Average Error: 0.2 → 0.1
Time: 12.6s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\sqrt[3]{{\left(\frac{x - 1}{\left(x + 1\right) + \sqrt{x} \cdot 4}\right)}^{3}} \cdot 6\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\sqrt[3]{{\left(\frac{x - 1}{\left(x + 1\right) + \sqrt{x} \cdot 4}\right)}^{3}} \cdot 6
double f(double x) {
        double r733463 = 6.0;
        double r733464 = x;
        double r733465 = 1.0;
        double r733466 = r733464 - r733465;
        double r733467 = r733463 * r733466;
        double r733468 = r733464 + r733465;
        double r733469 = 4.0;
        double r733470 = sqrt(r733464);
        double r733471 = r733469 * r733470;
        double r733472 = r733468 + r733471;
        double r733473 = r733467 / r733472;
        return r733473;
}
↓
double f(double x) {
        double r733474 = x;
        double r733475 = 1.0;
        double r733476 = r733474 - r733475;
        double r733477 = r733474 + r733475;
        double r733478 = sqrt(r733474);
        double r733479 = 4.0;
        double r733480 = r733478 * r733479;
        double r733481 = r733477 + r733480;
        double r733482 = r733476 / r733481;
        double r733483 = 3.0;
        double r733484 = pow(r733482, r733483);
        double r733485 = cbrt(r733484);
        double r733486 = 6.0;
        double r733487 = r733485 * r733486;
        return r733487;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original0.2
Target0.1
Herbie0.1
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

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

  2. Simplified0.0

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

  4. Applied add-cbrt-cube20.6

    \[\leadsto 6 \cdot \frac{x - 1}{\color{blue}{\sqrt[3]{\left(\left(\left(x + 1\right) + \sqrt{x} \cdot 4\right) \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot 4\right)\right) \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot 4\right)}}}\]

  5. Applied add-cbrt-cube21.2

    \[\leadsto 6 \cdot \frac{\color{blue}{\sqrt[3]{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right) \cdot \left(x - 1\right)}}}{\sqrt[3]{\left(\left(\left(x + 1\right) + \sqrt{x} \cdot 4\right) \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot 4\right)\right) \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot 4\right)}}\]

  6. Applied cbrt-undiv21.2

    \[\leadsto 6 \cdot \color{blue}{\sqrt[3]{\frac{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right) \cdot \left(x - 1\right)}{\left(\left(\left(x + 1\right) + \sqrt{x} \cdot 4\right) \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot 4\right)\right) \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot 4\right)}}}\]

  7. Simplified0.1

    \[\leadsto 6 \cdot \sqrt[3]{\color{blue}{{\left(\frac{x - 1}{\sqrt{x} \cdot 4 + \left(1 + x\right)}\right)}^{3}}}\]

  8. Final simplification0.1

    \[\leadsto \sqrt[3]{{\left(\frac{x - 1}{\left(x + 1\right) + \sqrt{x} \cdot 4}\right)}^{3}} \cdot 6\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :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)))))