Average Error: 30.0 → 0.3
Time: 10.4s
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 r94721 = x;
        double r94722 = 1.0;
        double r94723 = r94721 + r94722;
        double r94724 = sqrt(r94723);
        double r94725 = sqrt(r94721);
        double r94726 = r94724 - r94725;
        return r94726;
}
↓
double f(double x) {
        double r94727 = 1.0;
        double r94728 = x;
        double r94729 = cbrt(r94728);
        double r94730 = r94729 * r94729;
        double r94731 = sqrt(r94730);
        double r94732 = sqrt(r94728);
        double r94733 = cbrt(r94732);
        double r94734 = r94733 * r94733;
        double r94735 = sqrt(r94734);
        double r94736 = r94728 + r94727;
        double r94737 = sqrt(r94736);
        double r94738 = fma(r94731, r94735, r94737);
        double r94739 = r94727 / r94738;
        return r94739;
}

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 "2sqrt (example 3.1)"
  :precision binary64

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

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