Average Error: 25.5 → 8.2
Time: 8.5m
Precision: 64
Internal Precision: 1920
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -4.2533961552866985 \cdot 10^{-171}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\\ \mathbf{if}\;a \le 8.183905144226627 \cdot 10^{-137}:\\ \;\;\;\;t - \left(t - x\right) \cdot \frac{y}{z}\\ \mathbf{if}\;a \le 1.215880926283839 \cdot 10^{-41}:\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}\\ \mathbf{if}\;a \le 1820749428.2327986:\\ \;\;\;\;t - \left(t - x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original25.5
Target13.6
Herbie8.2
\[\begin{array}{l} \mathbf{if}\;z \lt -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{if}\;z \lt 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -4.2533961552866985e-171 or 1820749428.2327986 < a

    1. Initial program 21.9

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt22.2

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

    4. Applied times-frac8.9

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]

    if -4.2533961552866985e-171 < a < 8.183905144226627e-137 or 1.215880926283839e-41 < a < 1820749428.2327986

    1. Initial program 35.7

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]

    2. Taylor expanded around inf 5.9

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

    3. Applied simplify1.5

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

    if 8.183905144226627e-137 < a < 1.215880926283839e-41

    1. Initial program 23.9

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
    2. Using strategy rm

    3. Applied div-inv24.0

      \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 8.5m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))