Average Error: 0.4 → 0.3
Time: 8.3s
Precision: binary64
\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
↓
\[\mathsf{fma}\left(b, 1 + \mathsf{fma}\left(\frac{d}{b} + \left(\frac{c}{b} + \frac{e}{b}\right), 1, 0\right), a\right) \]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\mathsf{fma}\left(b, 1 + \mathsf{fma}\left(\frac{d}{b} + \left(\frac{c}{b} + \frac{e}{b}\right), 1, 0\right), a\right)

(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))
↓
(FPCore (a b c d e)
 :precision binary64
 (fma b (+ 1.0 (fma (+ (/ d b) (+ (/ c b) (/ e b))) 1.0 0.0)) a))
double code(double a, double b, double c, double d, double e) {
	return (((e + d) + c) + b) + a;
}
↓
double code(double a, double b, double c, double d, double e) {
	return fma(b, (1.0 + fma(((d / b) + ((c / b) + (e / b))), 1.0, 0.0)), a);
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Target

Original0.4
Target0.2
Herbie0.3
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e \]

Derivation

  1. Initial program 0.4

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

  2. Applied egg-rr0.5

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

  3. Applied egg-rr0.4

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

  4. Taylor expanded in b around 0 0.4

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

  5. Simplified0.3

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

  6. Applied egg-rr0.3

    \[\leadsto \mathsf{fma}\left(b, 1 + \color{blue}{\mathsf{fma}\left(\frac{d}{b} + \left(\frac{c}{b} + \frac{e}{b}\right), 1, 0\right)}, a\right) \]

  7. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(b, 1 + \mathsf{fma}\left(\frac{d}{b} + \left(\frac{c}{b} + \frac{e}{b}\right), 1, 0\right), a\right) \]

Reproduce

herbie shell --seed 2022127 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (and (and (and (and (and (and (and (and (and (<= 1.0 a) (<= a 2.0)) (<= 2.0 b)) (<= b 4.0)) (<= 4.0 c)) (<= c 8.0)) (<= 8.0 d)) (<= d 16.0)) (<= 16.0 e)) (<= e 32.0))

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

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