Average Error: 3.9 → 4.9
Time: 24.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot 1}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot 1}
double f(double alpha, double beta) {
        double r89211 = alpha;
        double r89212 = beta;
        double r89213 = r89211 + r89212;
        double r89214 = r89212 * r89211;
        double r89215 = r89213 + r89214;
        double r89216 = 1.0;
        double r89217 = r89215 + r89216;
        double r89218 = 2.0;
        double r89219 = r89218 * r89216;
        double r89220 = r89213 + r89219;
        double r89221 = r89217 / r89220;
        double r89222 = r89221 / r89220;
        double r89223 = r89220 + r89216;
        double r89224 = r89222 / r89223;
        return r89224;
}
↓
double f(double alpha, double beta) {
        double r89225 = alpha;
        double r89226 = beta;
        double r89227 = r89225 + r89226;
        double r89228 = r89226 * r89225;
        double r89229 = r89227 + r89228;
        double r89230 = 1.0;
        double r89231 = r89229 + r89230;
        double r89232 = 2.0;
        double r89233 = r89232 * r89230;
        double r89234 = r89227 + r89233;
        double r89235 = r89234 * r89234;
        double r89236 = r89231 / r89235;
        double r89237 = r89234 + r89230;
        double r89238 = 1.0;
        double r89239 = r89237 * r89238;
        double r89240 = r89236 / r89239;
        return r89240;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 3.9

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

  3. Applied add-sqr-sqrt4.4

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

  4. Applied add-sqr-sqrt4.4

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

  5. Applied times-frac4.4

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

  6. Applied associate-/l*4.0

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity4.0

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

  9. Applied sqrt-prod4.0

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

  10. Applied associate-/r*4.0

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

  11. Simplified4.0

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

  12. Final simplification4.9

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

Reproduce

herbie shell --seed 2019303 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))