Average Error: 23.9 → 11.8
Time: 18.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\frac{\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}
double f(double alpha, double beta, double i) {
        double r143204 = alpha;
        double r143205 = beta;
        double r143206 = r143204 + r143205;
        double r143207 = r143205 - r143204;
        double r143208 = r143206 * r143207;
        double r143209 = 2.0;
        double r143210 = i;
        double r143211 = r143209 * r143210;
        double r143212 = r143206 + r143211;
        double r143213 = r143208 / r143212;
        double r143214 = r143212 + r143209;
        double r143215 = r143213 / r143214;
        double r143216 = 1.0;
        double r143217 = r143215 + r143216;
        double r143218 = r143217 / r143209;
        return r143218;
}
↓
double f(double alpha, double beta, double i) {
        double r143219 = beta;
        double r143220 = i;
        double r143221 = 2.0;
        double r143222 = alpha;
        double r143223 = r143222 + r143219;
        double r143224 = fma(r143220, r143221, r143223);
        double r143225 = cbrt(r143223);
        double r143226 = r143225 * r143225;
        double r143227 = r143224 / r143226;
        double r143228 = 1.0;
        double r143229 = sqrt(r143228);
        double r143230 = r143229 / r143225;
        double r143231 = r143227 * r143230;
        double r143232 = r143219 / r143231;
        double r143233 = r143221 * r143220;
        double r143234 = r143223 + r143233;
        double r143235 = r143234 + r143221;
        double r143236 = r143232 / r143235;
        double r143237 = r143224 / r143223;
        double r143238 = r143222 / r143237;
        double r143239 = r143238 / r143235;
        double r143240 = 1.0;
        double r143241 = r143239 - r143240;
        double r143242 = r143236 - r143241;
        double r143243 = r143242 / r143221;
        return r143243;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.9

    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity23.9

    \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2}\]

  4. Applied associate-/r*23.9

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\]

  5. Simplified12.0

    \[\leadsto \frac{\frac{\color{blue}{\frac{\beta - \alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
  6. Using strategy rm

  7. Applied div-sub12.0

    \[\leadsto \frac{\frac{\color{blue}{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}} - \frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

  8. Applied div-sub12.0

    \[\leadsto \frac{\color{blue}{\left(\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)} + 1}{2}\]

  9. Applied associate-+l-11.7

    \[\leadsto \frac{\color{blue}{\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}}{2}\]
  10. Using strategy rm

  11. Applied div-inv11.7

    \[\leadsto \frac{\frac{\frac{\beta}{\color{blue}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \frac{1}{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]
  12. Using strategy rm

  13. Applied add-cube-cbrt11.8

    \[\leadsto \frac{\frac{\frac{\beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \frac{1}{\color{blue}{\left(\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}\right) \cdot \sqrt[3]{\alpha + \beta}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]

  14. Applied add-sqr-sqrt11.8

    \[\leadsto \frac{\frac{\frac{\beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}\right) \cdot \sqrt[3]{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]

  15. Applied times-frac11.8

    \[\leadsto \frac{\frac{\frac{\beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \color{blue}{\left(\frac{\sqrt{1}}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\alpha + \beta}}\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]

  16. Applied associate-*r*11.8

    \[\leadsto \frac{\frac{\frac{\beta}{\color{blue}{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \frac{\sqrt{1}}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}\right) \cdot \frac{\sqrt{1}}{\sqrt[3]{\alpha + \beta}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]

  17. Simplified11.8

    \[\leadsto \frac{\frac{\frac{\beta}{\color{blue}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]

  18. Final simplification11.8

    \[\leadsto \frac{\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))