\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.0016773000000000001 \land 0.0 \le d \le 0.0016773000000000001\]

\[a \cdot \left(\left(b + c\right) + d\right)\]

↓

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

a \cdot \left(\left(b + c\right) + d\right)

↓

\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)

double f(double a, double b, double c, double d) {
        double r85103 = a;
        double r85104 = b;
        double r85105 = c;
        double r85106 = r85104 + r85105;
        double r85107 = d;
        double r85108 = r85106 + r85107;
        double r85109 = r85103 * r85108;
        return r85109;
}

↓

double f(double a, double b, double c, double d) {
        double r85110 = d;
        double r85111 = a;
        double r85112 = b;
        double r85113 = c;
        double r85114 = r85111 * r85113;
        double r85115 = fma(r85111, r85112, r85114);
        double r85116 = fma(r85110, r85111, r85115);
        return r85116;
}

Error

The X axis uses an exponential scale — Bits error versus `a`

Original	0.0
Target	0.0
Herbie	0.0

Derivation

Initial program 0.0
\[a \cdot \left(\left(b + c\right) + d\right)\]
Taylor expanded around inf 0.0
\[\leadsto \color{blue}{d \cdot a + \left(a \cdot b + a \cdot c\right)}\]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)}\]
Final simplification0.0
\[\leadsto \mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (<= 56789 a 98765) (<= 0.0 b 1) (<= 0.0 c 0.0016773) (<= 0.0 d 0.0016773))

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

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

Error

Target

Derivation

Reproduce