Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.3

Time: 41.1s

Precision: binary64

Cost: 21952

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (* 6.0 (+ x -1.0)) (+ x (+ 1.0 (* 4.0 (sqrt x)))))))
   (* (/ 1.0 t_0) (* t_0 t_0))))

C

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

↓

double code(double x) {
	double t_0 = (6.0 * (x + -1.0)) / (x + (1.0 + (4.0 * sqrt(x))));
	return (1.0 / t_0) * (t_0 * t_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
    real(8) :: t_0
    t_0 = (6.0d0 * (x + (-1.0d0))) / (x + (1.0d0 + (4.0d0 * sqrt(x))))
    code = (1.0d0 / t_0) * (t_0 * t_0)
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) {
	double t_0 = (6.0 * (x + -1.0)) / (x + (1.0 + (4.0 * Math.sqrt(x))));
	return (1.0 / t_0) * (t_0 * t_0);
}

Python

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

↓

def code(x):
	t_0 = (6.0 * (x + -1.0)) / (x + (1.0 + (4.0 * math.sqrt(x))))
	return (1.0 / t_0) * (t_0 * t_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)
	t_0 = Float64(Float64(6.0 * Float64(x + -1.0)) / Float64(x + Float64(1.0 + Float64(4.0 * sqrt(x)))))
	return Float64(Float64(1.0 / t_0) * Float64(t_0 * t_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)
	t_0 = (6.0 * (x + -1.0)) / (x + (1.0 + (4.0 * sqrt(x))));
	tmp = (1.0 / t_0) * (t_0 * t_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_] := Block[{t$95$0 = N[(N[(6.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.2
Target	0.1
Herbie	0.3

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

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 0.3

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

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	1.6
Cost	7236

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

Alternative 2
Error	0.2
Cost	7232

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

Alternative 3
Error	0.2
Cost	7232

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

Alternative 4
Error	2.2
Cost	7108

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

Alternative 5
Error	3.0
Cost	580

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

Alternative 6
Error	3.0
Cost	196

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

Alternative 7
Error	33.2
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023077 
(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)))))