Average Error: 14.2 → 0.2
Time: 13.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{b + a} \cdot 1}{a \cdot b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\frac{\pi}{2}}{b + a} \cdot 1}{a \cdot b}
double f(double a, double b) {
        double r68084 = atan2(1.0, 0.0);
        double r68085 = 2.0;
        double r68086 = r68084 / r68085;
        double r68087 = 1.0;
        double r68088 = b;
        double r68089 = r68088 * r68088;
        double r68090 = a;
        double r68091 = r68090 * r68090;
        double r68092 = r68089 - r68091;
        double r68093 = r68087 / r68092;
        double r68094 = r68086 * r68093;
        double r68095 = r68087 / r68090;
        double r68096 = r68087 / r68088;
        double r68097 = r68095 - r68096;
        double r68098 = r68094 * r68097;
        return r68098;
}
↓
double f(double a, double b) {
        double r68099 = atan2(1.0, 0.0);
        double r68100 = 2.0;
        double r68101 = r68099 / r68100;
        double r68102 = b;
        double r68103 = a;
        double r68104 = r68102 + r68103;
        double r68105 = r68101 / r68104;
        double r68106 = 1.0;
        double r68107 = r68105 * r68106;
        double r68108 = r68103 * r68102;
        double r68109 = r68107 / r68108;
        return r68109;
}

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 14.2

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

  3. Applied difference-of-squares9.5

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

  4. Applied *-un-lft-identity9.5

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.0

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

  7. Simplified9.0

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

  9. Applied associate-*l*0.3

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

  10. Taylor expanded around 0 0.3

    \[\leadsto \frac{\frac{\pi}{2}}{b + a} \cdot \color{blue}{\frac{1}{a \cdot b}}\]
  11. Using strategy rm

  12. Applied associate-*r/0.2

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

  13. Final simplification0.2

    \[\leadsto \frac{\frac{\frac{\pi}{2}}{b + a} \cdot 1}{a \cdot b}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))