Average Error: 30.2 → 0.2
Time: 8.1s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt{\left|\sqrt[3]{x}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{x}}}, \sqrt{\sqrt{x}}, \sqrt{x + 1}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt{\left|\sqrt[3]{x}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{x}}}, \sqrt{\sqrt{x}}, \sqrt{x + 1}\right)}
double f(double x) {
        double r479284 = x;
        double r479285 = 1.0;
        double r479286 = r479284 + r479285;
        double r479287 = sqrt(r479286);
        double r479288 = sqrt(r479284);
        double r479289 = r479287 - r479288;
        return r479289;
}
↓
double f(double x) {
        double r479290 = 1.0;
        double r479291 = x;
        double r479292 = cbrt(r479291);
        double r479293 = fabs(r479292);
        double r479294 = sqrt(r479293);
        double r479295 = sqrt(r479292);
        double r479296 = sqrt(r479295);
        double r479297 = r479294 * r479296;
        double r479298 = sqrt(r479291);
        double r479299 = sqrt(r479298);
        double r479300 = r479291 + r479290;
        double r479301 = sqrt(r479300);
        double r479302 = fma(r479297, r479299, r479301);
        double r479303 = r479290 / r479302;
        return r479303;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 30.2

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

  3. Applied flip--29.9

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

  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-sqr-sqrt0.2

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

  8. Applied sqrt-prod0.3

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

  9. Applied fma-def0.2

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

  11. Applied add-cube-cbrt0.2

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

  12. Applied sqrt-prod0.2

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

  13. Applied sqrt-prod0.2

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

  14. Simplified0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +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)))