Average Error: 14.6 → 0.3
Time: 17.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{\frac{1 \cdot \left(b - a\right)}{b \cdot a}}{2 \cdot \left(a + b\right)} \cdot \pi\right) \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)}{\frac{b - a}{\sqrt[3]{1}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(\frac{\frac{1 \cdot \left(b - a\right)}{b \cdot a}}{2 \cdot \left(a + b\right)} \cdot \pi\right) \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)}{\frac{b - a}{\sqrt[3]{1}}}
double f(double a, double b) {
        double r50226 = atan2(1.0, 0.0);
        double r50227 = 2.0;
        double r50228 = r50226 / r50227;
        double r50229 = 1.0;
        double r50230 = b;
        double r50231 = r50230 * r50230;
        double r50232 = a;
        double r50233 = r50232 * r50232;
        double r50234 = r50231 - r50233;
        double r50235 = r50229 / r50234;
        double r50236 = r50228 * r50235;
        double r50237 = r50229 / r50232;
        double r50238 = r50229 / r50230;
        double r50239 = r50237 - r50238;
        double r50240 = r50236 * r50239;
        return r50240;
}
↓
double f(double a, double b) {
        double r50241 = 1.0;
        double r50242 = b;
        double r50243 = a;
        double r50244 = r50242 - r50243;
        double r50245 = r50241 * r50244;
        double r50246 = r50242 * r50243;
        double r50247 = r50245 / r50246;
        double r50248 = 2.0;
        double r50249 = r50243 + r50242;
        double r50250 = r50248 * r50249;
        double r50251 = r50247 / r50250;
        double r50252 = atan2(1.0, 0.0);
        double r50253 = r50251 * r50252;
        double r50254 = cbrt(r50241);
        double r50255 = r50254 * r50254;
        double r50256 = r50253 * r50255;
        double r50257 = r50244 / r50254;
        double r50258 = r50256 / r50257;
        return r50258;
}

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 14.6

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

  2. Simplified9.4

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

  4. Applied add-cube-cbrt9.4

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

  5. Applied times-frac9.4

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

  6. Applied associate-/r*0.3

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

  7. Simplified0.3

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

  9. Applied frac-sub0.3

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

  10. Simplified0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 
(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))))