Average Error: 30.1 → 0.2
Time: 16.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r27942498 = x;
        double r27942499 = 1.0;
        double r27942500 = r27942498 + r27942499;
        double r27942501 = sqrt(r27942500);
        double r27942502 = sqrt(r27942498);
        double r27942503 = r27942501 - r27942502;
        return r27942503;
}
↓
double f(double x) {
        double r27942504 = 1.0;
        double r27942505 = x;
        double r27942506 = r27942505 + r27942504;
        double r27942507 = sqrt(r27942506);
        double r27942508 = sqrt(r27942505);
        double r27942509 = r27942507 + r27942508;
        double r27942510 = r27942504 / r27942509;
        return r27942510;
}

Error

Bits error versus x

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Results

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Target

Original30.1
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.1

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

  3. Applied flip--29.8

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

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

    \[\leadsto \frac{1 + 0}{\color{blue}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]

  7. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{1 + 0}{1}}{\sqrt{x + 1} + \sqrt{x}}}\]

  8. Simplified0.2

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

  9. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019174 +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)))