Average Error: 14.7 → 0.3
Time: 5.3s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1 \cdot \frac{\frac{\pi}{a}}{b}}{2 \cdot \left(b + a\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{1 \cdot \frac{\frac{\pi}{a}}{b}}{2 \cdot \left(b + a\right)}
double f(double a, double b) {
        double r41170 = atan2(1.0, 0.0);
        double r41171 = 2.0;
        double r41172 = r41170 / r41171;
        double r41173 = 1.0;
        double r41174 = b;
        double r41175 = r41174 * r41174;
        double r41176 = a;
        double r41177 = r41176 * r41176;
        double r41178 = r41175 - r41177;
        double r41179 = r41173 / r41178;
        double r41180 = r41172 * r41179;
        double r41181 = r41173 / r41176;
        double r41182 = r41173 / r41174;
        double r41183 = r41181 - r41182;
        double r41184 = r41180 * r41183;
        return r41184;
}
↓
double f(double a, double b) {
        double r41185 = 1.0;
        double r41186 = atan2(1.0, 0.0);
        double r41187 = a;
        double r41188 = r41186 / r41187;
        double r41189 = b;
        double r41190 = r41188 / r41189;
        double r41191 = r41185 * r41190;
        double r41192 = 2.0;
        double r41193 = r41189 + r41187;
        double r41194 = r41192 * r41193;
        double r41195 = r41191 / r41194;
        return r41195;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.7

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  2. Using strategy rm

  3. Applied difference-of-squares9.8

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  4. Applied add-cube-cbrt9.8

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(b + a\right) \cdot \left(b - a\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  5. Applied times-frac9.3

    \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a} \cdot \frac{\sqrt[3]{1}}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  6. Applied associate-*r*9.3

    \[\leadsto \color{blue}{\left(\left(\frac{\pi}{2} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a}\right) \cdot \frac{\sqrt[3]{1}}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  7. Using strategy rm

  8. Applied frac-times9.2

    \[\leadsto \left(\color{blue}{\frac{\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)}{2 \cdot \left(b + a\right)}} \cdot \frac{\sqrt[3]{1}}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  9. Applied associate-*l/9.2

    \[\leadsto \color{blue}{\frac{\left(\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \frac{\sqrt[3]{1}}{b - a}}{2 \cdot \left(b + a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  10. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{\left(\left(\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \frac{\sqrt[3]{1}}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(b + a\right)}}\]

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{\pi}{a \cdot b}}}{2 \cdot \left(b + a\right)}\]
  12. Using strategy rm

  13. Applied associate-/r*0.3

    \[\leadsto \frac{1 \cdot \color{blue}{\frac{\frac{\pi}{a}}{b}}}{2 \cdot \left(b + a\right)}\]

  14. Final simplification0.3

    \[\leadsto \frac{1 \cdot \frac{\frac{\pi}{a}}{b}}{2 \cdot \left(b + a\right)}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))