Average Error: 29.5 → 0.2
Time: 16.8s
Precision: 64
\[\sqrt{x + 1.0} - \sqrt{x}\]
↓
\[\frac{1.0}{\sqrt{x} + \mathsf{hypot}\left(\sqrt{x}, \sqrt{1.0}\right)}\]
\sqrt{x + 1.0} - \sqrt{x}
↓
\frac{1.0}{\sqrt{x} + \mathsf{hypot}\left(\sqrt{x}, \sqrt{1.0}\right)}
double f(double x) {
        double r24466884 = x;
        double r24466885 = 1.0;
        double r24466886 = r24466884 + r24466885;
        double r24466887 = sqrt(r24466886);
        double r24466888 = sqrt(r24466884);
        double r24466889 = r24466887 - r24466888;
        return r24466889;
}
↓
double f(double x) {
        double r24466890 = 1.0;
        double r24466891 = x;
        double r24466892 = sqrt(r24466891);
        double r24466893 = sqrt(r24466890);
        double r24466894 = hypot(r24466892, r24466893);
        double r24466895 = r24466892 + r24466894;
        double r24466896 = r24466890 / r24466895;
        return r24466896;
}

Error

Bits error versus x

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Results

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Target

Original29.5
Target0.2
Herbie0.2
\[\frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}\]

Derivation

  1. Initial program 29.5

    \[\sqrt{x + 1.0} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.3

    \[\leadsto \color{blue}{\frac{\sqrt{x + 1.0} \cdot \sqrt{x + 1.0} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1.0} + \sqrt{x}}}\]

  4. Simplified28.9

    \[\leadsto \frac{\color{blue}{\left(x + 1.0\right) - x}}{\sqrt{x + 1.0} + \sqrt{x}}\]

  5. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{1.0}}{\sqrt{x + 1.0} + \sqrt{x}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{1.0}{\sqrt{x + \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}} + \sqrt{x}}\]

  8. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{1.0}{\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{1.0} \cdot \sqrt{1.0}} + \sqrt{x}}\]

  9. Applied hypot-def0.2

    \[\leadsto \frac{1.0}{\color{blue}{\mathsf{hypot}\left(\sqrt{x}, \sqrt{1.0}\right)} + \sqrt{x}}\]

  10. Final simplification0.2

    \[\leadsto \frac{1.0}{\sqrt{x} + \mathsf{hypot}\left(\sqrt{x}, \sqrt{1.0}\right)}\]

Reproduce

herbie shell --seed 2019162 +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)))