Average Error: 29.6 → 0.2
Time: 17.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]
\sqrt{x + 1} - \sqrt{x}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)
double f(double x) {
        double r20155098 = x;
        double r20155099 = 1.0;
        double r20155100 = r20155098 + r20155099;
        double r20155101 = sqrt(r20155100);
        double r20155102 = sqrt(r20155098);
        double r20155103 = r20155101 - r20155102;
        return r20155103;
}
↓
double f(double x) {
        double r20155104 = 1.0;
        double r20155105 = x;
        double r20155106 = r20155105 + r20155104;
        double r20155107 = sqrt(r20155106);
        double r20155108 = sqrt(r20155105);
        double r20155109 = r20155107 + r20155108;
        double r20155110 = r20155104 / r20155109;
        double r20155111 = log1p(r20155110);
        double r20155112 = expm1(r20155111);
        return r20155112;
}

Error

Bits error versus x

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Results

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Target

Original29.6
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.6

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

  3. Applied flip--29.4

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

  4. Simplified29.0

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

  5. Simplified29.0

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

  6. Taylor expanded around 0 0.2

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

  8. Applied expm1-log1p-u0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))