Average Error: 0.2 → 0.1
Time: 15.2s
Precision: 64
\[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]
↓
\[6.0 \cdot \left(\frac{\sqrt{x} + \sqrt{1.0}}{\sqrt{4.0 \cdot \sqrt{x} + \left(x + 1.0\right)}} \cdot \frac{\sqrt{x} - \sqrt{1.0}}{\sqrt{4.0 \cdot \sqrt{x} + \left(x + 1.0\right)}}\right)\]
\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}
↓
6.0 \cdot \left(\frac{\sqrt{x} + \sqrt{1.0}}{\sqrt{4.0 \cdot \sqrt{x} + \left(x + 1.0\right)}} \cdot \frac{\sqrt{x} - \sqrt{1.0}}{\sqrt{4.0 \cdot \sqrt{x} + \left(x + 1.0\right)}}\right)
double f(double x) {
        double r42474192 = 6.0;
        double r42474193 = x;
        double r42474194 = 1.0;
        double r42474195 = r42474193 - r42474194;
        double r42474196 = r42474192 * r42474195;
        double r42474197 = r42474193 + r42474194;
        double r42474198 = 4.0;
        double r42474199 = sqrt(r42474193);
        double r42474200 = r42474198 * r42474199;
        double r42474201 = r42474197 + r42474200;
        double r42474202 = r42474196 / r42474201;
        return r42474202;
}
↓
double f(double x) {
        double r42474203 = 6.0;
        double r42474204 = x;
        double r42474205 = sqrt(r42474204);
        double r42474206 = 1.0;
        double r42474207 = sqrt(r42474206);
        double r42474208 = r42474205 + r42474207;
        double r42474209 = 4.0;
        double r42474210 = r42474209 * r42474205;
        double r42474211 = r42474204 + r42474206;
        double r42474212 = r42474210 + r42474211;
        double r42474213 = sqrt(r42474212);
        double r42474214 = r42474208 / r42474213;
        double r42474215 = r42474205 - r42474207;
        double r42474216 = r42474215 / r42474213;
        double r42474217 = r42474214 * r42474216;
        double r42474218 = r42474203 * r42474217;
        return r42474218;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.2
Target0.1
Herbie0.1
\[\frac{6.0}{\frac{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}{x - 1.0}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.2

    \[\leadsto \frac{6.0 \cdot \left(x - 1.0\right)}{\color{blue}{1 \cdot \left(\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}\right)}}\]

  4. Applied times-frac0.0

    \[\leadsto \color{blue}{\frac{6.0}{1} \cdot \frac{x - 1.0}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}}\]

  5. Simplified0.0

    \[\leadsto \color{blue}{6.0} \cdot \frac{x - 1.0}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.4

    \[\leadsto 6.0 \cdot \frac{x - 1.0}{\color{blue}{\sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}} \cdot \sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}}}\]

  8. Applied add-sqr-sqrt0.4

    \[\leadsto 6.0 \cdot \frac{x - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}{\sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}} \cdot \sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}}\]

  9. Applied add-sqr-sqrt0.1

    \[\leadsto 6.0 \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{1.0} \cdot \sqrt{1.0}}{\sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}} \cdot \sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}}\]

  10. Applied difference-of-squares0.1

    \[\leadsto 6.0 \cdot \frac{\color{blue}{\left(\sqrt{x} + \sqrt{1.0}\right) \cdot \left(\sqrt{x} - \sqrt{1.0}\right)}}{\sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}} \cdot \sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}}\]

  11. Applied times-frac0.1

    \[\leadsto 6.0 \cdot \color{blue}{\left(\frac{\sqrt{x} + \sqrt{1.0}}{\sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}} \cdot \frac{\sqrt{x} - \sqrt{1.0}}{\sqrt{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}}\right)}\]

  12. Final simplification0.1

    \[\leadsto 6.0 \cdot \left(\frac{\sqrt{x} + \sqrt{1.0}}{\sqrt{4.0 \cdot \sqrt{x} + \left(x + 1.0\right)}} \cdot \frac{\sqrt{x} - \sqrt{1.0}}{\sqrt{4.0 \cdot \sqrt{x} + \left(x + 1.0\right)}}\right)\]

Reproduce

herbie shell --seed 2019164 
(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)))))