Average Error: 29.9 → 0.3
Time: 33.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\left(\sqrt{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt{\sqrt[3]{x + 1}}}\right) \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\left(\sqrt{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt{\sqrt[3]{x + 1}}}\right) \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}
double f(double x) {
        double r7750254 = x;
        double r7750255 = 1.0;
        double r7750256 = r7750254 + r7750255;
        double r7750257 = sqrt(r7750256);
        double r7750258 = sqrt(r7750254);
        double r7750259 = r7750257 - r7750258;
        return r7750259;
}
↓
double f(double x) {
        double r7750260 = 1.0;
        double r7750261 = x;
        double r7750262 = r7750261 + r7750260;
        double r7750263 = cbrt(r7750262);
        double r7750264 = r7750263 * r7750263;
        double r7750265 = sqrt(r7750264);
        double r7750266 = sqrt(r7750265);
        double r7750267 = sqrt(r7750263);
        double r7750268 = sqrt(r7750267);
        double r7750269 = r7750266 * r7750268;
        double r7750270 = sqrt(r7750262);
        double r7750271 = sqrt(r7750270);
        double r7750272 = r7750269 * r7750271;
        double r7750273 = sqrt(r7750261);
        double r7750274 = r7750272 + r7750273;
        double r7750275 = r7750260 / r7750274;
        return r7750275;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.7

    \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]

  4. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}} + \sqrt{x}}\]

  7. Applied sqrt-prod0.3

    \[\leadsto \frac{1}{\color{blue}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}} + \sqrt{x}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt0.3

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

  10. Applied sqrt-prod0.3

    \[\leadsto \frac{1}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}}} + \sqrt{x}}\]

  11. Applied sqrt-prod0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))