Expression, p14

Average Error: 0.0 → 0.0

Time: 7.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r62823 = a;
        double r62824 = b;
        double r62825 = c;
        double r62826 = r62824 + r62825;
        double r62827 = d;
        double r62828 = r62826 + r62827;
        double r62829 = r62823 * r62828;
        return r62829;
}

↓

double f(double a, double b, double c, double d) {
        double r62830 = b;
        double r62831 = c;
        double r62832 = r62830 + r62831;
        double r62833 = a;
        double r62834 = d;
        double r62835 = r62833 * r62834;
        double r62836 = fma(r62832, r62833, r62835);
        return r62836;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

3. Applied distribute-lft-in 0.0

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

4. Simplified 0.0

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

6. Applied fma-def 0.0

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

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020043 +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)))