Average Error: 30.1 → 0.3
Time: 19.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\sqrt[3]{x}}, \sqrt{x + 1}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\sqrt[3]{x}}, \sqrt{x + 1}\right)}
double f(double x) {
        double r99181 = x;
        double r99182 = 1.0;
        double r99183 = r99181 + r99182;
        double r99184 = sqrt(r99183);
        double r99185 = sqrt(r99181);
        double r99186 = r99184 - r99185;
        return r99186;
}
↓
double f(double x) {
        double r99187 = 1.0;
        double r99188 = x;
        double r99189 = cbrt(r99188);
        double r99190 = r99189 * r99189;
        double r99191 = sqrt(r99190);
        double r99192 = sqrt(r99189);
        double r99193 = r99188 + r99187;
        double r99194 = sqrt(r99193);
        double r99195 = fma(r99191, r99192, r99194);
        double r99196 = r99187 / r99195;
        return r99196;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 30.1

    \[\sqrt{x + 1} - \sqrt{x}\]
  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-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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))