Average Error: 30.2 → 0.2
Time: 8.2s
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 r141476 = x;
        double r141477 = 1.0;
        double r141478 = r141476 + r141477;
        double r141479 = sqrt(r141478);
        double r141480 = sqrt(r141476);
        double r141481 = r141479 - r141480;
        return r141481;
}
↓
double f(double x) {
        double r141482 = 1.0;
        double r141483 = x;
        double r141484 = cbrt(r141483);
        double r141485 = fabs(r141484);
        double r141486 = sqrt(r141485);
        double r141487 = sqrt(r141484);
        double r141488 = sqrt(r141487);
        double r141489 = r141486 * r141488;
        double r141490 = sqrt(r141483);
        double r141491 = sqrt(r141490);
        double r141492 = r141483 + r141482;
        double r141493 = sqrt(r141492);
        double r141494 = fma(r141489, r141491, r141493);
        double r141495 = r141482 / r141494;
        return r141495;
}

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

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied sqrt-prod0.3

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

  9. Applied fma-def0.2

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

  11. Applied add-cube-cbrt0.2

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

  12. Applied sqrt-prod0.2

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

  13. Applied sqrt-prod0.2

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

  14. Simplified0.2

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

  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 "2sqrt (example 3.1)"
  :precision binary64

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

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