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

Average Error: 4.8 → 3.1

Time: 16.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.429492536410088594385230239827933374577 \cdot 10^{62}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \le 2.286764800019861885579620504653914413686 \cdot 10^{-57}:\\ \;\;\;\;\left(x \cdot t\right) \cdot \frac{-1}{1 - z} + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r244953 = x;
        double r244954 = y;
        double r244955 = z;
        double r244956 = r244954 / r244955;
        double r244957 = t;
        double r244958 = 1.0;
        double r244959 = r244958 - r244955;
        double r244960 = r244957 / r244959;
        double r244961 = r244956 - r244960;
        double r244962 = r244953 * r244961;
        return r244962;
}

↓

double f(double x, double y, double z, double t) {
        double r244963 = x;
        double r244964 = -1.4294925364100886e+62;
        bool r244965 = r244963 <= r244964;
        double r244966 = z;
        double r244967 = y;
        double r244968 = r244966 / r244967;
        double r244969 = r244963 / r244968;
        double r244970 = t;
        double r244971 = 1.0;
        double r244972 = r244971 - r244966;
        double r244973 = r244970 / r244972;
        double r244974 = -r244973;
        double r244975 = r244963 * r244974;
        double r244976 = r244969 + r244975;
        double r244977 = 2.286764800019862e-57;
        bool r244978 = r244963 <= r244977;
        double r244979 = r244963 * r244970;
        double r244980 = -1.0;
        double r244981 = r244980 / r244972;
        double r244982 = r244979 * r244981;
        double r244983 = r244963 * r244967;
        double r244984 = r244983 / r244966;
        double r244985 = r244982 + r244984;
        double r244986 = r244967 / r244966;
        double r244987 = 1.0;
        double r244988 = r244972 / r244970;
        double r244989 = r244987 / r244988;
        double r244990 = r244986 - r244989;
        double r244991 = r244963 * r244990;
        double r244992 = r244978 ? r244985 : r244991;
        double r244993 = r244965 ? r244976 : r244992;
        return r244993;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.8
Target	4.3
Herbie	3.1

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.4294925364100886e+62

   1. Initial program 4.0

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

   3. Applied sub-neg 4.0

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

   4. Applied distribute-lft-in 4.0

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

   5. Simplified 10.8

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

   7. Applied associate-/l* 4.2

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

   if -1.4294925364100886e+62 < x < 2.286764800019862e-57

   1. Initial program 5.8

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

   3. Applied sub-neg 5.8

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

   4. Applied distribute-lft-in 5.8

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

   5. Simplified 2.7

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

   7. Applied div-inv 2.7

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

   8. Applied distribute-rgt-neg-in 2.7

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

   9. Applied associate-*r* 2.8

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

   if 2.286764800019862e-57 < x

   1. Initial program 2.8

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

   3. Applied clear-num 3.0

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.429492536410088594385230239827933374577 \cdot 10^{62}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \le 2.286764800019861885579620504653914413686 \cdot 10^{-57}:\\ \;\;\;\;\left(x \cdot t\right) \cdot \frac{-1}{1 - z} + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

  (* x (- (/ y z) (/ t (- 1 z)))))