Average Error: 14.2 → 1.0
Time: 7.7s
Precision: 64
Internal Precision: 128
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -3.6474706101836307 \cdot 10^{+179}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -5.600770411125762 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{y}}}{\frac{1}{x}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.8706786795667 \cdot 10^{-310}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 2.200763319221206 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{y}}}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Results

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Derivation

  1. Split input into 3 regimes

  2. if (/ y z) < -3.6474706101836307e+179

    1. Initial program 33.2

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Initial simplification2.2

      \[\leadsto y \cdot \frac{x}{z}\]

    if -3.6474706101836307e+179 < (/ y z) < -5.600770411125762e-165 or 1.8706786795667e-310 < (/ y z) < 2.200763319221206e+124

    1. Initial program 7.4

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Initial simplification9.1

      \[\leadsto y \cdot \frac{x}{z}\]
    3. Using strategy rm

    4. Applied associate-*r/8.9

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}}\]
    5. Using strategy rm

    6. Applied clear-num9.3

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{y \cdot x}}}\]
    7. Using strategy rm

    8. Applied associate-/r*0.8

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z}{y}}{x}}}\]
    9. Using strategy rm

    10. Applied div-inv0.9

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{y} \cdot \frac{1}{x}}}\]

    11. Applied associate-/r*0.5

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{z}{y}}}{\frac{1}{x}}}\]

    if -5.600770411125762e-165 < (/ y z) < 1.8706786795667e-310 or 2.200763319221206e+124 < (/ y z)

    1. Initial program 22.1

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Initial simplification1.4

      \[\leadsto y \cdot \frac{x}{z}\]
    3. Using strategy rm

    4. Applied associate-*r/1.6

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -3.6474706101836307 \cdot 10^{+179}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -5.600770411125762 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{y}}}{\frac{1}{x}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.8706786795667 \cdot 10^{-310}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 2.200763319221206 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{y}}}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Runtime

Time bar (total: 7.7s)Debug logProfile

herbie shell --seed 2018278 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))