Average Error: 4.0 → 1.9
Time: 37.3s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]
\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}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2303005 = x;
        double r2303006 = y;
        double r2303007 = 2.0;
        double r2303008 = z;
        double r2303009 = t;
        double r2303010 = a;
        double r2303011 = r2303009 + r2303010;
        double r2303012 = sqrt(r2303011);
        double r2303013 = r2303008 * r2303012;
        double r2303014 = r2303013 / r2303009;
        double r2303015 = b;
        double r2303016 = c;
        double r2303017 = r2303015 - r2303016;
        double r2303018 = 5.0;
        double r2303019 = 6.0;
        double r2303020 = r2303018 / r2303019;
        double r2303021 = r2303010 + r2303020;
        double r2303022 = 3.0;
        double r2303023 = r2303009 * r2303022;
        double r2303024 = r2303007 / r2303023;
        double r2303025 = r2303021 - r2303024;
        double r2303026 = r2303017 * r2303025;
        double r2303027 = r2303014 - r2303026;
        double r2303028 = r2303007 * r2303027;
        double r2303029 = exp(r2303028);
        double r2303030 = r2303006 * r2303029;
        double r2303031 = r2303005 + r2303030;
        double r2303032 = r2303005 / r2303031;
        return r2303032;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2303033 = x;
        double r2303034 = y;
        double r2303035 = 2.0;
        double r2303036 = c;
        double r2303037 = b;
        double r2303038 = r2303036 - r2303037;
        double r2303039 = 5.0;
        double r2303040 = 6.0;
        double r2303041 = r2303039 / r2303040;
        double r2303042 = t;
        double r2303043 = r2303035 / r2303042;
        double r2303044 = 3.0;
        double r2303045 = r2303043 / r2303044;
        double r2303046 = a;
        double r2303047 = r2303045 - r2303046;
        double r2303048 = r2303041 - r2303047;
        double r2303049 = z;
        double r2303050 = cbrt(r2303049);
        double r2303051 = r2303050 * r2303050;
        double r2303052 = r2303046 + r2303042;
        double r2303053 = sqrt(r2303052);
        double r2303054 = r2303051 * r2303053;
        double r2303055 = r2303042 / r2303050;
        double r2303056 = r2303054 / r2303055;
        double r2303057 = fma(r2303038, r2303048, r2303056);
        double r2303058 = r2303035 * r2303057;
        double r2303059 = exp(r2303058);
        double r2303060 = fma(r2303034, r2303059, r2303033);
        double r2303061 = r2303033 / r2303060;
        return r2303061;
}

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 4.0

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

  2. Simplified1.9

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

  4. Applied add-cube-cbrt1.9

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

  5. Applied *-un-lft-identity1.9

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

  6. Applied times-frac1.9

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

  7. Applied associate-/r*1.9

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

  8. Simplified1.9

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

  9. Final simplification1.9

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))