Average Error: 14.2 → 0.7
Time: 11.3s
Precision: 64
Internal Precision: 384
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot x \le -8.848462821600082 \cdot 10^{+285}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{if}\;y \cdot x \le -4.2284236289604585 \cdot 10^{-290}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{if}\;y \cdot x \le 7.585235682791941 \cdot 10^{-295}:\\ \;\;\;\;\log_* (1 + (e^{\frac{y}{\frac{z}{x}}} - 1)^*)\\ \mathbf{if}\;y \cdot x \le 7.861310228265737 \cdot 10^{+86}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{if}\;y \cdot x \le +\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

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 x) < -8.848462821600082e+285

    1. Initial program 7.9

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

    2. Applied simplify0.2

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

    4. Applied associate-/r/0.3

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

    if -8.848462821600082e+285 < (* y x) < -4.2284236289604585e-290 or 7.585235682791941e-295 < (* y x) < 7.861310228265737e+86

    1. Initial program 16.9

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

    2. Applied simplify7.8

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

    3. Taylor expanded around 0 0.2

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

    if -4.2284236289604585e-290 < (* y x) < 7.585235682791941e-295

    1. Initial program 2.4

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

    2. Applied simplify0.1

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

    4. Applied div-inv0.1

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

    5. Applied associate-/r*0.1

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

    7. Applied log1p-expm1-u0.1

      \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{\frac{x}{z}}{\frac{1}{y}}} - 1)^*)}\]

    8. Applied simplify0.1

      \[\leadsto \log_* (1 + \color{blue}{(e^{\frac{y}{\frac{z}{x}}} - 1)^*})\]

    if 7.861310228265737e+86 < (* y x)

    1. Initial program 16.2

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

    2. Applied simplify3.7

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

Runtime

Time bar (total: 11.3s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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)))