Average Error: 29.9 → 0.2
Time: 17.7s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]
\sqrt{x + 1} - \sqrt{x}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)
double f(double x) {
        double r1636046 = x;
        double r1636047 = 1.0;
        double r1636048 = r1636046 + r1636047;
        double r1636049 = sqrt(r1636048);
        double r1636050 = sqrt(r1636046);
        double r1636051 = r1636049 - r1636050;
        return r1636051;
}
↓
double f(double x) {
        double r1636052 = 1.0;
        double r1636053 = x;
        double r1636054 = r1636053 + r1636052;
        double r1636055 = sqrt(r1636054);
        double r1636056 = sqrt(r1636053);
        double r1636057 = r1636055 + r1636056;
        double r1636058 = r1636052 / r1636057;
        double r1636059 = expm1(r1636058);
        double r1636060 = log1p(r1636059);
        return r1636060;
}

Error

Bits error versus x

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Results

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Target

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

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. Simplified29.3

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

  5. Taylor expanded around 0 0.2

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

  7. Applied add-sqr-sqrt0.3

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

  8. Applied associate-/r*0.3

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

  10. Applied log1p-expm1-u0.3

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\right)\right)}\]

  11. Simplified0.2

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}\right)\]

  12. Final simplification0.2

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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