Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Error: 2.0 → 0.7

Time: 30.4s

Precision: 64

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

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \le 52.279119054873007:\\ \;\;\;\;\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \le 567.847783482109776:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)}^{1}}}{\frac{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}{\frac{\sqrt[3]{x}}{{\left(\sqrt[3]{a}\right)}^{1}}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \left(\sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}} \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\right)\right) \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r544988 = x;
        double r544989 = y;
        double r544990 = z;
        double r544991 = log(r544990);
        double r544992 = r544989 * r544991;
        double r544993 = t;
        double r544994 = 1.0;
        double r544995 = r544993 - r544994;
        double r544996 = a;
        double r544997 = log(r544996);
        double r544998 = r544995 * r544997;
        double r544999 = r544992 + r544998;
        double r545000 = b;
        double r545001 = r544999 - r545000;
        double r545002 = exp(r545001);
        double r545003 = r544988 * r545002;
        double r545004 = r545003 / r544989;
        return r545004;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r545005 = t;
        double r545006 = 1.0;
        double r545007 = r545005 - r545006;
        double r545008 = a;
        double r545009 = log(r545008);
        double r545010 = r545007 * r545009;
        double r545011 = 52.27911905487301;
        bool r545012 = r545010 <= r545011;
        double r545013 = x;
        double r545014 = y;
        double r545015 = z;
        double r545016 = log(r545015);
        double r545017 = -r545016;
        double r545018 = -r545009;
        double r545019 = b;
        double r545020 = fma(r545018, r545005, r545019);
        double r545021 = fma(r545014, r545017, r545020);
        double r545022 = exp(r545021);
        double r545023 = sqrt(r545022);
        double r545024 = r545013 / r545023;
        double r545025 = 1.0;
        double r545026 = pow(r545008, r545006);
        double r545027 = r545025 / r545026;
        double r545028 = r545027 / r545023;
        double r545029 = r545028 / r545014;
        double r545030 = r545024 * r545029;
        double r545031 = 567.8477834821098;
        bool r545032 = r545010 <= r545031;
        double r545033 = cbrt(r545013);
        double r545034 = r545033 * r545033;
        double r545035 = cbrt(r545008);
        double r545036 = r545035 * r545035;
        double r545037 = pow(r545036, r545006);
        double r545038 = r545034 / r545037;
        double r545039 = pow(r545035, r545006);
        double r545040 = r545033 / r545039;
        double r545041 = r545022 / r545040;
        double r545042 = r545038 / r545041;
        double r545043 = r545042 / r545014;
        double r545044 = cbrt(r545029);
        double r545045 = r545044 * r545044;
        double r545046 = r545024 * r545045;
        double r545047 = r545046 * r545044;
        double r545048 = r545032 ? r545043 : r545047;
        double r545049 = r545012 ? r545030 : r545048;
        return r545049;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	2.0
Target	11.2
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;t \lt -0.88458485041274715:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.22883740731:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* (- t 1.0) (log a)) < 52.27911905487301

   1. Initial program 2.5

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

   2. Taylor expanded around inf 2.5

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

   3. Simplified 6.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 6.1

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\color{blue}{1 \cdot y}}\]

   6. Applied add-sqr-sqrt 6.1

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{\color{blue}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}{1 \cdot y}\]

   7. Applied div-inv 6.1

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{1}{{a}^{1}}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{1 \cdot y}\]

   8. Applied times-frac 1.9

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}{1 \cdot y}\]

   9. Applied times-frac 0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{1} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\]

   10. Simplified 0.1

       \[\leadsto \color{blue}{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}\]

   if 52.27911905487301 < (* (- t 1.0) (log a)) < 567.8477834821098

   1. Initial program 1.5

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

   2. Taylor expanded around inf 1.5

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

   3. Simplified 11.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 11.5

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

   6. Applied unpow-prod-down 11.5

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

   7. Applied add-cube-cbrt 11.6

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

   8. Applied times-frac 11.6

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

   9. Applied associate-/l* 1.7

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

   if 567.8477834821098 < (* (- t 1.0) (log a))

   1. Initial program 0.3

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

   2. Taylor expanded around inf 0.3

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

   3. Simplified 14.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 14.2

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\color{blue}{1 \cdot y}}\]

   6. Applied add-sqr-sqrt 14.2

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{\color{blue}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}{1 \cdot y}\]

   7. Applied div-inv 14.2

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{1}{{a}^{1}}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{1 \cdot y}\]

   8. Applied times-frac 0.0

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}{1 \cdot y}\]

   9. Applied times-frac 2.2

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{1} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\]

   10. Simplified 2.2

       \[\leadsto \color{blue}{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 2.2

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

   13. Applied associate-*r* 2.2

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \le 52.279119054873007:\\ \;\;\;\;\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \le 567.847783482109776:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)}^{1}}}{\frac{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}{\frac{\sqrt[3]{x}}{{\left(\sqrt[3]{a}\right)}^{1}}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \left(\sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}} \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\right)\right) \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))