Average Error: 14.0 → 1.8
Time: 21.2s
Precision: 64
Internal Precision: 128
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \left(y \cdot \left(\left(\sqrt[3]{\frac{1}{z}} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\frac{1}{z}} \cdot \sqrt[3]{x}\right)\right)\right)\]

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. Initial program 14.0

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

  2. Initial simplification6.0

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

  4. Applied add-cube-cbrt6.8

    \[\leadsto y \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}\right) \cdot \sqrt[3]{\frac{x}{z}}\right)}\]

  5. Applied associate-*r*6.8

    \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}\right)\right) \cdot \sqrt[3]{\frac{x}{z}}}\]
  6. Using strategy rm

  7. Applied div-inv6.8

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

  8. Applied cbrt-prod6.7

    \[\leadsto \left(y \cdot \left(\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{\frac{1}{z}}\right)} \cdot \sqrt[3]{\frac{x}{z}}\right)\right) \cdot \sqrt[3]{\frac{x}{z}}\]
  9. Using strategy rm

  10. Applied cbrt-div6.7

    \[\leadsto \left(y \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{\frac{1}{z}}\right) \cdot \sqrt[3]{\frac{x}{z}}\right)\right) \cdot \color{blue}{\frac{\sqrt[3]{x}}{\sqrt[3]{z}}}\]
  11. Using strategy rm

  12. Applied div-inv6.7

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

  13. Applied cbrt-prod1.8

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

  14. Final simplification1.8

    \[\leadsto \frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \left(y \cdot \left(\left(\sqrt[3]{\frac{1}{z}} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\frac{1}{z}} \cdot \sqrt[3]{x}\right)\right)\right)\]

Runtime

Time bar (total: 21.2s)Debug logProfile

herbie shell --seed 2018257 +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)))