Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.2

Time: 6.4s

Precision: binary64

Cost: 7616

Math

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

↓

\[\frac{6}{\frac{x + \frac{1 + x \cdot -16}{1 + \sqrt{x} \cdot -4}}{x + -1}} \]

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 -16.0)) (+ 1.0 (* (sqrt x) -4.0)))) (+ 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 * -16.0)) / (1.0 + (sqrt(x) * -4.0)))) / (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 * (-16.0d0))) / (1.0d0 + (sqrt(x) * (-4.0d0))))) / (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 * -16.0)) / (1.0 + (Math.sqrt(x) * -4.0)))) / (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 * -16.0)) / (1.0 + (math.sqrt(x) * -4.0)))) / (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 + Float64(Float64(1.0 + Float64(x * -16.0)) / Float64(1.0 + Float64(sqrt(x) * -4.0)))) / Float64(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 * -16.0)) / (1.0 + (sqrt(x) * -4.0)))) / (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 + N[(N[(1.0 + N[(x * -16.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.0
Herbie	0.2

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{6}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{x + -1}}} \]
   Proof
   (/.f64 6 (/.f64 (+.f64 x (+.f64 1 (*.f64 4 (sqrt.f64 x)))) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 6 (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 x 1) (*.f64 4 (sqrt.f64 x)))) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 6 (/.f64 (+.f64 (+.f64 x 1) (*.f64 4 (sqrt.f64 x))) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 6 (/.f64 (+.f64 (+.f64 x 1) (*.f64 4 (sqrt.f64 x))) (Rewrite<= sub-neg_binary64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 6 (-.f64 x 1)) (+.f64 (+.f64 x 1) (*.f64 4 (sqrt.f64 x))))): 25 points increase in error, 5 points decrease in error

3. Applied egg-rr 0.2

   \[\leadsto \frac{6}{\frac{\color{blue}{\mathsf{fma}\left(1 - x \cdot 16, \frac{1}{1 - 4 \cdot \sqrt{x}}, x\right)}}{x + -1}} \]

4. Simplified 0.2

   \[\leadsto \frac{6}{\frac{\color{blue}{x + \frac{1 + x \cdot -16}{1 + \sqrt{x} \cdot -4}}}{x + -1}} \]
   Proof
   (+.f64 x (/.f64 (+.f64 1 (*.f64 x -16)) (+.f64 1 (*.f64 (sqrt.f64 x) -4)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (+.f64 1 (*.f64 x (Rewrite<= metadata-eval (neg.f64 16)))) (+.f64 1 (*.f64 (sqrt.f64 x) -4)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (+.f64 1 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x 16)))) (+.f64 1 (*.f64 (sqrt.f64 x) -4)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 x 16))) (+.f64 1 (*.f64 (sqrt.f64 x) -4)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (-.f64 1 (*.f64 x 16)) 1)) (+.f64 1 (*.f64 (sqrt.f64 x) -4)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (*.f64 (-.f64 1 (*.f64 x 16)) 1) (+.f64 1 (Rewrite<= *-commutative_binary64 (*.f64 -4 (sqrt.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (*.f64 (-.f64 1 (*.f64 x 16)) 1) (+.f64 1 (*.f64 (Rewrite<= metadata-eval (neg.f64 4)) (sqrt.f64 x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (*.f64 (-.f64 1 (*.f64 x 16)) 1) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 4 (sqrt.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 1 (*.f64 x 16)) (/.f64 1 (-.f64 1 (*.f64 4 (sqrt.f64 x))))))): 2 points increase in error, 1 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (-.f64 1 (*.f64 x 16)) (/.f64 1 (-.f64 1 (*.f64 4 (sqrt.f64 x))))) x)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-udef_binary64 (fma.f64 (-.f64 1 (*.f64 x 16)) (/.f64 1 (-.f64 1 (*.f64 4 (sqrt.f64 x)))) x)): 1 points increase in error, 0 points decrease in error

5. Final simplification 0.2

   \[\leadsto \frac{6}{\frac{x + \frac{1 + x \cdot -16}{1 + \sqrt{x} \cdot -4}}{x + -1}} \]

Alternatives

Alternative 1
Error	1.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;x \cdot 12 + -6\\ \mathbf{elif}\;x \leq 10^{+33}:\\ \;\;\;\;\frac{6 \cdot x}{\sqrt{x} \cdot 4 + \left(x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7232

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

Alternative 3
Error	0.0
Cost	7232

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

Alternative 4
Error	2.6
Cost	704

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

Alternative 5
Error	2.6
Cost	576

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

Alternative 6
Error	2.6
Cost	452

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

Alternative 7
Error	2.6
Cost	452

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

Alternative 8
Error	2.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 0.49:\\ \;\;\;\;x \cdot 12 + -6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 9
Error	2.6
Cost	196

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

Alternative 10
Error	32.4
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2022337 
(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)))))