Average Error: 14.5 → 0.3
Time: 28.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi \cdot \frac{1}{b + a}}{2} \cdot \frac{1}{a \cdot b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi \cdot \frac{1}{b + a}}{2} \cdot \frac{1}{a \cdot b}
double f(double a, double b) {
        double r4795205 = atan2(1.0, 0.0);
        double r4795206 = 2.0;
        double r4795207 = r4795205 / r4795206;
        double r4795208 = 1.0;
        double r4795209 = b;
        double r4795210 = r4795209 * r4795209;
        double r4795211 = a;
        double r4795212 = r4795211 * r4795211;
        double r4795213 = r4795210 - r4795212;
        double r4795214 = r4795208 / r4795213;
        double r4795215 = r4795207 * r4795214;
        double r4795216 = r4795208 / r4795211;
        double r4795217 = r4795208 / r4795209;
        double r4795218 = r4795216 - r4795217;
        double r4795219 = r4795215 * r4795218;
        return r4795219;
}
↓
double f(double a, double b) {
        double r4795220 = atan2(1.0, 0.0);
        double r4795221 = 1.0;
        double r4795222 = b;
        double r4795223 = a;
        double r4795224 = r4795222 + r4795223;
        double r4795225 = r4795221 / r4795224;
        double r4795226 = r4795220 * r4795225;
        double r4795227 = 2.0;
        double r4795228 = r4795226 / r4795227;
        double r4795229 = r4795223 * r4795222;
        double r4795230 = r4795221 / r4795229;
        double r4795231 = r4795228 * r4795230;
        return r4795231;
}

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-squares9.8

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

  4. Applied 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-times9.3

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

  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

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

Reproduce

herbie shell --seed 2019173 
(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))))