Average Error: 30.0 → 0.3
Time: 10.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}}, \sqrt{x + 1}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}}, \sqrt{x + 1}\right)}
double f(double x) {
        double r440305 = x;
        double r440306 = 1.0;
        double r440307 = r440305 + r440306;
        double r440308 = sqrt(r440307);
        double r440309 = sqrt(r440305);
        double r440310 = r440308 - r440309;
        return r440310;
}
↓
double f(double x) {
        double r440311 = 1.0;
        double r440312 = x;
        double r440313 = cbrt(r440312);
        double r440314 = r440313 * r440313;
        double r440315 = sqrt(r440314);
        double r440316 = sqrt(r440312);
        double r440317 = cbrt(r440316);
        double r440318 = r440317 * r440317;
        double r440319 = sqrt(r440318);
        double r440320 = r440312 + r440311;
        double r440321 = sqrt(r440320);
        double r440322 = fma(r440315, r440319, r440321);
        double r440323 = r440311 / r440322;
        return r440323;
}

Error

Bits error versus x

Target

Original30.0
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.0

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

  3. Applied flip--29.7

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-cube-cbrt0.3

    \[\leadsto \frac{1 + 0}{\sqrt{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} + \sqrt{x + 1}}\]

  8. Applied sqrt-prod0.3

    \[\leadsto \frac{1 + 0}{\color{blue}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}} + \sqrt{x + 1}}\]

  9. Applied fma-def0.3

    \[\leadsto \frac{1 + 0}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\sqrt[3]{x}}, \sqrt{x + 1}\right)}}\]
  10. Using strategy rm

  11. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}, \sqrt{x + 1}\right)}\]

  12. Applied cbrt-prod0.3

    \[\leadsto \frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\color{blue}{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}}}, \sqrt{x + 1}\right)}\]

  13. Final simplification0.3

    \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}}, \sqrt{x + 1}\right)}\]

Reproduce

herbie shell --seed 2020045 +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)))