Expression 1, p15

Average Error: 0.4 → 0.3

Time: 5.5s

Precision: 64

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

\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]

Math

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

↓

\[\left(e + d\right) + \mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, b + c\right)\]

C

double f(double a, double b, double c, double d, double e) {
        double r116028 = e;
        double r116029 = d;
        double r116030 = r116028 + r116029;
        double r116031 = c;
        double r116032 = r116030 + r116031;
        double r116033 = b;
        double r116034 = r116032 + r116033;
        double r116035 = a;
        double r116036 = r116034 + r116035;
        return r116036;
}

↓

double f(double a, double b, double c, double d, double e) {
        double r116037 = e;
        double r116038 = d;
        double r116039 = r116037 + r116038;
        double r116040 = a;
        double r116041 = cbrt(r116040);
        double r116042 = r116041 * r116041;
        double r116043 = b;
        double r116044 = c;
        double r116045 = r116043 + r116044;
        double r116046 = fma(r116042, r116041, r116045);
        double r116047 = r116039 + r116046;
        return r116047;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Target

Original	0.4
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 0.4

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

3. Applied associate-+l+ 0.4

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

4. Simplified 0.4

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

6. Applied associate-+l+ 0.3

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

7. Simplified 0.3

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

9. Applied add-cube-cbrt 0.3

   \[\leadsto \left(e + d\right) + \left(\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}} + \left(b + c\right)\right)\]

10. Applied fma-def 0.3

    \[\leadsto \left(e + d\right) + \color{blue}{\mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, b + c\right)}\]

11. Final simplification 0.3

    \[\leadsto \left(e + d\right) + \mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, b + c\right)\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

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

  (+ (+ (+ (+ e d) c) b) a))