Average Error: 6.5 → 1.8
Time: 22.9s
Precision: 64
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\]
↓
\[2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, -\left(i \cdot c\right) \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)\]
2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)
↓
2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, -\left(i \cdot c\right) \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r756181 = 2.0;
        double r756182 = x;
        double r756183 = y;
        double r756184 = r756182 * r756183;
        double r756185 = z;
        double r756186 = t;
        double r756187 = r756185 * r756186;
        double r756188 = r756184 + r756187;
        double r756189 = a;
        double r756190 = b;
        double r756191 = c;
        double r756192 = r756190 * r756191;
        double r756193 = r756189 + r756192;
        double r756194 = r756193 * r756191;
        double r756195 = i;
        double r756196 = r756194 * r756195;
        double r756197 = r756188 - r756196;
        double r756198 = r756181 * r756197;
        return r756198;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r756199 = 2.0;
        double r756200 = y;
        double r756201 = x;
        double r756202 = z;
        double r756203 = t;
        double r756204 = i;
        double r756205 = c;
        double r756206 = r756204 * r756205;
        double r756207 = b;
        double r756208 = a;
        double r756209 = fma(r756205, r756207, r756208);
        double r756210 = r756206 * r756209;
        double r756211 = -r756210;
        double r756212 = fma(r756202, r756203, r756211);
        double r756213 = fma(r756200, r756201, r756212);
        double r756214 = r756199 * r756213;
        return r756214;
}

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

Original6.5
Target1.8
Herbie1.8
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right)\]

Derivation

  1. Initial program 6.5

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

  2. Simplified6.5

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

  3. Taylor expanded around inf 8.8

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

  4. Simplified6.5

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

  5. Taylor expanded around inf 8.8

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

  6. Simplified1.8

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

  7. Final simplification1.8

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

Reproduce

herbie shell --seed 2019350 +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))))