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

Average Error: 4.9 → 4.4

Time: 7.1s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.74740209752635133 \cdot 10^{-102}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.9
Target	4.5
Herbie	4.4

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

Derivation

1. Split input into 2 regimes

2. if z < -2.74740209752635133e-102

   1. Initial program 2.6

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

   3. Applied div-inv 2.6

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

   5. Applied sub-neg 2.6

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

   6. Applied distribute-lft-in 2.6

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

   7. Simplified 6.1

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

   8. Simplified 6.1

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

   10. Applied associate-/l* 2.5

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

   if -2.74740209752635133e-102 < z

   1. Initial program 6.3

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

   3. Applied div-inv 6.3

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

   5. Applied sub-neg 6.3

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

   6. Applied distribute-lft-in 6.3

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

   7. Simplified 5.4

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

   8. Simplified 5.4

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

   10. Applied div-inv 5.5

       \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}} + \left(-\frac{t}{1 - z}\right) \cdot x\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.74740209752635133 \cdot 10^{-102}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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) (neg (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

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