \frac{x + -1}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{24}} \cdot \color{blue}{0.25} \]

5. Final simplification 0.0

   \[\leadsto \frac{x + -1}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{24}} \cdot 0.25 \]

Alternatives

Alternative 1
Error	2.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;-6 \cdot \left(\left(-1 - x\right) \cdot \left(3 \cdot x - 1\right)\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 2
Error	2.1
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;-6 \cdot \left(\left(-1 - x\right) \cdot \left(3 \cdot x - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot 6\\ \end{array} \]

Alternative 3
Error	0.0
Cost	7232

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

Alternative 4
Error	2.9
Cost	2112

\[-6 \cdot \left(-1 + \left(1 - \frac{x + \left(x + 2\right)}{\left(1 + x\right) \cdot \frac{\frac{1 + x}{x + -1}}{x + -1}} \cdot \frac{0.5}{x + -1}\right)\right) \]

Alternative 5
Error	2.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;-6 \cdot \left(-1 + \left(2 + -2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 6
Error	2.8
Cost	704

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

Alternative 7
Error	2.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;-6 \cdot \left(1 + -2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 8
Error	2.8
Cost	576

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

Alternative 9
Error	2.8
Cost	452

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

Alternative 10
Error	2.8
Cost	452

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

Alternative 11
Error	2.8
Cost	452

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

Alternative 12
Error	2.8
Cost	196

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

Alternative 13
Error	32.8
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023074 
(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)))))