Average Error: 14.1 → 0.3
Time: 22.8s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\mathsf{fma}\left(\sqrt{1}, \frac{\sqrt{1}}{a}, -\frac{1}{b}\right) \cdot \frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a} + \frac{\pi}{\left(b + a\right) \cdot 2} \cdot \left(\frac{1}{b - a} \cdot \left(\left(-\frac{1}{b}\right) + \frac{1}{b}\right)\right)\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\mathsf{fma}\left(\sqrt{1}, \frac{\sqrt{1}}{a}, -\frac{1}{b}\right) \cdot \frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a} + \frac{\pi}{\left(b + a\right) \cdot 2} \cdot \left(\frac{1}{b - a} \cdot \left(\left(-\frac{1}{b}\right) + \frac{1}{b}\right)\right)
double f(double a, double b) {
        double r65083 = atan2(1.0, 0.0);
        double r65084 = 2.0;
        double r65085 = r65083 / r65084;
        double r65086 = 1.0;
        double r65087 = b;
        double r65088 = r65087 * r65087;
        double r65089 = a;
        double r65090 = r65089 * r65089;
        double r65091 = r65088 - r65090;
        double r65092 = r65086 / r65091;
        double r65093 = r65085 * r65092;
        double r65094 = r65086 / r65089;
        double r65095 = r65086 / r65087;
        double r65096 = r65094 - r65095;
        double r65097 = r65093 * r65096;
        return r65097;
}
↓
double f(double a, double b) {
        double r65098 = 1.0;
        double r65099 = sqrt(r65098);
        double r65100 = a;
        double r65101 = r65099 / r65100;
        double r65102 = b;
        double r65103 = r65098 / r65102;
        double r65104 = -r65103;
        double r65105 = fma(r65099, r65101, r65104);
        double r65106 = atan2(1.0, 0.0);
        double r65107 = 2.0;
        double r65108 = r65106 / r65107;
        double r65109 = r65105 * r65108;
        double r65110 = r65102 + r65100;
        double r65111 = r65109 / r65110;
        double r65112 = r65102 - r65100;
        double r65113 = r65098 / r65112;
        double r65114 = r65111 * r65113;
        double r65115 = r65110 * r65107;
        double r65116 = r65106 / r65115;
        double r65117 = r65104 + r65103;
        double r65118 = r65113 * r65117;
        double r65119 = r65116 * r65118;
        double r65120 = r65114 + r65119;
        return r65120;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.1

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

  3. Applied difference-of-squares9.6

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

  4. Applied *-un-lft-identity9.6

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.1

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

  7. Simplified9.0

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

  9. Applied add-cube-cbrt9.2

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{1}{a} - \color{blue}{\left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{b}}}\right)\]

  10. Applied *-un-lft-identity9.2

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{1}{\color{blue}{1 \cdot a}} - \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{b}}\right)\]

  11. Applied add-sqr-sqrt9.2

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot a} - \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{b}}\right)\]

  12. Applied times-frac9.2

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\color{blue}{\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{a}} - \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{b}}\right)\]

  13. Applied prod-diff9.2

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\sqrt{1}}{1}, \frac{\sqrt{1}}{a}, -\sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right)\right)}\]

  14. Applied distribute-lft-in9.2

    \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \mathsf{fma}\left(\frac{\sqrt{1}}{1}, \frac{\sqrt{1}}{a}, -\sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right) + \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \mathsf{fma}\left(-\sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right)}\]

  15. Simplified0.3

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{1}, \frac{\sqrt{1}}{a}, -\frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b + a}\right) \cdot \frac{1}{b - a}} + \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \mathsf{fma}\left(-\sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right)\]

  16. Simplified0.3

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

  18. Applied associate-*r/0.3

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

  19. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))