Average Error: 29.6 → 0.2
Time: 18.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}
double f(double x) {
        double r310076 = x;
        double r310077 = 1.0;
        double r310078 = r310076 + r310077;
        double r310079 = sqrt(r310078);
        double r310080 = sqrt(r310076);
        double r310081 = r310079 - r310080;
        return r310081;
}
↓
double f(double x) {
        double r310082 = 1.0;
        double r310083 = x;
        double r310084 = sqrt(r310083);
        double r310085 = r310083 + r310082;
        double r310086 = sqrt(r310085);
        double r310087 = r310084 + r310086;
        double r310088 = r310082 / r310087;
        double r310089 = sqrt(r310088);
        double r310090 = sqrt(r310082);
        double r310091 = r310089 * r310090;
        double r310092 = sqrt(r310087);
        double r310093 = r310091 / r310092;
        return r310093;
}

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. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.3

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

  9. Applied sqrt-div0.3

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

  10. Applied associate-*r/0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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