Average Error: 0.0 → 0.0
Time: 7.8s
Precision: 64
\[56789 \le a \le 98765 \land 0 \le b \le 1 \land 0 \le c \le 0.0016773 \land 0 \le d \le 0.0016773\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(b, a, \left(c \cdot a + a \cdot d\right)\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(b, a, \left(c \cdot a + a \cdot d\right)\right)
double f(double a, double b, double c, double d) {
        double r11471848 = a;
        double r11471849 = b;
        double r11471850 = c;
        double r11471851 = r11471849 + r11471850;
        double r11471852 = d;
        double r11471853 = r11471851 + r11471852;
        double r11471854 = r11471848 * r11471853;
        return r11471854;
}
↓
double f(double a, double b, double c, double d) {
        double r11471855 = b;
        double r11471856 = a;
        double r11471857 = c;
        double r11471858 = r11471857 * r11471856;
        double r11471859 = d;
        double r11471860 = r11471856 * r11471859;
        double r11471861 = r11471858 + r11471860;
        double r11471862 = fma(r11471855, r11471856, r11471861);
        return r11471862;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original0.0
Target0.0
Herbie0.0
\[a \cdot b + a \cdot \left(c + d\right)\]

Derivation

  1. Initial program 0.0

    \[a \cdot \left(\left(b + c\right) + d\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.0

    \[\leadsto \color{blue}{a \cdot \left(b + c\right) + a \cdot d}\]
  4. Using strategy rm

  5. Applied distribute-rgt-in0.0

    \[\leadsto \color{blue}{\left(b \cdot a + c \cdot a\right)} + a \cdot d\]

  6. Applied associate-+l+0.0

    \[\leadsto \color{blue}{b \cdot a + \left(c \cdot a + a \cdot d\right)}\]
  7. Using strategy rm

  8. Applied fma-def0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \left(c \cdot a + a \cdot d\right)\right)}\]

  9. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(b, a, \left(c \cdot a + a \cdot d\right)\right)\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :pre (and (<= 56789 a 98765) (<= 0 b 1) (<= 0 c 0.0016773) (<= 0 d 0.0016773))

  :herbie-target
  (+ (* a b) (* a (+ c d)))

  (* a (+ (+ b c) d)))