Average Error: 14.4 → 1.0
Time: 23.3s
Precision: 64
Internal Precision: 128
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -2.234125294553967 \cdot 10^{-243}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 4.30802458720323 \cdot 10^{-186}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 8.964702872818498 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if (/ y z) < -inf.0

    1. Initial program 60.1

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

    2. Simplified60.1

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

    4. Applied div-inv60.1

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

    5. Applied associate-*r*0.3

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

    if -inf.0 < (/ y z) < -2.234125294553967e-243 or 4.30802458720323e-186 < (/ y z) < 8.964702872818498e+71

    1. Initial program 8.5

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

    2. Simplified0.2

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

    if -2.234125294553967e-243 < (/ y z) < 4.30802458720323e-186 or 8.964702872818498e+71 < (/ y z)

    1. Initial program 19.9

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

    2. Simplified10.6

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

    4. Applied div-inv10.6

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

    5. Applied associate-*r*2.1

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

    7. Applied pow12.1

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

    8. Applied pow12.1

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

    9. Applied pow12.1

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

    10. Applied pow-prod-down2.1

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

    11. Applied pow-prod-down2.1

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

    12. Simplified11.1

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

    13. Taylor expanded around -inf 2.1

      \[\leadsto {\color{blue}{\left(\frac{x \cdot y}{z}\right)}}^{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -2.234125294553967 \cdot 10^{-243}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 4.30802458720323 \cdot 10^{-186}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 8.964702872818498 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 +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)))