Expression 1, p15

Average Error: 0.4 → 0.2

Time: 4.9s

Precision: binary64

Cost: 576

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

Math

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

↓

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))

↓

(FPCore (a b c d e) :precision binary64 (+ e (+ d (+ a (+ c b)))))

C

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

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	0.2

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

Alternatives

Alternative 1
Error	1.2
Cost	20800

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

Alternative 2
Error	1.1
Cost	20544

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

Alternative 3
Error	0.5
Cost	15296

\[a + \frac{\left(e - d\right) \cdot \left(\left(e + d\right) \cdot \left(c \cdot c + b \cdot \left(b - c\right)\right) + \left({c}^{3} + {b}^{3}\right)\right)}{\left(e - d\right) \cdot \left(c \cdot c + b \cdot \left(b - c\right)\right)}\]

Alternative 4
Error	0.6
Cost	14912

\[\frac{{\left(b + \left(\left(e + d\right) + c\right)\right)}^{3} + {a}^{3}}{a \cdot a + \left(b + \left(\left(e + d\right) + c\right)\right) \cdot \left(\left(b + \left(\left(e + d\right) + c\right)\right) - a\right)}\]

Alternative 5
Error	0.6
Cost	14656

\[a + \frac{{\left(\left(e + d\right) + c\right)}^{3} + {b}^{3}}{b \cdot b + \left(\left(e + d\right) + c\right) \cdot \left(\left(\left(e + d\right) + c\right) - b\right)}\]

Alternative 6
Error	0.6
Cost	14400

\[a + \left(b + \frac{{\left(e + d\right)}^{3} + {c}^{3}}{c \cdot c + \left(e + d\right) \cdot \left(\left(e + d\right) - c\right)}\right)\]

Alternative 7
Error	0.6
Cost	14016

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

Alternative 8
Error	0.6
Cost	13888

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

Alternative 9
Error	0.6
Cost	13760

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

Alternative 10
Error	0.7
Cost	13440

\[\sqrt[3]{{\left(a + \left(b + \left(\left(e + d\right) + c\right)\right)\right)}^{3}}\]

Alternative 11
Error	0.6
Cost	13440

\[a + \left(b + \sqrt[3]{{\left(\left(e + d\right) + c\right)}^{3}}\right)\]

Alternative 12
Error	0.7
Cost	13440

\[a + \sqrt[3]{{\left(b + \left(\left(e + d\right) + c\right)\right)}^{3}}\]

Alternative 13
Error	0.8
Cost	13376

\[e^{\log \left(a + \left(b + \left(\left(e + d\right) + c\right)\right)\right)}\]

Alternative 14
Error	0.7
Cost	13376

\[a + \left(b + e^{\log \left(\left(e + d\right) + c\right)}\right)\]

Alternative 15
Error	0.8
Cost	13376

\[a + e^{\log \left(b + \left(\left(e + d\right) + c\right)\right)}\]

Alternative 16
Error	0.5
Cost	1856

\[\frac{\left(e + d\right) \cdot \left(e + d\right) - \left(a + \left(c + b\right)\right) \cdot \left(a + \left(c + b\right)\right)}{\left(e + d\right) - \left(a + \left(c + b\right)\right)}\]

Alternative 17
Error	0.5
Cost	1856

\[\frac{\left(b + \left(\left(e + d\right) + c\right)\right) \cdot \left(b + \left(\left(e + d\right) + c\right)\right) - a \cdot a}{\left(b + \left(\left(e + d\right) + c\right)\right) - a}\]

Alternative 18
Error	0.5
Cost	1600

\[a + \frac{\left(\left(e + d\right) + c\right) \cdot \left(\left(e + d\right) + c\right) - b \cdot b}{\left(\left(e + d\right) + c\right) - b}\]

Alternative 19
Error	0.5
Cost	1600

\[a + \frac{\left(e + d\right) \cdot \left(e + d\right) - \left(c + b\right) \cdot \left(c + b\right)}{\left(e + d\right) - \left(c + b\right)}\]

Alternative 20
Error	0.6
Cost	1600

\[a + \frac{\left(c - b\right) \cdot \left(\left(\left(e + d\right) + \left(c + b\right)\right) \cdot \left(e - d\right)\right)}{\left(c - b\right) \cdot \left(e - d\right)}\]

Alternative 21
Error	0.5
Cost	1344

\[a + \left(b + \frac{\left(e + d\right) \cdot \left(e + d\right) - c \cdot c}{\left(e + d\right) - c}\right)\]

Alternative 22
Error	0.2
Cost	576

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

Alternative 23
Error	0.3
Cost	576

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

Alternative 24
Error	0.4
Cost	576

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

Alternative 25
Error	0.3
Cost	576

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

Alternative 26
Error	0.3
Cost	576

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

Alternative 27
Error	0.4
Cost	576

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

Alternative 28
Error	0.3
Cost	576

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

Alternative 29
Error	0.3
Cost	576

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

Alternative 30
Error	0.3
Cost	576

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

Alternative 31
Error	54.4
Cost	64

\[1\]

Alternative 32
Error	62.0
Cost	64

\[0\]

Alternative 33
Error	63.0
Cost	64

\[-1\]

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+_binary64_670 0.3

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

5. Applied associate-+l+_binary64_670 0.3

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

6. Simplified 0.3

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

8. Applied associate-+l+_binary64_670 0.2

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

9. Simplified 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2021042 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (<= 1.0 a 2.0 b 4.0 c 8.0 d 16.0 e 32.0)

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

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