Average Error: 29.8 → 0.2
Time: 4.3s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt{x + 1}}, \sqrt{\sqrt{x + 1}}, \sqrt{x}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt{x + 1}}, \sqrt{\sqrt{x + 1}}, \sqrt{x}\right)}
double f(double x) {
        double r490480 = x;
        double r490481 = 1.0;
        double r490482 = r490480 + r490481;
        double r490483 = sqrt(r490482);
        double r490484 = sqrt(r490480);
        double r490485 = r490483 - r490484;
        return r490485;
}
↓
double f(double x) {
        double r490486 = 1.0;
        double r490487 = 0.0;
        double r490488 = r490486 + r490487;
        double r490489 = x;
        double r490490 = r490489 + r490486;
        double r490491 = sqrt(r490490);
        double r490492 = sqrt(r490491);
        double r490493 = sqrt(r490489);
        double r490494 = fma(r490492, r490492, r490493);
        double r490495 = r490488 / r490494;
        return r490495;
}

Error

Bits error versus x

Target

Original29.8
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.8

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

  3. Applied flip--29.6

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020064 +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)))