Average Error: 15.0 → 0.3
Time: 37.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{a + b} \cdot \pi}{2} \cdot \frac{1}{b \cdot a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{1}{a + b} \cdot \pi}{2} \cdot \frac{1}{b \cdot a}
double f(double a, double b) {
        double r2334807 = atan2(1.0, 0.0);
        double r2334808 = 2.0;
        double r2334809 = r2334807 / r2334808;
        double r2334810 = 1.0;
        double r2334811 = b;
        double r2334812 = r2334811 * r2334811;
        double r2334813 = a;
        double r2334814 = r2334813 * r2334813;
        double r2334815 = r2334812 - r2334814;
        double r2334816 = r2334810 / r2334815;
        double r2334817 = r2334809 * r2334816;
        double r2334818 = r2334810 / r2334813;
        double r2334819 = r2334810 / r2334811;
        double r2334820 = r2334818 - r2334819;
        double r2334821 = r2334817 * r2334820;
        return r2334821;
}
↓
double f(double a, double b) {
        double r2334822 = 1.0;
        double r2334823 = a;
        double r2334824 = b;
        double r2334825 = r2334823 + r2334824;
        double r2334826 = r2334822 / r2334825;
        double r2334827 = atan2(1.0, 0.0);
        double r2334828 = r2334826 * r2334827;
        double r2334829 = 2.0;
        double r2334830 = r2334828 / r2334829;
        double r2334831 = r2334824 * r2334823;
        double r2334832 = r2334822 / r2334831;
        double r2334833 = r2334830 * r2334832;
        return r2334833;
}

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 15.0

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

  3. Applied difference-of-squares10.0

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

  4. Applied associate-/r*9.4

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

  6. Applied frac-times9.4

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

  7. Applied associate-*l/0.3

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

  9. Applied times-frac0.3

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

  10. Taylor expanded around 0 0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))