Average Error: 30.6 → 0.3
Time: 7.7s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}
double f(double x) {
        double r2114 = x;
        double r2115 = 1.0;
        double r2116 = r2114 + r2115;
        double r2117 = sqrt(r2116);
        double r2118 = sqrt(r2114);
        double r2119 = r2117 - r2118;
        return r2119;
}
↓
double f(double x) {
        double r2120 = 1.0;
        double r2121 = 0.0;
        double r2122 = r2120 + r2121;
        double r2123 = x;
        double r2124 = r2123 + r2120;
        double r2125 = cbrt(r2124);
        double r2126 = r2125 * r2125;
        double r2127 = sqrt(r2126);
        double r2128 = sqrt(r2125);
        double r2129 = sqrt(r2123);
        double r2130 = fma(r2127, r2128, r2129);
        double r2131 = r2122 / r2130;
        return r2131;
}

Error

Bits error versus x

Target

Original30.6
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.6

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

  3. Applied flip--30.4

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

  4. Simplified0.2

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

  6. Applied add-cube-cbrt0.3

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

  7. Applied sqrt-prod0.3

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

  8. Applied fma-def0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020025 +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)))