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

Average Error: 3.9 → 6.0

Time: 9.4s

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.55405474180701056 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \log \left(e^{\frac{2}{t \cdot 3}}\right)\right)\right)}}\\ \mathbf{elif}\;t \le 1.379871604667855 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \sqrt[3]{t} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\sqrt[3]{t} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\log \left(e^{\sqrt[3]{t}}\right)} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r415908 = x;
        double r415909 = y;
        double r415910 = 2.0;
        double r415911 = z;
        double r415912 = t;
        double r415913 = a;
        double r415914 = r415912 + r415913;
        double r415915 = sqrt(r415914);
        double r415916 = r415911 * r415915;
        double r415917 = r415916 / r415912;
        double r415918 = b;
        double r415919 = c;
        double r415920 = r415918 - r415919;
        double r415921 = 5.0;
        double r415922 = 6.0;
        double r415923 = r415921 / r415922;
        double r415924 = r415913 + r415923;
        double r415925 = 3.0;
        double r415926 = r415912 * r415925;
        double r415927 = r415910 / r415926;
        double r415928 = r415924 - r415927;
        double r415929 = r415920 * r415928;
        double r415930 = r415917 - r415929;
        double r415931 = r415910 * r415930;
        double r415932 = exp(r415931);
        double r415933 = r415909 * r415932;
        double r415934 = r415908 + r415933;
        double r415935 = r415908 / r415934;
        return r415935;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r415936 = t;
        double r415937 = -3.5540547418070106e-127;
        bool r415938 = r415936 <= r415937;
        double r415939 = x;
        double r415940 = y;
        double r415941 = 2.0;
        double r415942 = z;
        double r415943 = a;
        double r415944 = r415936 + r415943;
        double r415945 = sqrt(r415944);
        double r415946 = r415942 * r415945;
        double r415947 = r415946 / r415936;
        double r415948 = b;
        double r415949 = c;
        double r415950 = r415948 - r415949;
        double r415951 = 5.0;
        double r415952 = 6.0;
        double r415953 = r415951 / r415952;
        double r415954 = r415943 + r415953;
        double r415955 = 3.0;
        double r415956 = r415936 * r415955;
        double r415957 = r415941 / r415956;
        double r415958 = exp(r415957);
        double r415959 = log(r415958);
        double r415960 = r415954 - r415959;
        double r415961 = r415950 * r415960;
        double r415962 = r415947 - r415961;
        double r415963 = r415941 * r415962;
        double r415964 = exp(r415963);
        double r415965 = r415940 * r415964;
        double r415966 = r415939 + r415965;
        double r415967 = r415939 / r415966;
        double r415968 = 1.379871604667855e-48;
        bool r415969 = r415936 <= r415968;
        double r415970 = cbrt(r415936);
        double r415971 = r415970 * r415970;
        double r415972 = r415942 / r415971;
        double r415973 = r415972 * r415945;
        double r415974 = r415943 - r415953;
        double r415975 = r415974 * r415956;
        double r415976 = r415973 * r415975;
        double r415977 = r415943 * r415943;
        double r415978 = r415953 * r415953;
        double r415979 = r415977 - r415978;
        double r415980 = r415979 * r415956;
        double r415981 = r415974 * r415941;
        double r415982 = r415980 - r415981;
        double r415983 = r415950 * r415982;
        double r415984 = r415970 * r415983;
        double r415985 = r415976 - r415984;
        double r415986 = r415970 * r415975;
        double r415987 = r415985 / r415986;
        double r415988 = r415941 * r415987;
        double r415989 = exp(r415988);
        double r415990 = r415940 * r415989;
        double r415991 = r415939 + r415990;
        double r415992 = r415939 / r415991;
        double r415993 = exp(r415970);
        double r415994 = log(r415993);
        double r415995 = r415945 / r415994;
        double r415996 = r415972 * r415995;
        double r415997 = r415954 - r415957;
        double r415998 = r415950 * r415997;
        double r415999 = r415996 - r415998;
        double r416000 = r415941 * r415999;
        double r416001 = exp(r416000);
        double r416002 = r415940 * r416001;
        double r416003 = r415939 + r416002;
        double r416004 = r415939 / r416003;
        double r416005 = r415969 ? r415992 : r416004;
        double r416006 = r415938 ? r415967 : r416005;
        return r416006;
}

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

Bits error versus c

Target

Original	3.9
Target	3.0
Herbie	6.0

\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -3.5540547418070106e-127

   1. Initial program 2.9

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied add-log-exp 7.7

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{\log \left(e^{\frac{2}{t \cdot 3}}\right)}\right)\right)}}\]

   if -3.5540547418070106e-127 < t < 1.379871604667855e-48

   1. Initial program 6.1

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 6.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   4. Applied times-frac 6.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   5. Using strategy rm

   6. Applied flip-+ 9.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5}{6} \cdot \frac{5}{6}}{a - \frac{5}{6}}} - \frac{2}{t \cdot 3}\right)\right)}}\]

   7. Applied frac-sub 9.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   8. Applied associate-*r/ 9.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   9. Applied associate-*r/ 9.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt{t + a}}{\sqrt[3]{t}}} - \frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}\right)}}\]

   10. Applied frac-sub 7.4

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \sqrt[3]{t} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\sqrt[3]{t} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}}\]

   if 1.379871604667855e-48 < t

   1. Initial program 2.5

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 2.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   4. Applied times-frac 0.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   5. Using strategy rm

   6. Applied add-log-exp 4.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\color{blue}{\log \left(e^{\sqrt[3]{t}}\right)}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.55405474180701056 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \log \left(e^{\frac{2}{t \cdot 3}}\right)\right)\right)}}\\ \mathbf{elif}\;t \le 1.379871604667855 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \sqrt[3]{t} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\sqrt[3]{t} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\log \left(e^{\sqrt[3]{t}}\right)} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))