Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Accuracy: 99.6% → 99.9%

Time: 9.4s

Precision: binary64

Cost: 13504

Math

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

↓

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

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

Target

Original	99.6%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 99.6%

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

2. Simplified 99.9%

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

   [Start] 99.6	\[ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   associate-*l/ [<=] 99.9	\[ \color{blue}{\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot \left(x - 1\right)} \]
   +-commutative [=>] 99.9	\[ \frac{6}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \cdot \left(x - 1\right) \]
   fma-def [=>] 99.9	\[ \frac{6}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \cdot \left(x - 1\right) \]
   sub-neg [=>] 99.9	\[ \frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \cdot \color{blue}{\left(x + \left(-1\right)\right)} \]
   metadata-eval [=>] 99.9	\[ \frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \cdot \left(x + \color{blue}{-1}\right) \]

3. Applied egg-rr 99.7%

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

   [Start] 99.9	\[ \frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \cdot \left(x + -1\right) \]
   *-commutative [=>] 99.9	\[ \color{blue}{\left(x + -1\right) \cdot \frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]
   clear-num [=>] 99.7	\[ \left(x + -1\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{6}}} \]
   un-div-inv [=>] 99.9	\[ \color{blue}{\frac{x + -1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{6}}} \]
   div-inv [=>] 99.7	\[ \frac{x + -1}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot \frac{1}{6}}} \]
   fma-udef [=>] 99.7	\[ \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)} \cdot \frac{1}{6}} \]
   associate-+r+ [=>] 99.7	\[ \frac{x + -1}{\color{blue}{\left(\left(4 \cdot \sqrt{x} + x\right) + 1\right)} \cdot \frac{1}{6}} \]
   fma-def [=>] 99.7	\[ \frac{x + -1}{\left(\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right)} + 1\right) \cdot \frac{1}{6}} \]
   metadata-eval [=>] 99.7	\[ \frac{x + -1}{\left(\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1\right) \cdot \color{blue}{0.16666666666666666}} \]

4. Simplified 99.9%

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

   [Start] 99.7	\[ \frac{x + -1}{\left(\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1\right) \cdot 0.16666666666666666} \]
   /-rgt-identity [<=] 99.7	\[ \frac{x + -1}{\color{blue}{\frac{\left(\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1\right) \cdot 0.16666666666666666}{1}}} \]
   associate-/l* [=>] 99.9	\[ \frac{x + -1}{\color{blue}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}{\frac{1}{0.16666666666666666}}}} \]
   metadata-eval [=>] 99.9	\[ \frac{x + -1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}{\color{blue}{6}}} \]
   associate-/r/ [=>] 99.9	\[ \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \cdot 6} \]
   *-lft-identity [<=] 99.9	\[ \frac{x + -1}{\color{blue}{1 \cdot \left(\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1\right)}} \cdot 6 \]
   *-lft-identity [=>] 99.9	\[ \frac{x + -1}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}} \cdot 6 \]
   fma-udef [=>] 99.9	\[ \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right)} + 1} \cdot 6 \]
   +-commutative [=>] 99.9	\[ \frac{x + -1}{\color{blue}{\left(x + 4 \cdot \sqrt{x}\right)} + 1} \cdot 6 \]
   associate-+r+ [<=] 99.9	\[ \frac{x + -1}{\color{blue}{x + \left(4 \cdot \sqrt{x} + 1\right)}} \cdot 6 \]
   fma-udef [<=] 99.9	\[ \frac{x + -1}{x + \color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \cdot 6 \]

5. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \frac{x + -1}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6 \]
   associate-*l/ [=>] 99.6	\[ \color{blue}{\frac{\left(x + -1\right) \cdot 6}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]
   associate-/l* [=>] 99.9	\[ \color{blue}{\frac{x + -1}{\frac{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}{6}}} \]

6. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13504

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

Alternative 2
Accuracy	96.6%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;6 \cdot \frac{1}{\frac{x + 1}{x + -1}}\\ \mathbf{elif}\;x \leq 10^{+32}:\\ \;\;\;\;\frac{x \cdot 6}{x + \left(1 + 4 \cdot \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	7232

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

Alternative 4
Accuracy	99.9%
Cost	7232

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

Alternative 5
Accuracy	95.4%
Cost	704

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

Alternative 6
Accuracy	95.3%
Cost	576

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

Alternative 7
Accuracy	95.4%
Cost	452

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

Alternative 8
Accuracy	95.4%
Cost	452

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

Alternative 9
Accuracy	95.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{x + -1}{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 10
Accuracy	95.4%
Cost	452

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

Alternative 11
Accuracy	95.4%
Cost	196

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

Alternative 12
Accuracy	48.5%
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023147 
(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)))))