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

Average Error: 4.7 → 2.4

Time: 17.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.273245738592455952784286915322615210697 \cdot 10^{61}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.308282208361249760659009261849846214049 \cdot 10^{177}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{\frac{1 - z}{t}}\right) + \frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r299934 = x;
        double r299935 = y;
        double r299936 = z;
        double r299937 = r299935 / r299936;
        double r299938 = t;
        double r299939 = 1.0;
        double r299940 = r299939 - r299936;
        double r299941 = r299938 / r299940;
        double r299942 = r299937 - r299941;
        double r299943 = r299934 * r299942;
        return r299943;
}

↓

double f(double x, double y, double z, double t) {
        double r299944 = y;
        double r299945 = z;
        double r299946 = r299944 / r299945;
        double r299947 = t;
        double r299948 = 1.0;
        double r299949 = r299948 - r299945;
        double r299950 = r299947 / r299949;
        double r299951 = r299946 - r299950;
        double r299952 = -1.273245738592456e+61;
        bool r299953 = r299951 <= r299952;
        double r299954 = x;
        double r299955 = r299954 * r299944;
        double r299956 = r299955 / r299945;
        double r299957 = -r299947;
        double r299958 = r299954 * r299957;
        double r299959 = r299958 / r299949;
        double r299960 = r299956 + r299959;
        double r299961 = 7.30828220836125e+177;
        bool r299962 = r299951 <= r299961;
        double r299963 = r299945 / r299944;
        double r299964 = r299954 / r299963;
        double r299965 = -r299950;
        double r299966 = r299954 * r299965;
        double r299967 = r299964 + r299966;
        double r299968 = r299949 / r299947;
        double r299969 = r299954 / r299968;
        double r299970 = -r299969;
        double r299971 = r299970 + r299956;
        double r299972 = r299962 ? r299967 : r299971;
        double r299973 = r299953 ? r299960 : r299972;
        return r299973;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.7
Target	4.5
Herbie	2.4

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

Derivation

1. Split input into 3 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -1.273245738592456e+61

   1. Initial program 8.6

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

   3. Applied sub-neg 8.6

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

   4. Applied distribute-lft-in 8.6

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

   5. Simplified 3.5

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

   7. Applied distribute-neg-frac 3.5

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

   8. Applied associate-*r/ 5.1

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

   if -1.273245738592456e+61 < (- (/ y z) (/ t (- 1.0 z))) < 7.30828220836125e+177

   1. Initial program 1.6

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

   3. Applied sub-neg 1.6

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

   4. Applied distribute-lft-in 1.6

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

   5. Simplified 6.8

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

   7. Applied associate-/l* 1.7

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

   if 7.30828220836125e+177 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 16.4

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

   3. Applied sub-neg 16.4

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

   4. Applied distribute-lft-in 16.4

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

   5. Simplified 1.3

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

   7. Applied clear-num 1.3

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

   9. Applied pow1 1.3

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

   10. Applied pow1 1.3

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

   11. Applied pow-prod-down 1.3

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

   12. Simplified 1.4

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.273245738592455952784286915322615210697 \cdot 10^{61}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.308282208361249760659009261849846214049 \cdot 10^{177}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{\frac{1 - z}{t}}\right) + \frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))))