Average Error: 25.3 → 5.6
Time: 1.2m
Precision: 64
Internal Precision: 384
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.2709557362568119 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{if}\;z \le 3.0149748654828578 \cdot 10^{+60}:\\ \;\;\;\;\frac{y \cdot x}{\left|\sqrt[3]{z \cdot z - a \cdot t}\right|} \cdot \frac{z}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original25.3
Target6.8
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{if}\;z \lt 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.2709557362568119e+112

    1. Initial program 45.9

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

    2. Taylor expanded around -inf 0

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

    3. Applied simplify0

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

    if -1.2709557362568119e+112 < z < 3.0149748654828578e+60

    1. Initial program 10.6

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

    3. Applied add-cube-cbrt11.0

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

    4. Applied sqrt-prod11.0

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

    5. Applied times-frac10.4

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

    6. Applied simplify10.4

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

    if 3.0149748654828578e+60 < z

    1. Initial program 39.8

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

    2. Taylor expanded around inf 0

      \[\leadsto \color{blue}{y \cdot x}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.2m)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 "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

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