Average Error: 6.4 → 1.7
Time: 7.8s
Precision: 64
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\]
↓
\[\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\]
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(t, z, x \cdot y\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 r571825 = 2.0;
        double r571826 = x;
        double r571827 = y;
        double r571828 = r571826 * r571827;
        double r571829 = z;
        double r571830 = t;
        double r571831 = r571829 * r571830;
        double r571832 = r571828 + r571831;
        double r571833 = a;
        double r571834 = b;
        double r571835 = c;
        double r571836 = r571834 * r571835;
        double r571837 = r571833 + r571836;
        double r571838 = r571837 * r571835;
        double r571839 = i;
        double r571840 = r571838 * r571839;
        double r571841 = r571832 - r571840;
        double r571842 = r571825 * r571841;
        return r571842;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r571843 = 2.0;
        double r571844 = t;
        double r571845 = z;
        double r571846 = x;
        double r571847 = y;
        double r571848 = r571846 * r571847;
        double r571849 = fma(r571844, r571845, r571848);
        double r571850 = c;
        double r571851 = b;
        double r571852 = a;
        double r571853 = fma(r571850, r571851, r571852);
        double r571854 = -r571853;
        double r571855 = i;
        double r571856 = r571850 * r571855;
        double r571857 = r571854 * r571856;
        double r571858 = r571843 * r571857;
        double r571859 = fma(r571843, r571849, r571858);
        return r571859;
}

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

Original6.4
Target1.7
Herbie1.7
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right)\]

Derivation

  1. Initial program 6.4

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

  2. Simplified1.7

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

  3. Taylor expanded around inf 1.7

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

  4. Simplified1.7

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

  5. Final simplification1.7

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

Reproduce

herbie shell --seed 2020062 +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))))