Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Average Error: 24.1 → 6.6

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.09785828767813579 \cdot 10^{102}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 1.56172699756646245 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot \left(x \cdot y\right)}}\\ \mathbf{elif}\;z \le 2.4622880997986805 \cdot 10^{126}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.1
Target	7.6
Herbie	6.6

\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if z < -4.09785828767813579e102

   1. Initial program 44.1

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

   2. Taylor expanded around -inf 2.1

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

   3. Simplified 2.1

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

   if -4.09785828767813579e102 < z < 1.56172699756646245e-233

   1. Initial program 11.4

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

   3. Applied clear-num 11.8

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

   4. Simplified 11.8

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

   if 1.56172699756646245e-233 < z < 2.4622880997986805e126

   1. Initial program 9.9

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

   2. Simplified 7.1

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

   if 2.4622880997986805e126 < z

   1. Initial program 48.1

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

   2. Taylor expanded around inf 1.3

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.09785828767813579 \cdot 10^{102}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 1.56172699756646245 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot \left(x \cdot y\right)}}\\ \mathbf{elif}\;z \le 2.4622880997986805 \cdot 10^{126}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (neg (* 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)))))