Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Average Error: 4.4 → 1.6

Time: 6.8s

Precision: binary64

Math

\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.835758328218971 \cdot 10^{274}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \sqrt[3]{1 - z} - z \cdot \left(\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot t\right)\right)}{z \cdot \sqrt[3]{1 - z}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.0260174088170734 \cdot 10^{99}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \frac{-\frac{t}{1 - z}}{1}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.4
Target	4.2
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -2.835758328218971e+274

   1. Initial program 40.0

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 40.0

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{t}{\color{blue}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}}\right)\]

   4. Applied *-un-lft-identity 40.0

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}\right)\]

   5. Applied times-frac 40.0

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}}\right)\]
   6. Using strategy rm

   7. Applied associate-*r/ 40.0

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot t}{\sqrt[3]{1 - z}}}\right)\]

   8. Applied frac-sub 40.0

      \[\leadsto x \cdot \color{blue}{\frac{y \cdot \sqrt[3]{1 - z} - z \cdot \left(\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot t\right)}{z \cdot \sqrt[3]{1 - z}}}\]

   9. Applied associate-*r/ 0.3

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot \sqrt[3]{1 - z} - z \cdot \left(\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot t\right)\right)}{z \cdot \sqrt[3]{1 - z}}}\]

   if -2.835758328218971e+274 < (- (/ y z) (/ t (- 1.0 z))) < 1.0260174088170734e+99

   1. Initial program 1.3

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 1.6

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{t}{\color{blue}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}}\right)\]

   4. Applied *-un-lft-identity 1.6

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}\right)\]

   5. Applied times-frac 1.6

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}}\right)\]

   if 1.0260174088170734e+99 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 10.3

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 11.3

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]

   4. Applied associate-*l* 11.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\right)}\]
   5. Using strategy rm

   6. Applied sub-neg 11.3

      \[\leadsto \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\left(\frac{y}{z} + \left(-\frac{t}{1 - z}\right)\right)}\right)\]

   7. Applied distribute-lft-in 11.3

      \[\leadsto \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \frac{y}{z} + \sqrt[3]{x} \cdot \left(-\frac{t}{1 - z}\right)\right)}\]

   8. Applied distribute-lft-in 11.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(-\frac{t}{1 - z}\right)\right)}\]

   9. Simplified 2.5

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(-\frac{t}{1 - z}\right)\right)\]

   10. Simplified 2.1

       \[\leadsto \frac{x \cdot y}{z} + \color{blue}{x \cdot \frac{-\frac{t}{1 - z}}{1}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.835758328218971 \cdot 10^{274}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \sqrt[3]{1 - z} - z \cdot \left(\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot t\right)\right)}{z \cdot \sqrt[3]{1 - z}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.0260174088170734 \cdot 10^{99}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \frac{-\frac{t}{1 - z}}{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))