Average Error: 9.7 → 4.9
Time: 8.5m
Precision: 64
Internal Precision: 1344
\[x + \frac{y - z}{\left(t + 1.0\right) - z} \cdot \left(a - x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y - z}{\left(t + 1.0\right) - z} \le 0.9999999999999978:\\ \;\;\;\;x + \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{1} \cdot \left(\frac{\sqrt[3]{y - z}}{\left(t + 1.0\right) - z} \cdot \left(a - x\right)\right)\\ \mathbf{if}\;\frac{y - z}{\left(t + 1.0\right) - z} \le 1.0138787963625333:\\ \;\;\;\;\frac{1.0}{z} \cdot \left(a - x\right) + a\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{1} \cdot \left(\frac{\sqrt[3]{y - z}}{\left(t + 1.0\right) - z} \cdot \left(a - x\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if (/ (- y z) (- (+ t 1.0) z)) < 0.9999999999999978 or 1.0138787963625333 < (/ (- y z) (- (+ t 1.0) z))

    1. Initial program 0.6

      \[x + \frac{y - z}{\left(t + 1.0\right) - z} \cdot \left(a - x\right)\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.6

      \[\leadsto x + \frac{y - z}{\color{blue}{1 \cdot \left(\left(t + 1.0\right) - z\right)}} \cdot \left(a - x\right)\]

    4. Applied add-cube-cbrt0.9

      \[\leadsto x + \frac{\color{blue}{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \sqrt[3]{y - z}}}{1 \cdot \left(\left(t + 1.0\right) - z\right)} \cdot \left(a - x\right)\]

    5. Applied times-frac0.9

      \[\leadsto x + \color{blue}{\left(\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{1} \cdot \frac{\sqrt[3]{y - z}}{\left(t + 1.0\right) - z}\right)} \cdot \left(a - x\right)\]

    6. Applied associate-*l*1.0

      \[\leadsto x + \color{blue}{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{1} \cdot \left(\frac{\sqrt[3]{y - z}}{\left(t + 1.0\right) - z} \cdot \left(a - x\right)\right)}\]

    if 0.9999999999999978 < (/ (- y z) (- (+ t 1.0) z)) < 1.0138787963625333

    1. Initial program 29.4

      \[x + \frac{y - z}{\left(t + 1.0\right) - z} \cdot \left(a - x\right)\]
    2. Using strategy rm

    3. Applied *-un-lft-identity29.4

      \[\leadsto x + \frac{y - z}{\color{blue}{1 \cdot \left(\left(t + 1.0\right) - z\right)}} \cdot \left(a - x\right)\]

    4. Applied add-cube-cbrt29.7

      \[\leadsto x + \frac{\color{blue}{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \sqrt[3]{y - z}}}{1 \cdot \left(\left(t + 1.0\right) - z\right)} \cdot \left(a - x\right)\]

    5. Applied times-frac29.7

      \[\leadsto x + \color{blue}{\left(\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{1} \cdot \frac{\sqrt[3]{y - z}}{\left(t + 1.0\right) - z}\right)} \cdot \left(a - x\right)\]

    6. Applied associate-*l*31.0

      \[\leadsto x + \color{blue}{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{1} \cdot \left(\frac{\sqrt[3]{y - z}}{\left(t + 1.0\right) - z} \cdot \left(a - x\right)\right)}\]

    7. Taylor expanded around inf 13.4

      \[\leadsto \color{blue}{\left(1.0 \cdot \frac{a}{z} + a\right) - 1.0 \cdot \frac{x}{z}}\]

    8. Applied simplify13.4

      \[\leadsto \color{blue}{\frac{1.0}{z} \cdot \left(a - x\right) + a}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 8.5m)Debug logProfile

herbie shell --seed '#(341049388 4193966283 3776730818 3836052170 128576249 3840315966)' 
(FPCore (x y z t a)
  :name "Hakyll.Web.Tags:renderTagCloud from hakyll-4.7.2.3"
  (+ x (* (/ (- y z) (- (+ t 1.0) z)) (- a x))))