Average Error: 13.9 → 0.3
Time: 5.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{\frac{b + a}{\frac{\pi}{2}}} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{1}{\frac{b + a}{\frac{\pi}{2}}} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r46096 = atan2(1.0, 0.0);
        double r46097 = 2.0;
        double r46098 = r46096 / r46097;
        double r46099 = 1.0;
        double r46100 = b;
        double r46101 = r46100 * r46100;
        double r46102 = a;
        double r46103 = r46102 * r46102;
        double r46104 = r46101 - r46103;
        double r46105 = r46099 / r46104;
        double r46106 = r46098 * r46105;
        double r46107 = r46099 / r46102;
        double r46108 = r46099 / r46100;
        double r46109 = r46107 - r46108;
        double r46110 = r46106 * r46109;
        return r46110;
}
↓
double f(double a, double b) {
        double r46111 = 1.0;
        double r46112 = b;
        double r46113 = a;
        double r46114 = r46112 + r46113;
        double r46115 = atan2(1.0, 0.0);
        double r46116 = 2.0;
        double r46117 = r46115 / r46116;
        double r46118 = r46114 / r46117;
        double r46119 = r46111 / r46118;
        double r46120 = 1.0;
        double r46121 = r46119 * r46120;
        double r46122 = r46112 - r46113;
        double r46123 = r46120 / r46113;
        double r46124 = r46120 / r46112;
        double r46125 = r46123 - r46124;
        double r46126 = r46122 / r46125;
        double r46127 = r46121 / r46126;
        return r46127;
}

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 13.9

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

  3. Applied difference-of-squares8.9

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

  4. Applied *-un-lft-identity8.9

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

  5. Applied times-frac8.6

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

  6. Applied associate-*r*8.6

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

  7. Simplified8.5

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

  9. Applied associate-*r/8.5

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

  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 clear-num0.3

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

  14. Applied associate-/l*0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020056 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))