Average Error: 3.6 → 0
Time: 5.4s
Precision: binary64
Cost: 7104
\[-14 \leq a \land a \leq -13 \land -3 \leq b \land b \leq -2 \land 3 \leq c \land c \leq 3.5 \land 12.5 \leq d \land d \leq 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[2 \cdot \left(\left(\left(b + c\right) + \left(a + d\right)\right) \cdot \sqrt[3]{1}\right)\]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
2 \cdot \left(\left(\left(b + c\right) + \left(a + d\right)\right) \cdot \sqrt[3]{1}\right)
(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))
↓
(FPCore (a b c d)
 :precision binary64
 (* 2.0 (* (+ (+ b c) (+ a d)) (cbrt 1.0))))
double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}
↓
double code(double a, double b, double c, double d) {
	return 2.0 * (((b + c) + (a + d)) * cbrt(1.0));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.6
Target3.8
Herbie0
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Alternatives

Alternative 1
Error22.9
Cost39616
\[2 \cdot \left(\sqrt[3]{\sqrt{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}}\right)\]
Alternative 2
Error3.1
Cost39616
\[2 \cdot \sqrt[3]{{\left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right)}^{6}}\right)}^{3}}\]
Alternative 3
Error3.3
Cost27520
\[2 \cdot \sqrt[3]{{\left(\frac{{a}^{3} + {\left(\left(b + c\right) + d\right)}^{3}}{a \cdot a + \left(\left(b + c\right) + d\right) \cdot \left(\left(\left(b + c\right) + d\right) - a\right)}\right)}^{3}}\]
Alternative 4
Error22.9
Cost26816
\[2 \cdot \left(\sqrt[3]{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{1.5}}\right)\]
Alternative 5
Error3.1
Cost26752
\[2 \cdot \left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right)}^{6}}\right)\]
Alternative 6
Error1.5
Cost26752
\[2 \cdot \left(\sqrt[3]{\left(b + c\right) + \left(a + d\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{\left(b + c\right) + \left(a + d\right)}\right)}^{6}}\right)\]
Alternative 7
Error2.9
Cost26304
\[2 \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}\right)}^{3}}\]
Alternative 8
Error22.8
Cost26240
\[2 \cdot \sqrt[3]{{\left(e^{\log \left(a + \left(\left(b + c\right) + d\right)\right)}\right)}^{3}}\]
Alternative 9
Error3.0
Cost26240
\[2 \cdot \sqrt[3]{{\log \left(e^{a + \left(\left(b + c\right) + d\right)}\right)}^{3}}\]
Alternative 10
Error3.8
Cost20800
\[2 \cdot \left(\sqrt[3]{a + \left(b + \left(c + d\right)\right)} \cdot \left(\sqrt[3]{a + \left(b + \left(c + d\right)\right)} \cdot \sqrt[3]{a + \left(b + \left(c + d\right)\right)}\right)\right)\]
Alternative 11
Error3.0
Cost20800
\[2 \cdot \left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)} \cdot \left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)} \cdot \sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right)\right)\]
Alternative 12
Error5.8
Cost20544
\[2 \cdot \left(a + \sqrt[3]{b + \left(c + d\right)} \cdot \left(\sqrt[3]{b + \left(c + d\right)} \cdot \sqrt[3]{b + \left(c + d\right)}\right)\right)\]
Alternative 13
Error1.5
Cost15296
\[2 \cdot \frac{\left(a - d\right) \cdot \left(\left(a + d\right) \cdot \left(b \cdot b + c \cdot \left(c - b\right)\right) + \left({b}^{3} + {c}^{3}\right)\right)}{\left(a - d\right) \cdot \left(b \cdot b + c \cdot \left(c - b\right)\right)}\]
Alternative 14
Error2.6
Cost15296
\[2 \cdot \frac{\left(b - c\right) \cdot \left(\left({a}^{3} + {d}^{3}\right) + \left(b + c\right) \cdot \left(a \cdot a + d \cdot \left(d - a\right)\right)\right)}{\left(b - c\right) \cdot \left(a \cdot a + d \cdot \left(d - a\right)\right)}\]
Alternative 15
Error3.9
Cost14656
\[2 \cdot \frac{{\left(b + \left(c + d\right)\right)}^{3} + {a}^{3}}{a \cdot a + \left(b + \left(c + d\right)\right) \cdot \left(\left(b + \left(c + d\right)\right) - a\right)}\]
Alternative 16
Error3.2
Cost14656
\[2 \cdot \frac{{a}^{3} + {\left(\left(b + c\right) + d\right)}^{3}}{a \cdot a + \left(\left(b + c\right) + d\right) \cdot \left(\left(\left(b + c\right) + d\right) - a\right)}\]
Alternative 17
Error3.4
Cost14464
\[2 \cdot \sqrt[3]{{\left(\frac{a \cdot a - \left(\left(b + c\right) + d\right) \cdot \left(\left(b + c\right) + d\right)}{a - \left(\left(b + c\right) + d\right)}\right)}^{3}}\]
Alternative 18
Error4.5
Cost14400
\[2 \cdot \left(a + \frac{{b}^{3} + {\left(c + d\right)}^{3}}{b \cdot b + \left(c + d\right) \cdot \left(\left(c + d\right) - b\right)}\right)\]
Alternative 19
Error23.4
Cost13888
\[2 \cdot \left(\sqrt{a + \left(b + \left(c + d\right)\right)} \cdot \sqrt{a + \left(b + \left(c + d\right)\right)}\right)\]
Alternative 20
Error22.8
Cost13888
\[2 \cdot \left(\sqrt{a + \left(\left(b + c\right) + d\right)} \cdot \sqrt{a + \left(\left(b + c\right) + d\right)}\right)\]
Alternative 21
Error21.3
Cost13888
\[2 \cdot \left(\sqrt{\left(b + c\right) + \left(a + d\right)} \cdot \sqrt{\left(b + c\right) + \left(a + d\right)}\right)\]
Alternative 22
Error4.2
Cost13760
\[2 \cdot \left(a + \sqrt{b + \left(c + d\right)} \cdot \sqrt{b + \left(c + d\right)}\right)\]
Alternative 23
Error21.1
Cost13504
\[2 \cdot {\left({\left(\left(b + c\right) + \left(a + d\right)\right)}^{3}\right)}^{0.3333333333333333}\]
Alternative 24
Error4.2
Cost13440
\[2 \cdot \left(a + \sqrt[3]{{\left(b + \left(c + d\right)\right)}^{3}}\right)\]
Alternative 25
Error3.7
Cost13440
\[2 \cdot \sqrt[3]{{\left(a + \left(b + \left(c + d\right)\right)\right)}^{3}}\]
Alternative 26
Error2.9
Cost13440
\[2 \cdot \sqrt[3]{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}\]
Alternative 27
Error0.6
Cost13440
\[2 \cdot \sqrt[3]{{\left(\left(b + c\right) + \left(a + d\right)\right)}^{3}}\]
Alternative 28
Error4.6
Cost13376
\[2 \cdot \left(a + e^{\log \left(b + \left(c + d\right)\right)}\right)\]
Alternative 29
Error23.3
Cost13376
\[2 \cdot e^{\log \left(a + \left(b + \left(c + d\right)\right)\right)}\]
Alternative 30
Error3.8
Cost13376
\[2 \cdot \log \left(e^{a + \left(b + \left(c + d\right)\right)}\right)\]
Alternative 31
Error22.7
Cost13376
\[2 \cdot e^{\log \left(a + \left(\left(b + c\right) + d\right)\right)}\]
Alternative 32
Error3.0
Cost13376
\[2 \cdot \log \left(e^{a + \left(\left(b + c\right) + d\right)}\right)\]
Alternative 33
Error21.1
Cost13376
\[2 \cdot e^{\log \left(\left(b + c\right) + \left(a + d\right)\right)}\]
Alternative 34
Error0.9
Cost13376
\[2 \cdot \log \left(e^{\left(b + c\right) + \left(a + d\right)}\right)\]
Alternative 35
Error4.0
Cost1600
\[2 \cdot \frac{a \cdot a - \left(b + \left(c + d\right)\right) \cdot \left(b + \left(c + d\right)\right)}{a - \left(b + \left(c + d\right)\right)}\]
Alternative 36
Error0.6
Cost1600
\[2 \cdot \frac{\left(a + d\right) \cdot \left(a + d\right) - \left(b + c\right) \cdot \left(b + c\right)}{\left(a + d\right) - \left(b + c\right)}\]
Alternative 37
Error0.1
Cost1600
\[2 \cdot \frac{\left(\left(b - c\right) \cdot \left(a - d\right)\right) \cdot \left(\left(b + c\right) + \left(a + d\right)\right)}{\left(b - c\right) \cdot \left(a - d\right)}\]
Alternative 38
Error3.3
Cost1600
\[2 \cdot \frac{a \cdot a - \left(\left(b + c\right) + d\right) \cdot \left(\left(b + c\right) + d\right)}{a - \left(\left(b + c\right) + d\right)}\]
Alternative 39
Error4.2
Cost1344
\[2 \cdot \left(a + \frac{b \cdot b - \left(c + d\right) \cdot \left(c + d\right)}{b - \left(c + d\right)}\right)\]
Alternative 40
Error2.8
Cost576
\[\left(a + \left(\left(b + c\right) + d\right)\right) \cdot 2\]
Alternative 41
Error3.8
Cost576
\[2 \cdot \left(\left(c + d\right) + \left(a + b\right)\right)\]
Alternative 42
Error3.6
Cost576
\[2 \cdot \left(a + \left(b + \left(c + d\right)\right)\right)\]
Alternative 43
Error0
Cost576
\[2 \cdot \left(\left(b + c\right) + \left(a + d\right)\right)\]
Alternative 44
Error55.2
Cost64
\[1\]
Alternative 45
Error62.0
Cost64
\[0\]
Alternative 46
Error59.3
Cost64
\[-1\]

Error

Derivation

  1. Initial program 3.6

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

  3. Applied associate-+r+_binary64_39992.8

    \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
  4. Using strategy rm

  5. Applied add-cbrt-cube_binary64_41032.9

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)}} \cdot 2\]

  6. Simplified2.9

    \[\leadsto \sqrt[3]{\color{blue}{{\left(a + \left(d + \left(b + c\right)\right)\right)}^{3}}} \cdot 2\]
  7. Using strategy rm

  8. Applied associate-+r+_binary64_39990.6

    \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(a + d\right) + \left(b + c\right)\right)}}^{3}} \cdot 2\]
  9. Using strategy rm

  10. Applied *-un-lft-identity_binary64_40670.6

    \[\leadsto \sqrt[3]{{\color{blue}{\left(1 \cdot \left(\left(a + d\right) + \left(b + c\right)\right)\right)}}^{3}} \cdot 2\]

  11. Applied unpow-prod-down_binary64_41460.6

    \[\leadsto \sqrt[3]{\color{blue}{{1}^{3} \cdot {\left(\left(a + d\right) + \left(b + c\right)\right)}^{3}}} \cdot 2\]

  12. Applied cbrt-prod_binary64_40980.6

    \[\leadsto \color{blue}{\left(\sqrt[3]{{1}^{3}} \cdot \sqrt[3]{{\left(\left(a + d\right) + \left(b + c\right)\right)}^{3}}\right)} \cdot 2\]

  13. Simplified0.6

    \[\leadsto \left(\color{blue}{\sqrt[3]{1}} \cdot \sqrt[3]{{\left(\left(a + d\right) + \left(b + c\right)\right)}^{3}}\right) \cdot 2\]

  14. Simplified0

    \[\leadsto \left(\sqrt[3]{1} \cdot \color{blue}{\left(\left(a + d\right) + \left(b + c\right)\right)}\right) \cdot 2\]

  15. Simplified0

    \[\leadsto \color{blue}{2 \cdot \left(\left(\left(b + c\right) + \left(a + d\right)\right) \cdot \sqrt[3]{1}\right)}\]

  16. Final simplification0

    \[\leadsto 2 \cdot \left(\left(\left(b + c\right) + \left(a + d\right)\right) \cdot \sqrt[3]{1}\right)\]

Reproduce

herbie shell --seed 2021042 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (<= -14.0 a -13.0) (<= -3.0 b -2.0) (<= 3.0 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2.0) (* (+ c d) 2.0))

  (* (+ a (+ b (+ c d))) 2.0))