Expression, p6

Average Accuracy: 94.3% → 100.0%

Time: 8.1s

Precision: binary64

Cost: 576

\[\left(\left(\left(-14 \leq a \land a \leq -13\right) \land \left(-3 \leq b \land b \leq -2\right)\right) \land \left(3 \leq c \land c \leq 3.5\right)\right) \land \left(12.5 \leq d \land d \leq 13.5\right)\]

Math

\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]

↓

\[\left(\left(c + \left(a + d\right)\right) + b\right) \cdot 2 \]

FPCore

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))

↓

(FPCore (a b c d) :precision binary64 (* (+ (+ c (+ a d)) b) 2.0))

C

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

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = (a + (b + (c + d))) * 2.0d0
end function

↓

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((c + (a + d)) + b) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

↓

public static double code(double a, double b, double c, double d) {
	return ((c + (a + d)) + b) * 2.0;
}

Python

def code(a, b, c, d):
	return (a + (b + (c + d))) * 2.0

↓

def code(a, b, c, d):
	return ((c + (a + d)) + b) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(a + Float64(b + Float64(c + d))) * 2.0)
end

↓

function code(a, b, c, d)
	return Float64(Float64(Float64(c + Float64(a + d)) + b) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (a + (b + (c + d))) * 2.0;
end

↓

function tmp = code(a, b, c, d)
	tmp = ((c + (a + d)) + b) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(a + N[(b + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

↓

code[a_, b_, c_, d_] := N[(N[(N[(c + N[(a + d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] * 2.0), $MachinePrecision]

Target

Original	94.3%
Target	94.0%
Herbie	100.0%

\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2 \]

Derivation

1. Initial program 94.3%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]

2. Applied egg-rr 93.8%

   \[\leadsto \color{blue}{\left(\left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)}\right)} \cdot 2 \]
   Proof

   [Start] 94.3	\[ \left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
   flip3-+ [=>] 93.8	\[ \color{blue}{\frac{{a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}}{a \cdot a + \left(\left(b + \left(c + d\right)\right) \cdot \left(b + \left(c + d\right)\right) - a \cdot \left(b + \left(c + d\right)\right)\right)}} \cdot 2 \]
   div-inv [=>] 93.8	\[ \color{blue}{\left(\left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right) \cdot \frac{1}{a \cdot a + \left(\left(b + \left(c + d\right)\right) \cdot \left(b + \left(c + d\right)\right) - a \cdot \left(b + \left(c + d\right)\right)\right)}\right)} \cdot 2 \]
   fma-def [=>] 93.8	\[ \left(\left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(c + d\right)\right) - a \cdot \left(b + \left(c + d\right)\right)\right)}}\right) \cdot 2 \]
   distribute-rgt-out-- [=>] 93.8	\[ \left(\left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(a, a, \color{blue}{\left(b + \left(c + d\right)\right) \cdot \left(\left(b + \left(c + d\right)\right) - a\right)}\right)}\right) \cdot 2 \]
   associate--l+ [=>] 93.8	\[ \left(\left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \color{blue}{\left(b + \left(\left(c + d\right) - a\right)\right)}\right)}\right) \cdot 2 \]

3. Simplified 94.4%

   \[\leadsto \color{blue}{\frac{\left({a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}\right) \cdot 1}{\mathsf{fma}\left(a, a, \left(c + \left(d + b\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)}} \cdot 2 \]
   Proof

   [Start] 93.8	\[ \left(\left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)}\right) \cdot 2 \]
   associate-*r/ [=>] 93.8	\[ \color{blue}{\frac{\left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right) \cdot 1}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)}} \cdot 2 \]
   +-commutative [=>] 93.8	\[ \frac{\left({a}^{3} + {\color{blue}{\left(\left(c + d\right) + b\right)}}^{3}\right) \cdot 1}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   associate-+r+ [<=] 94.4	\[ \frac{\left({a}^{3} + {\color{blue}{\left(c + \left(d + b\right)\right)}}^{3}\right) \cdot 1}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   +-commutative [=>] 94.4	\[ \frac{\left({a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}\right) \cdot 1}{\mathsf{fma}\left(a, a, \color{blue}{\left(\left(c + d\right) + b\right)} \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   associate-+r+ [<=] 94.4	\[ \frac{\left({a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}\right) \cdot 1}{\mathsf{fma}\left(a, a, \color{blue}{\left(c + \left(d + b\right)\right)} \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]

4. Applied egg-rr 100.0%

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

   [Start] 94.4	\[ \frac{\left({a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}\right) \cdot 1}{\mathsf{fma}\left(a, a, \left(c + \left(d + b\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   *-commutative [=>] 94.4	\[ \frac{\color{blue}{1 \cdot \left({a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}\right)}}{\mathsf{fma}\left(a, a, \left(c + \left(d + b\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   *-un-lft-identity [<=] 94.4	\[ \frac{\color{blue}{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}}{\mathsf{fma}\left(a, a, \left(c + \left(d + b\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   fma-udef [=>] 94.4	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{\color{blue}{a \cdot a + \left(c + \left(d + b\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)}} \cdot 2 \]
   associate-+r- [=>] 94.4	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{a \cdot a + \left(c + \left(d + b\right)\right) \cdot \color{blue}{\left(\left(b + \left(c + d\right)\right) - a\right)}} \cdot 2 \]
   +-commutative [=>] 94.4	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{a \cdot a + \left(c + \left(d + b\right)\right) \cdot \left(\color{blue}{\left(\left(c + d\right) + b\right)} - a\right)} \cdot 2 \]
   associate-+r+ [<=] 94.4	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{a \cdot a + \left(c + \left(d + b\right)\right) \cdot \left(\color{blue}{\left(c + \left(d + b\right)\right)} - a\right)} \cdot 2 \]
   distribute-rgt-out-- [<=] 94.4	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{a \cdot a + \color{blue}{\left(\left(c + \left(d + b\right)\right) \cdot \left(c + \left(d + b\right)\right) - a \cdot \left(c + \left(d + b\right)\right)\right)}} \cdot 2 \]
   flip3-+ [<=] 95.1	\[ \color{blue}{\left(a + \left(c + \left(d + b\right)\right)\right)} \cdot 2 \]
   associate-+r+ [=>] 94.3	\[ \left(a + \color{blue}{\left(\left(c + d\right) + b\right)}\right) \cdot 2 \]
   associate-+r+ [=>] 94.5	\[ \color{blue}{\left(\left(a + \left(c + d\right)\right) + b\right)} \cdot 2 \]
   +-commutative [<=] 94.5	\[ \left(\color{blue}{\left(\left(c + d\right) + a\right)} + b\right) \cdot 2 \]
   associate-+l+ [=>] 100.0	\[ \left(\color{blue}{\left(c + \left(d + a\right)\right)} + b\right) \cdot 2 \]
   +-commutative [=>] 100.0	\[ \left(\left(c + \color{blue}{\left(a + d\right)}\right) + b\right) \cdot 2 \]

5. Final simplification 100.0%

   \[\leadsto \left(\left(c + \left(a + d\right)\right) + b\right) \cdot 2 \]

Alternatives

Alternative 1
Accuracy	94.3%
Cost	576

\[2 \cdot \left(a + \left(b + \left(c + d\right)\right)\right) \]

Alternative 2
Accuracy	95.7%
Cost	576

\[2 \cdot \left(c + \left(a + \left(d + b\right)\right)\right) \]

Alternative 3
Accuracy	13.8%
Cost	320

\[2 \cdot \left(c + b\right) \]

Alternative 4
Accuracy	6.3%
Cost	192

\[b \cdot 2 \]

Alternative 5
Accuracy	11.6%
Cost	192

\[c \cdot 2 \]

Reproduce

herbie shell --seed 2023126 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (and (and (and (<= -14.0 a) (<= a -13.0)) (and (<= -3.0 b) (<= b -2.0))) (and (<= 3.0 c) (<= c 3.5))) (and (<= 12.5 d) (<= d 13.5)))

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

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