Average Error: 3.8 → 2.1
Time: 10.7s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\]
\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)}}
↓
\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r96030 = x;
        double r96031 = y;
        double r96032 = 2.0;
        double r96033 = z;
        double r96034 = t;
        double r96035 = a;
        double r96036 = r96034 + r96035;
        double r96037 = sqrt(r96036);
        double r96038 = r96033 * r96037;
        double r96039 = r96038 / r96034;
        double r96040 = b;
        double r96041 = c;
        double r96042 = r96040 - r96041;
        double r96043 = 5.0;
        double r96044 = 6.0;
        double r96045 = r96043 / r96044;
        double r96046 = r96035 + r96045;
        double r96047 = 3.0;
        double r96048 = r96034 * r96047;
        double r96049 = r96032 / r96048;
        double r96050 = r96046 - r96049;
        double r96051 = r96042 * r96050;
        double r96052 = r96039 - r96051;
        double r96053 = r96032 * r96052;
        double r96054 = exp(r96053);
        double r96055 = r96031 * r96054;
        double r96056 = r96030 + r96055;
        double r96057 = r96030 / r96056;
        return r96057;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r96058 = x;
        double r96059 = y;
        double r96060 = 2.0;
        double r96061 = z;
        double r96062 = t;
        double r96063 = a;
        double r96064 = r96062 + r96063;
        double r96065 = sqrt(r96064);
        double r96066 = r96065 / r96062;
        double r96067 = b;
        double r96068 = c;
        double r96069 = r96067 - r96068;
        double r96070 = 5.0;
        double r96071 = 6.0;
        double r96072 = r96070 / r96071;
        double r96073 = r96063 + r96072;
        double r96074 = 3.0;
        double r96075 = r96062 * r96074;
        double r96076 = r96060 / r96075;
        double r96077 = r96073 - r96076;
        double r96078 = r96069 * r96077;
        double r96079 = -r96078;
        double r96080 = fma(r96061, r96066, r96079);
        double r96081 = -r96069;
        double r96082 = r96081 + r96069;
        double r96083 = r96077 * r96082;
        double r96084 = r96080 + r96083;
        double r96085 = r96060 * r96084;
        double r96086 = exp(r96085);
        double r96087 = r96059 * r96086;
        double r96088 = r96058 + r96087;
        double r96089 = r96058 / r96088;
        return r96089;
}

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

Derivation

  1. Initial program 3.8

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

  3. Applied *-un-lft-identity3.8

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

  4. Applied times-frac3.2

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

  5. Applied prod-diff21.4

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \mathsf{fma}\left(-\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right), b - c, \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right)\right)}}}\]

  6. Simplified21.4

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

  7. Simplified2.1

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

  8. Final simplification2.1

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))