Average Error: 23.1 → 12.1
Time: 22.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\mathsf{log1p}\left(e^{\log \left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, \frac{\alpha + \beta}{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}}, 1.0\right)\right)\right)}\right)}{2.0}\]
\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) + 2.0} + 1.0}{2.0}
↓
\frac{\mathsf{log1p}\left(e^{\log \left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, \frac{\alpha + \beta}{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}}, 1.0\right)\right)\right)}\right)}{2.0}
double f(double alpha, double beta, double i) {
        double r4442118 = alpha;
        double r4442119 = beta;
        double r4442120 = r4442118 + r4442119;
        double r4442121 = r4442119 - r4442118;
        double r4442122 = r4442120 * r4442121;
        double r4442123 = 2.0;
        double r4442124 = i;
        double r4442125 = r4442123 * r4442124;
        double r4442126 = r4442120 + r4442125;
        double r4442127 = r4442122 / r4442126;
        double r4442128 = 2.0;
        double r4442129 = r4442126 + r4442128;
        double r4442130 = r4442127 / r4442129;
        double r4442131 = 1.0;
        double r4442132 = r4442130 + r4442131;
        double r4442133 = r4442132 / r4442128;
        return r4442133;
}
↓
double f(double alpha, double beta, double i) {
        double r4442134 = beta;
        double r4442135 = alpha;
        double r4442136 = r4442134 - r4442135;
        double r4442137 = i;
        double r4442138 = 2.0;
        double r4442139 = r4442135 + r4442134;
        double r4442140 = fma(r4442137, r4442138, r4442139);
        double r4442141 = r4442136 / r4442140;
        double r4442142 = 2.0;
        double r4442143 = r4442140 + r4442142;
        double r4442144 = sqrt(r4442143);
        double r4442145 = r4442139 / r4442144;
        double r4442146 = 1.0;
        double r4442147 = r4442146 / r4442144;
        double r4442148 = r4442145 * r4442147;
        double r4442149 = 1.0;
        double r4442150 = fma(r4442141, r4442148, r4442149);
        double r4442151 = expm1(r4442150);
        double r4442152 = log(r4442151);
        double r4442153 = exp(r4442152);
        double r4442154 = log1p(r4442153);
        double r4442155 = r4442154 / r4442142;
        return r4442155;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.1

    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

  2. Simplified19.2

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}, \beta + \alpha, 1.0\right)}{2.0}}\]
  3. Using strategy rm

  4. Applied log1p-expm1-u19.2

    \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}, \beta + \alpha, 1.0\right)\right)\right)}}{2.0}\]

  5. Simplified12.0

    \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}, 1.0\right)\right)}\right)}{2.0}\]
  6. Using strategy rm

  7. Applied add-exp-log12.0

    \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}, 1.0\right)\right)\right)}}\right)}{2.0}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt12.1

    \[\leadsto \frac{\mathsf{log1p}\left(e^{\log \left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta + \alpha}{\color{blue}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}, 1.0\right)\right)\right)}\right)}{2.0}\]

  10. Applied *-un-lft-identity12.1

    \[\leadsto \frac{\mathsf{log1p}\left(e^{\log \left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\color{blue}{1 \cdot \left(\beta + \alpha\right)}}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}, 1.0\right)\right)\right)}\right)}{2.0}\]

  11. Applied times-frac12.1

    \[\leadsto \frac{\mathsf{log1p}\left(e^{\log \left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \color{blue}{\frac{1}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \cdot \frac{\beta + \alpha}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}, 1.0\right)\right)\right)}\right)}{2.0}\]

  12. Final simplification12.1

    \[\leadsto \frac{\mathsf{log1p}\left(e^{\log \left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, \frac{\alpha + \beta}{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}}, 1.0\right)\right)\right)}\right)}{2.0}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))