Average Error: 52.7 → 37.0
Time: 2.9m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\sqrt[3]{(\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*} \cdot \left(\frac{i}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{\beta + \left(i + \alpha\right)}{(i \cdot 2 + \alpha)_* + \beta}\right)\right) \cdot \left(\sqrt[3]{(\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*} \cdot \sqrt[3]{(\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}\right) \le 0.06268573961976906:\\ \;\;\;\;\frac{(\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*} \cdot \left(\frac{i}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{\beta + \left(i + \alpha\right)}{(i \cdot 2 + \alpha)_* + \beta}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (* (cbrt (fma (+ (+ alpha i) beta) i (* beta alpha))) (cbrt (fma (+ (+ alpha i) beta) i (* beta alpha)))) 1) (* (/ (cbrt (fma (+ (+ alpha i) beta) i (* beta alpha))) (fma (+ (fma i 2 alpha) beta) (+ (fma i 2 alpha) beta) (- 1.0))) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 alpha) beta)) (/ i (+ (fma i 2 alpha) beta))))) < 0.06268573961976906

    1. Initial program 38.9

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

    2. Applied simplify5.7

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

    if 0.06268573961976906 < (* (/ (* (cbrt (fma (+ (+ alpha i) beta) i (* beta alpha))) (cbrt (fma (+ (+ alpha i) beta) i (* beta alpha)))) 1) (* (/ (cbrt (fma (+ (+ alpha i) beta) i (* beta alpha))) (fma (+ (fma i 2 alpha) beta) (+ (fma i 2 alpha) beta) (- 1.0))) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 alpha) beta)) (/ i (+ (fma i 2 alpha) beta)))))

    1. Initial program 62.0

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

    2. Applied simplify62.0

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

    3. Taylor expanded around inf 58.2

      \[\leadsto \color{blue}{0} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\]

    4. Applied simplify58.2

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

  4. Applied simplify37.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\frac{\sqrt[3]{(\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*} \cdot \left(\frac{i}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{\beta + \left(i + \alpha\right)}{(i \cdot 2 + \alpha)_* + \beta}\right)\right) \cdot \left(\sqrt[3]{(\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*} \cdot \sqrt[3]{(\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}\right) \le 0.06268573961976906:\\ \;\;\;\;\frac{(\left(\beta + \left(i + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*} \cdot \left(\frac{i}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{\beta + \left(i + \alpha\right)}{(i \cdot 2 + \alpha)_* + \beta}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018198 +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)))