Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.0016773000000000001 \land 0.0 \le d \le 0.0016773000000000001\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)\]
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 r94105 = a;
        double r94106 = b;
        double r94107 = c;
        double r94108 = r94106 + r94107;
        double r94109 = d;
        double r94110 = r94108 + r94109;
        double r94111 = r94105 * r94110;
        return r94111;
}
↓
double f(double a, double b, double c, double d) {
        double r94112 = d;
        double r94113 = a;
        double r94114 = b;
        double r94115 = c;
        double r94116 = r94113 * r94115;
        double r94117 = fma(r94113, r94114, r94116);
        double r94118 = fma(r94112, r94113, r94117);
        return r94118;
}

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 pow10.0

    \[\leadsto a \cdot \color{blue}{{\left(\left(b + c\right) + d\right)}^{1}}\]

  4. Applied pow10.0

    \[\leadsto \color{blue}{{a}^{1}} \cdot {\left(\left(b + c\right) + d\right)}^{1}\]

  5. Applied pow-prod-down0.0

    \[\leadsto \color{blue}{{\left(a \cdot \left(\left(b + c\right) + d\right)\right)}^{1}}\]

  6. Simplified0.0

    \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(d, a, \mathsf{fma}\left(a, b, a \cdot c\right)\right)\right)}}^{1}\]

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020049 +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)))