Average Error: 6.2 → 1.6
Time: 6.6s
Precision: 64
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\]
↓
\[\mathsf{fma}\left(2, \mathsf{fma}\left(x, y, z \cdot t\right), 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\]
2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)
↓
\mathsf{fma}\left(2, \mathsf{fma}\left(x, y, z \cdot t\right), 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r641885 = 2.0;
        double r641886 = x;
        double r641887 = y;
        double r641888 = r641886 * r641887;
        double r641889 = z;
        double r641890 = t;
        double r641891 = r641889 * r641890;
        double r641892 = r641888 + r641891;
        double r641893 = a;
        double r641894 = b;
        double r641895 = c;
        double r641896 = r641894 * r641895;
        double r641897 = r641893 + r641896;
        double r641898 = r641897 * r641895;
        double r641899 = i;
        double r641900 = r641898 * r641899;
        double r641901 = r641892 - r641900;
        double r641902 = r641885 * r641901;
        return r641902;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r641903 = 2.0;
        double r641904 = x;
        double r641905 = y;
        double r641906 = z;
        double r641907 = t;
        double r641908 = r641906 * r641907;
        double r641909 = fma(r641904, r641905, r641908);
        double r641910 = c;
        double r641911 = b;
        double r641912 = a;
        double r641913 = fma(r641910, r641911, r641912);
        double r641914 = -r641913;
        double r641915 = i;
        double r641916 = r641910 * r641915;
        double r641917 = r641914 * r641916;
        double r641918 = r641903 * r641917;
        double r641919 = fma(r641903, r641909, r641918);
        return r641919;
}

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

Target

Original6.2
Target1.6
Herbie1.6
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right)\]

Derivation

  1. Initial program 6.2

    \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\]

  2. Simplified1.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(x, y, z \cdot t\right), 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)}\]

  3. Final simplification1.6

    \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(x, y, z \cdot t\right), 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (* 2 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i))))

  (* 2 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))