Average Error: 30.3 → 0.2
Time: 16.4s
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 r6040491 = x;
        double r6040492 = 1.0;
        double r6040493 = r6040491 + r6040492;
        double r6040494 = sqrt(r6040493);
        double r6040495 = sqrt(r6040491);
        double r6040496 = r6040494 - r6040495;
        return r6040496;
}
↓
double f(double x) {
        double r6040497 = 1.0;
        double r6040498 = x;
        double r6040499 = r6040498 + r6040497;
        double r6040500 = sqrt(r6040499);
        double r6040501 = sqrt(r6040498);
        double r6040502 = r6040500 + r6040501;
        double r6040503 = r6040497 / r6040502;
        return r6040503;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.3

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

  3. Applied flip--30.0

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

  4. Simplified29.6

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

  5. Simplified29.6

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

  6. Taylor expanded around 0 0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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