Average Error: 29.1 → 29.1
Time: 3.8m
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right)\right), t\right)}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(a + y\right), y, b\right)\right), c\right)\right), y, i\right)}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\frac{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right)\right), t\right)}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(a + y\right), y, b\right)\right), c\right)\right), y, i\right)}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r21449024 = x;
        double r21449025 = y;
        double r21449026 = r21449024 * r21449025;
        double r21449027 = z;
        double r21449028 = r21449026 + r21449027;
        double r21449029 = r21449028 * r21449025;
        double r21449030 = 27464.7644705;
        double r21449031 = r21449029 + r21449030;
        double r21449032 = r21449031 * r21449025;
        double r21449033 = 230661.510616;
        double r21449034 = r21449032 + r21449033;
        double r21449035 = r21449034 * r21449025;
        double r21449036 = t;
        double r21449037 = r21449035 + r21449036;
        double r21449038 = a;
        double r21449039 = r21449025 + r21449038;
        double r21449040 = r21449039 * r21449025;
        double r21449041 = b;
        double r21449042 = r21449040 + r21449041;
        double r21449043 = r21449042 * r21449025;
        double r21449044 = c;
        double r21449045 = r21449043 + r21449044;
        double r21449046 = r21449045 * r21449025;
        double r21449047 = i;
        double r21449048 = r21449046 + r21449047;
        double r21449049 = r21449037 / r21449048;
        return r21449049;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r21449050 = y;
        double r21449051 = x;
        double r21449052 = z;
        double r21449053 = fma(r21449050, r21449051, r21449052);
        double r21449054 = 27464.7644705;
        double r21449055 = fma(r21449050, r21449053, r21449054);
        double r21449056 = 230661.510616;
        double r21449057 = fma(r21449050, r21449055, r21449056);
        double r21449058 = t;
        double r21449059 = fma(r21449050, r21449057, r21449058);
        double r21449060 = a;
        double r21449061 = r21449060 + r21449050;
        double r21449062 = b;
        double r21449063 = fma(r21449061, r21449050, r21449062);
        double r21449064 = c;
        double r21449065 = fma(r21449050, r21449063, r21449064);
        double r21449066 = i;
        double r21449067 = fma(r21449065, r21449050, r21449066);
        double r21449068 = r21449059 / r21449067;
        return r21449068;
}

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

Bits error versus i

Derivation

  1. Initial program 29.1

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
  2. Using strategy rm

  3. Applied clear-num29.3

    \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}}}\]
  4. Using strategy rm

  5. Applied div-inv29.4

    \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right) \cdot \frac{1}{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}}}\]

  6. Applied associate-/r*29.2

    \[\leadsto \color{blue}{\frac{\frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}{\frac{1}{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}}}\]

  7. Simplified29.2

    \[\leadsto \frac{\frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}{\color{blue}{\frac{1}{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(y, x, z\right)\right), y, 27464.7644705\right)\right), y, 230661.510616\right)\right), t\right)}}}\]
  8. Using strategy rm

  9. Applied div-inv29.2

    \[\leadsto \frac{\frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}{\color{blue}{1 \cdot \frac{1}{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(y, x, z\right)\right), y, 27464.7644705\right)\right), y, 230661.510616\right)\right), t\right)}}}\]

  10. Applied *-un-lft-identity29.2

    \[\leadsto \frac{\frac{1}{\color{blue}{1 \cdot \left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right)}}}{1 \cdot \frac{1}{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(y, x, z\right)\right), y, 27464.7644705\right)\right), y, 230661.510616\right)\right), t\right)}}\]

  11. Applied add-cube-cbrt29.2

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right)}}{1 \cdot \frac{1}{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(y, x, z\right)\right), y, 27464.7644705\right)\right), y, 230661.510616\right)\right), t\right)}}\]

  12. Applied times-frac29.2

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}}{1 \cdot \frac{1}{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(y, x, z\right)\right), y, 27464.7644705\right)\right), y, 230661.510616\right)\right), t\right)}}\]

  13. Applied times-frac29.2

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{1} \cdot \frac{\frac{\sqrt[3]{1}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}{\frac{1}{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(y, x, z\right)\right), y, 27464.7644705\right)\right), y, 230661.510616\right)\right), t\right)}}}\]

  14. Simplified29.2

    \[\leadsto \color{blue}{1} \cdot \frac{\frac{\sqrt[3]{1}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}{\frac{1}{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(y, x, z\right)\right), y, 27464.7644705\right)\right), y, 230661.510616\right)\right), t\right)}}\]

  15. Simplified29.1

    \[\leadsto 1 \cdot \color{blue}{\frac{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right)\right), t\right)}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(a + y\right), y, b\right)\right), c\right)\right), y, i\right)}}\]

  16. Final simplification29.1

    \[\leadsto \frac{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right)\right), t\right)}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(a + y\right), y, b\right)\right), c\right)\right), y, i\right)}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))