Average Error: 14.1 → 0.8
Time: 1.4m
Precision: 64
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.6936446320575042 \cdot 10^{+262}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.7219844561894888 \cdot 10^{-291}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 0.0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.585203053785728 \cdot 10^{+83}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if (/ y z) < -2.6936446320575042e+262

    1. Initial program 51.1

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

    2. Simplified38.4

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

    4. Applied div-inv38.4

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

    5. Applied associate-*r*0.4

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

    if -2.6936446320575042e+262 < (/ y z) < -1.7219844561894888e-291 or 0.0 < (/ y z) < 1.585203053785728e+83

    1. Initial program 8.7

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

    2. Simplified0.4

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

    if -1.7219844561894888e-291 < (/ y z) < 0.0

    1. Initial program 19.0

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

    2. Simplified17.8

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

    4. Applied *-un-lft-identity17.8

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

    5. Applied add-cube-cbrt17.8

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

    6. Applied times-frac17.8

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

    7. Applied associate-*r*4.2

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

    9. Applied associate-*r/4.2

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

    10. Applied associate-*l/4.2

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

    11. Simplified0.1

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

    if 1.585203053785728e+83 < (/ y z)

    1. Initial program 25.8

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

    2. Simplified11.7

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

    4. Applied add-cube-cbrt12.6

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

    5. Applied *-un-lft-identity12.6

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

    6. Applied times-frac12.6

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

    7. Applied associate-*r*5.9

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

    8. Taylor expanded around 0 4.1

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.6936446320575042 \cdot 10^{+262}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.7219844561894888 \cdot 10^{-291}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 0.0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.585203053785728 \cdot 10^{+83}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))