Average Error: 14.3 → 0.3
Time: 18.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{2} \cdot \frac{\frac{1}{a \cdot b} \cdot 1}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi}{2} \cdot \frac{\frac{1}{a \cdot b} \cdot 1}{b + a}
double f(double a, double b) {
        double r45661 = atan2(1.0, 0.0);
        double r45662 = 2.0;
        double r45663 = r45661 / r45662;
        double r45664 = 1.0;
        double r45665 = b;
        double r45666 = r45665 * r45665;
        double r45667 = a;
        double r45668 = r45667 * r45667;
        double r45669 = r45666 - r45668;
        double r45670 = r45664 / r45669;
        double r45671 = r45663 * r45670;
        double r45672 = r45664 / r45667;
        double r45673 = r45664 / r45665;
        double r45674 = r45672 - r45673;
        double r45675 = r45671 * r45674;
        return r45675;
}
↓
double f(double a, double b) {
        double r45676 = atan2(1.0, 0.0);
        double r45677 = 2.0;
        double r45678 = r45676 / r45677;
        double r45679 = 1.0;
        double r45680 = a;
        double r45681 = b;
        double r45682 = r45680 * r45681;
        double r45683 = r45679 / r45682;
        double r45684 = r45683 * r45679;
        double r45685 = r45681 + r45680;
        double r45686 = r45684 / r45685;
        double r45687 = r45678 * r45686;
        return r45687;
}

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 14.3

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

  3. Applied associate-*l*14.3

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

  4. Simplified0.3

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

  5. Taylor expanded around 0 0.3

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

  7. Applied associate-*r/0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))