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

Average Error: 4.5 → 1.5

Time: 6.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.10707959431727866 \cdot 10^{304}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \left(\left(-\frac{t}{\sqrt[3]{1 - z}}\right) + \frac{t}{\sqrt[3]{1 - z}}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r470052 = x;
        double r470053 = y;
        double r470054 = z;
        double r470055 = r470053 / r470054;
        double r470056 = t;
        double r470057 = 1.0;
        double r470058 = r470057 - r470054;
        double r470059 = r470056 / r470058;
        double r470060 = r470055 - r470059;
        double r470061 = r470052 * r470060;
        return r470061;
}

↓

double f(double x, double y, double z, double t) {
        double r470062 = y;
        double r470063 = z;
        double r470064 = r470062 / r470063;
        double r470065 = t;
        double r470066 = 1.0;
        double r470067 = r470066 - r470063;
        double r470068 = r470065 / r470067;
        double r470069 = r470064 - r470068;
        double r470070 = -inf.0;
        bool r470071 = r470069 <= r470070;
        double r470072 = 1.1070795943172787e+304;
        bool r470073 = r470069 <= r470072;
        double r470074 = !r470073;
        bool r470075 = r470071 || r470074;
        double r470076 = x;
        double r470077 = r470076 * r470062;
        double r470078 = r470077 / r470063;
        double r470079 = 1.0;
        double r470080 = cbrt(r470067);
        double r470081 = r470065 / r470080;
        double r470082 = r470080 * r470080;
        double r470083 = r470079 / r470082;
        double r470084 = r470081 * r470083;
        double r470085 = -r470084;
        double r470086 = fma(r470079, r470064, r470085);
        double r470087 = -r470081;
        double r470088 = r470087 + r470081;
        double r470089 = r470083 * r470088;
        double r470090 = r470086 + r470089;
        double r470091 = r470076 * r470090;
        double r470092 = r470075 ? r470078 : r470091;
        return r470092;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.5
Target	4.2
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or 1.1070795943172787e+304 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 61.8

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

   3. Applied add-cube-cbrt 61.8

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

   4. Applied *-un-lft-identity 61.8

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

   5. Applied times-frac 61.8

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

   6. Applied add-sqr-sqrt 62.1

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

   7. Applied prod-diff 62.1

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

   8. Simplified 61.8

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

   9. Simplified 61.8

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

   10. Taylor expanded around 0 1.3

       \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]

   if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < 1.1070795943172787e+304

   1. Initial program 1.2

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

   3. Applied add-cube-cbrt 1.5

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

   4. Applied *-un-lft-identity 1.5

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

   5. Applied times-frac 1.5

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

   6. Applied add-sqr-sqrt 28.9

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

   7. Applied prod-diff 28.9

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

   8. Simplified 1.5

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

   9. Simplified 1.5

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.10707959431727866 \cdot 10^{304}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \left(\left(-\frac{t}{\sqrt[3]{1 - z}}\right) + \frac{t}{\sqrt[3]{1 - z}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))