Average Error: 14.4 → 0.3
Time: 13.8s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\left(1 \cdot \left(b - a\right)\right) \cdot \frac{\frac{\pi}{2}}{b + a}\right) \cdot 1}{a \cdot b}}{b - a}\]
\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 \frac{\frac{\pi}{2}}{b + a}\right) \cdot 1}{a \cdot b}}{b - a}
double f(double a, double b) {
        double r45207 = atan2(1.0, 0.0);
        double r45208 = 2.0;
        double r45209 = r45207 / r45208;
        double r45210 = 1.0;
        double r45211 = b;
        double r45212 = r45211 * r45211;
        double r45213 = a;
        double r45214 = r45213 * r45213;
        double r45215 = r45212 - r45214;
        double r45216 = r45210 / r45215;
        double r45217 = r45209 * r45216;
        double r45218 = r45210 / r45213;
        double r45219 = r45210 / r45211;
        double r45220 = r45218 - r45219;
        double r45221 = r45217 * r45220;
        return r45221;
}
↓
double f(double a, double b) {
        double r45222 = 1.0;
        double r45223 = b;
        double r45224 = a;
        double r45225 = r45223 - r45224;
        double r45226 = r45222 * r45225;
        double r45227 = atan2(1.0, 0.0);
        double r45228 = 2.0;
        double r45229 = r45227 / r45228;
        double r45230 = r45223 + r45224;
        double r45231 = r45229 / r45230;
        double r45232 = r45226 * r45231;
        double r45233 = r45232 * r45222;
        double r45234 = r45224 * r45223;
        double r45235 = r45233 / r45234;
        double r45236 = r45235 / r45225;
        return r45236;
}

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 14.4

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

  3. Applied difference-of-squares9.4

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

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

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

  5. Applied times-frac8.9

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

  6. Applied associate-*r*8.9

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

  7. Simplified8.8

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

  9. Applied associate-*r/8.8

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

  10. Applied associate-*l/0.3

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

  12. Applied frac-sub0.3

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

  13. Applied associate-*r/0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019209 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))