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

Average Error: 4.4 → 1.5

Time: 17.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -3.36200500075499702763179380350065140813 \cdot 10^{241} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.983801995551396888093353762194298415423 \cdot 10^{200}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r301123 = x;
        double r301124 = y;
        double r301125 = z;
        double r301126 = r301124 / r301125;
        double r301127 = t;
        double r301128 = 1.0;
        double r301129 = r301128 - r301125;
        double r301130 = r301127 / r301129;
        double r301131 = r301126 - r301130;
        double r301132 = r301123 * r301131;
        return r301132;
}

↓

double f(double x, double y, double z, double t) {
        double r301133 = y;
        double r301134 = z;
        double r301135 = r301133 / r301134;
        double r301136 = t;
        double r301137 = 1.0;
        double r301138 = r301137 - r301134;
        double r301139 = r301136 / r301138;
        double r301140 = r301135 - r301139;
        double r301141 = -3.362005000754997e+241;
        bool r301142 = r301140 <= r301141;
        double r301143 = 1.983801995551397e+200;
        bool r301144 = r301140 <= r301143;
        double r301145 = !r301144;
        bool r301146 = r301142 || r301145;
        double r301147 = x;
        double r301148 = r301147 * r301133;
        double r301149 = 1.0;
        double r301150 = r301149 / r301134;
        double r301151 = r301148 * r301150;
        double r301152 = -r301139;
        double r301153 = r301147 * r301152;
        double r301154 = r301151 + r301153;
        double r301155 = r301134 / r301133;
        double r301156 = r301147 / r301155;
        double r301157 = r301156 + r301153;
        double r301158 = r301146 ? r301154 : r301157;
        return r301158;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.4
Target	4.0
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -3.362005000754997e+241 or 1.983801995551397e+200 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 21.8

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

   3. Applied sub-neg 21.8

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

   4. Applied distribute-lft-in 21.8

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

   6. Applied div-inv 21.8

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

   7. Applied associate-*r* 0.8

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

   if -3.362005000754997e+241 < (- (/ y z) (/ t (- 1.0 z))) < 1.983801995551397e+200

   1. Initial program 1.5

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

   3. Applied sub-neg 1.5

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

   4. Applied distribute-lft-in 1.5

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

   6. Applied div-inv 1.6

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

   7. Applied associate-*r* 5.9

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

   8. Taylor expanded around 0 5.9

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

   9. Simplified 1.6

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -3.36200500075499702763179380350065140813 \cdot 10^{241} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.983801995551396888093353762194298415423 \cdot 10^{200}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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)))) -7.62322630331204244e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.41339449277023022e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

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