Average Error: 30.2 → 0.2
Time: 4.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt{\sqrt[3]{x + 1}}}, \sqrt{\sqrt{x + 1}}, \sqrt{x}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt{\sqrt[3]{x + 1}}}, \sqrt{\sqrt{x + 1}}, \sqrt{x}\right)}
double f(double x) {
        double r477789 = x;
        double r477790 = 1.0;
        double r477791 = r477789 + r477790;
        double r477792 = sqrt(r477791);
        double r477793 = sqrt(r477789);
        double r477794 = r477792 - r477793;
        return r477794;
}
↓
double f(double x) {
        double r477795 = 1.0;
        double r477796 = 0.0;
        double r477797 = r477795 + r477796;
        double r477798 = x;
        double r477799 = r477798 + r477795;
        double r477800 = cbrt(r477799);
        double r477801 = r477800 * r477800;
        double r477802 = sqrt(r477801);
        double r477803 = sqrt(r477802);
        double r477804 = sqrt(r477800);
        double r477805 = sqrt(r477804);
        double r477806 = r477803 * r477805;
        double r477807 = sqrt(r477799);
        double r477808 = sqrt(r477807);
        double r477809 = sqrt(r477798);
        double r477810 = fma(r477806, r477808, r477809);
        double r477811 = r477797 / r477810;
        return r477811;
}

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

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied sqrt-prod0.3

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

  8. Applied fma-def0.2

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

  10. Applied add-cube-cbrt0.2

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

  11. Applied sqrt-prod0.2

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

  12. Applied sqrt-prod0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +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)))