Average Error: 19.7 → 0.4
Time: 19.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\right)\right)}{\sqrt{\sqrt{x + 1}} \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\right)\right)}{\sqrt{\sqrt{x + 1}} \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right)}
double f(double x) {
        double r2294069 = 1.0;
        double r2294070 = x;
        double r2294071 = sqrt(r2294070);
        double r2294072 = r2294069 / r2294071;
        double r2294073 = r2294070 + r2294069;
        double r2294074 = sqrt(r2294073);
        double r2294075 = r2294069 / r2294074;
        double r2294076 = r2294072 - r2294075;
        return r2294076;
}
↓
double f(double x) {
        double r2294077 = 1.0;
        double r2294078 = x;
        double r2294079 = r2294078 + r2294077;
        double r2294080 = sqrt(r2294079);
        double r2294081 = sqrt(r2294078);
        double r2294082 = r2294080 + r2294081;
        double r2294083 = r2294077 / r2294082;
        double r2294084 = log1p(r2294083);
        double r2294085 = expm1(r2294084);
        double r2294086 = sqrt(r2294080);
        double r2294087 = r2294081 * r2294086;
        double r2294088 = r2294086 * r2294087;
        double r2294089 = r2294085 / r2294088;
        return r2294089;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied frac-sub19.7

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

  4. Simplified19.7

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

  6. Applied flip--19.5

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

  7. Simplified19.1

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

  9. Applied expm1-log1p-u19.1

    \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{\left(1 + x\right) - x}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\right)\right)}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

  10. Simplified0.4

    \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(\mathsf{log1p}\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\right)}\right)}{\sqrt{x} \cdot \sqrt{x + 1}}\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\right)\right)}{\sqrt{x} \cdot \sqrt{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}}\]

  13. Applied sqrt-prod0.4

    \[\leadsto \frac{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\right)\right)}{\sqrt{x} \cdot \color{blue}{\left(\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}\right)}}\]

  14. Applied associate-*r*0.4

    \[\leadsto \frac{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\right)\right)}{\color{blue}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}}\]

  15. Final simplification0.4

    \[\leadsto \frac{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\right)\right)}{\sqrt{\sqrt{x + 1}} \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right)}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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