Average Error: 28.8 → 28.8
Time: 35.5s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)\]
\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{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r1923064 = x;
        double r1923065 = y;
        double r1923066 = r1923064 * r1923065;
        double r1923067 = z;
        double r1923068 = r1923066 + r1923067;
        double r1923069 = r1923068 * r1923065;
        double r1923070 = 27464.7644705;
        double r1923071 = r1923069 + r1923070;
        double r1923072 = r1923071 * r1923065;
        double r1923073 = 230661.510616;
        double r1923074 = r1923072 + r1923073;
        double r1923075 = r1923074 * r1923065;
        double r1923076 = t;
        double r1923077 = r1923075 + r1923076;
        double r1923078 = a;
        double r1923079 = r1923065 + r1923078;
        double r1923080 = r1923079 * r1923065;
        double r1923081 = b;
        double r1923082 = r1923080 + r1923081;
        double r1923083 = r1923082 * r1923065;
        double r1923084 = c;
        double r1923085 = r1923083 + r1923084;
        double r1923086 = r1923085 * r1923065;
        double r1923087 = i;
        double r1923088 = r1923086 + r1923087;
        double r1923089 = r1923077 / r1923088;
        return r1923089;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r1923090 = 1.0;
        double r1923091 = a;
        double r1923092 = y;
        double r1923093 = r1923091 + r1923092;
        double r1923094 = b;
        double r1923095 = fma(r1923093, r1923092, r1923094);
        double r1923096 = c;
        double r1923097 = fma(r1923095, r1923092, r1923096);
        double r1923098 = i;
        double r1923099 = fma(r1923097, r1923092, r1923098);
        double r1923100 = r1923090 / r1923099;
        double r1923101 = x;
        double r1923102 = z;
        double r1923103 = fma(r1923092, r1923101, r1923102);
        double r1923104 = 27464.7644705;
        double r1923105 = fma(r1923092, r1923103, r1923104);
        double r1923106 = 230661.510616;
        double r1923107 = fma(r1923092, r1923105, r1923106);
        double r1923108 = t;
        double r1923109 = fma(r1923092, r1923107, r1923108);
        double r1923110 = r1923100 * r1923109;
        return r1923110;
}

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 28.8

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

  3. Applied clear-num29.0

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

  4. Simplified29.0

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

  6. Applied div-inv29.0

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

  8. Applied associate-*r/29.0

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

  9. Applied associate-/r/28.8

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

  10. Simplified28.8

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

  11. Final simplification28.8

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

Reproduce

herbie shell --seed 2019152 +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)))