Average Error: 0.2 → 0.1
Time: 14.0s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{1}{\frac{\sqrt{x} \cdot 4 + \left(x + 1\right)}{x - 1}} \cdot 6\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{1}{\frac{\sqrt{x} \cdot 4 + \left(x + 1\right)}{x - 1}} \cdot 6
double f(double x) {
        double r704300 = 6.0;
        double r704301 = x;
        double r704302 = 1.0;
        double r704303 = r704301 - r704302;
        double r704304 = r704300 * r704303;
        double r704305 = r704301 + r704302;
        double r704306 = 4.0;
        double r704307 = sqrt(r704301);
        double r704308 = r704306 * r704307;
        double r704309 = r704305 + r704308;
        double r704310 = r704304 / r704309;
        return r704310;
}
↓
double f(double x) {
        double r704311 = 1.0;
        double r704312 = x;
        double r704313 = sqrt(r704312);
        double r704314 = 4.0;
        double r704315 = r704313 * r704314;
        double r704316 = 1.0;
        double r704317 = r704312 + r704316;
        double r704318 = r704315 + r704317;
        double r704319 = r704312 - r704316;
        double r704320 = r704318 / r704319;
        double r704321 = r704311 / r704320;
        double r704322 = 6.0;
        double r704323 = r704321 * r704322;
        return r704323;
}

Error

Bits error versus x

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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 clear-num0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019195 
(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)))))