Octave 3.8, jcobi/4

Average Error: 52.8 → 35.9

Time: 25.2s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.547494373636242 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(i + \left(\alpha + \beta\right), i, \alpha \cdot \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \sqrt{\mathsf{fma}\left(i + \left(\alpha + \beta\right), i, \alpha \cdot \beta\right)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r3695136 = i;
        double r3695137 = alpha;
        double r3695138 = beta;
        double r3695139 = r3695137 + r3695138;
        double r3695140 = r3695139 + r3695136;
        double r3695141 = r3695136 * r3695140;
        double r3695142 = r3695138 * r3695137;
        double r3695143 = r3695142 + r3695141;
        double r3695144 = r3695141 * r3695143;
        double r3695145 = 2.0;
        double r3695146 = r3695145 * r3695136;
        double r3695147 = r3695139 + r3695146;
        double r3695148 = r3695147 * r3695147;
        double r3695149 = r3695144 / r3695148;
        double r3695150 = 1.0;
        double r3695151 = r3695148 - r3695150;
        double r3695152 = r3695149 / r3695151;
        return r3695152;
}

↓

double f(double alpha, double beta, double i) {
        double r3695153 = beta;
        double r3695154 = 1.547494373636242e+216;
        bool r3695155 = r3695153 <= r3695154;
        double r3695156 = i;
        double r3695157 = alpha;
        double r3695158 = r3695157 + r3695153;
        double r3695159 = r3695156 + r3695158;
        double r3695160 = r3695157 * r3695153;
        double r3695161 = fma(r3695159, r3695156, r3695160);
        double r3695162 = sqrt(r3695161);
        double r3695163 = 2.0;
        double r3695164 = fma(r3695163, r3695156, r3695158);
        double r3695165 = r3695162 / r3695164;
        double r3695166 = r3695165 * r3695162;
        double r3695167 = 1.0;
        double r3695168 = sqrt(r3695167);
        double r3695169 = r3695168 + r3695164;
        double r3695170 = r3695166 / r3695169;
        double r3695171 = r3695156 * r3695159;
        double r3695172 = r3695171 / r3695164;
        double r3695173 = r3695164 - r3695168;
        double r3695174 = r3695172 / r3695173;
        double r3695175 = r3695170 * r3695174;
        double r3695176 = 0.0;
        double r3695177 = r3695155 ? r3695175 : r3695176;
        return r3695177;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 1.547494373636242e+216

   1. Initial program 51.8

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

   2. Simplified 51.8

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

   4. Applied add-sqr-sqrt 51.8

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

   5. Applied difference-of-squares 51.8

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

   6. Applied times-frac 37.6

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

   7. Applied times-frac 35.3

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

   9. Applied *-un-lft-identity 35.3

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

   10. Applied add-sqr-sqrt 35.3

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

   11. Applied times-frac 35.3

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

   if 1.547494373636242e+216 < beta

   1. Initial program 62.7

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

   2. Simplified 62.7

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

   3. Taylor expanded around inf 41.0

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.547494373636242 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(i + \left(\alpha + \beta\right), i, \alpha \cdot \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \sqrt{\mathsf{fma}\left(i + \left(\alpha + \beta\right), i, \alpha \cdot \beta\right)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))