Average Error: 9.4 → 6.2
Time: 2.0m
Precision: 64
Internal Precision: 1408
\[x + \frac{y - z}{\left(t + 1.0\right) - z} \cdot \left(a - x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y - z}{\left(t + 1.0\right) - z} \cdot \left(a - x\right) \le -6.810126330032964 \cdot 10^{-283}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{1} \cdot \left(\frac{\sqrt[3]{y - z}}{\left(t + 1.0\right) - z} \cdot \left(a - x\right)\right)\\ \mathbf{if}\;x + \frac{y - z}{\left(t + 1.0\right) - z} \cdot \left(a - x\right) \le 4.313248978008767 \cdot 10^{-283}:\\ \;\;\;\;a - \left(a - x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y - z}{\left(t + 1.0\right) - z} \cdot \left(\sqrt[3]{a - x} \cdot \sqrt[3]{a - x}\right)\right) \cdot \sqrt[3]{a - x}\\ \end{array}\]

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 3 regimes

  2. if (+ x (* (/ (- y z) (- (+ t 1.0) z)) (- a x))) < -6.810126330032964e-283

    1. Initial program 0.8

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

    3. Applied *-un-lft-identity0.8

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

    4. Applied add-cube-cbrt1.6

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

    5. Applied times-frac1.6

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

    6. Applied associate-*l*2.2

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

    if -6.810126330032964e-283 < (+ x (* (/ (- y z) (- (+ t 1.0) z)) (- a x))) < 4.313248978008767e-283

    1. Initial program 59.2

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

    2. Taylor expanded around inf 35.9

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

    3. Applied simplify30.7

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

    if 4.313248978008767e-283 < (+ x (* (/ (- y z) (- (+ t 1.0) z)) (- a x)))

    1. Initial program 0.8

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

    3. Applied add-cube-cbrt1.6

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

    4. Applied associate-*r*1.6

      \[\leadsto x + \color{blue}{\left(\frac{y - z}{\left(t + 1.0\right) - z} \cdot \left(\sqrt[3]{a - x} \cdot \sqrt[3]{a - x}\right)\right) \cdot \sqrt[3]{a - x}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(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))))