Average Error: 14.5 → 0.5
Time: 4.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\left(1 \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{b + a}}{2 \cdot \left(a \cdot b\right)}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\left(\left(1 \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{b + a}}{2 \cdot \left(a \cdot b\right)}}{b - a}
double f(double a, double b) {
        double r54073 = atan2(1.0, 0.0);
        double r54074 = 2.0;
        double r54075 = r54073 / r54074;
        double r54076 = 1.0;
        double r54077 = b;
        double r54078 = r54077 * r54077;
        double r54079 = a;
        double r54080 = r54079 * r54079;
        double r54081 = r54078 - r54080;
        double r54082 = r54076 / r54081;
        double r54083 = r54075 * r54082;
        double r54084 = r54076 / r54079;
        double r54085 = r54076 / r54077;
        double r54086 = r54084 - r54085;
        double r54087 = r54083 * r54086;
        return r54087;
}
↓
double f(double a, double b) {
        double r54088 = 1.0;
        double r54089 = b;
        double r54090 = a;
        double r54091 = r54089 - r54090;
        double r54092 = r54088 * r54091;
        double r54093 = atan2(1.0, 0.0);
        double r54094 = r54092 * r54093;
        double r54095 = r54089 + r54090;
        double r54096 = r54088 / r54095;
        double r54097 = r54094 * r54096;
        double r54098 = 2.0;
        double r54099 = r54090 * r54089;
        double r54100 = r54098 * r54099;
        double r54101 = r54097 / r54100;
        double r54102 = r54101 / r54091;
        return r54102;
}

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 14.5

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

  3. Applied difference-of-squares9.7

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

  4. Applied associate-/r*9.2

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

  6. Applied associate-*r/9.2

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

  7. Applied associate-*l/0.3

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

  9. Applied frac-sub0.3

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

  10. Applied associate-*l/0.3

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

  11. Applied frac-times0.3

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020057 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))