Average Error: 19.3 → 0.5
Time: 1.4m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{\sqrt{1 + \left(x - x\right)}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}}{\frac{\sqrt{x} \cdot \sqrt{x + 1}}{\frac{\sqrt{1 + \left(x - x\right)}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{\sqrt{1 + \left(x - x\right)}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}}{\frac{\sqrt{x} \cdot \sqrt{x + 1}}{\frac{\sqrt{1 + \left(x - x\right)}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}}}
double f(double x) {
        double r5037947 = 1.0;
        double r5037948 = x;
        double r5037949 = sqrt(r5037948);
        double r5037950 = r5037947 / r5037949;
        double r5037951 = r5037948 + r5037947;
        double r5037952 = sqrt(r5037951);
        double r5037953 = r5037947 / r5037952;
        double r5037954 = r5037950 - r5037953;
        return r5037954;
}
↓
double f(double x) {
        double r5037955 = 1.0;
        double r5037956 = x;
        double r5037957 = r5037956 - r5037956;
        double r5037958 = r5037955 + r5037957;
        double r5037959 = sqrt(r5037958);
        double r5037960 = sqrt(r5037956);
        double r5037961 = r5037956 + r5037955;
        double r5037962 = sqrt(r5037961);
        double r5037963 = r5037960 + r5037962;
        double r5037964 = sqrt(r5037963);
        double r5037965 = r5037959 / r5037964;
        double r5037966 = r5037960 * r5037962;
        double r5037967 = r5037966 / r5037965;
        double r5037968 = r5037965 / r5037967;
        return r5037968;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.3
Target0.6
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.3

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

  3. Applied frac-sub19.3

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

  4. Simplified19.3

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

  6. Applied flip--19.1

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

  7. Simplified0.4

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

  9. Applied add-sqr-sqrt0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied times-frac0.5

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

  12. Applied associate-/l*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019130 +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)))))