Average Error: 19.8 → 0.5
Time: 7.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}, \sqrt[3]{\sqrt{x + 1}}, \sqrt{x}\right)}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}, \sqrt[3]{\sqrt{x + 1}}, \sqrt{x}\right)}}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r156091 = 1.0;
        double r156092 = x;
        double r156093 = sqrt(r156092);
        double r156094 = r156091 / r156093;
        double r156095 = r156092 + r156091;
        double r156096 = sqrt(r156095);
        double r156097 = r156091 / r156096;
        double r156098 = r156094 - r156097;
        return r156098;
}
↓
double f(double x) {
        double r156099 = 1.0;
        double r156100 = x;
        double r156101 = r156100 + r156099;
        double r156102 = sqrt(r156101);
        double r156103 = cbrt(r156102);
        double r156104 = r156103 * r156103;
        double r156105 = sqrt(r156100);
        double r156106 = fma(r156104, r156103, r156105);
        double r156107 = r156099 / r156106;
        double r156108 = r156099 * r156107;
        double r156109 = r156105 * r156102;
        double r156110 = r156108 / r156109;
        return r156110;
}

Error

Bits error versus x

Target

Original19.8
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

    \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{x + 1} - \sqrt{x}\right)}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
  5. Using strategy rm

  6. Applied flip--19.6

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

  7. Simplified19.2

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

  8. Taylor expanded around 0 0.4

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

  10. Applied add-cube-cbrt0.5

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

  11. Applied fma-def0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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