Average Error: 0.0 → 0.0
Time: 10.1s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.001677300000000000058247850986958837893326 \land 0.0 \le d \le 0.001677300000000000058247850986958837893326\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(a, b + c, d \cdot a\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(a, b + c, d \cdot a\right)
double f(double a, double b, double c, double d) {
        double r4841805 = a;
        double r4841806 = b;
        double r4841807 = c;
        double r4841808 = r4841806 + r4841807;
        double r4841809 = d;
        double r4841810 = r4841808 + r4841809;
        double r4841811 = r4841805 * r4841810;
        return r4841811;
}
↓
double f(double a, double b, double c, double d) {
        double r4841812 = a;
        double r4841813 = b;
        double r4841814 = c;
        double r4841815 = r4841813 + r4841814;
        double r4841816 = d;
        double r4841817 = r4841816 * r4841812;
        double r4841818 = fma(r4841812, r4841815, r4841817);
        return r4841818;
}

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 add-sqr-sqrt0.5

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

  4. Taylor expanded around inf 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :pre (and (<= 56789.0 a 98765.0) (<= 0.0 b 1.0) (<= 0.0 c 0.0016773) (<= 0.0 d 0.0016773))

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

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