Expression, p6

Percentage Accurate: 94.4% → 100.0%

Time: 3.7s

Precision: binary64

Cost: 576

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

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

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(c + Float64(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[(c + N[(N[(a + d), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Target

Original	94.4%
Target	94.1%
Herbie	100.0%

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

Derivation

1. Initial program 94.2%

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

2. Taylor expanded in a around 0 95.6%

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

3. Applied egg-rr 98.1%

   \[\leadsto \left(c + \color{blue}{\frac{\mathsf{fma}\left(a + d, a + d, -b \cdot b\right)}{\left(a + d\right) - b}}\right) \cdot 2 \]
   Step-by-step derivation

   [Start] 95.6%	\[ \left(c + \left(a + \left(d + b\right)\right)\right) \cdot 2 \]
   associate-+r+ [=>] 100.0%	\[ \left(c + \color{blue}{\left(\left(a + d\right) + b\right)}\right) \cdot 2 \]
   flip-+ [=>] 98.2%	\[ \left(c + \color{blue}{\frac{\left(a + d\right) \cdot \left(a + d\right) - b \cdot b}{\left(a + d\right) - b}}\right) \cdot 2 \]
   +-commutative [=>] 98.2%	\[ \left(c + \frac{\color{blue}{\left(d + a\right)} \cdot \left(a + d\right) - b \cdot b}{\left(a + d\right) - b}\right) \cdot 2 \]
   +-commutative [=>] 98.2%	\[ \left(c + \frac{\left(d + a\right) \cdot \color{blue}{\left(d + a\right)} - b \cdot b}{\left(a + d\right) - b}\right) \cdot 2 \]
   fma-neg [=>] 98.1%	\[ \left(c + \frac{\color{blue}{\mathsf{fma}\left(d + a, d + a, -b \cdot b\right)}}{\left(a + d\right) - b}\right) \cdot 2 \]
   +-commutative [<=] 98.1%	\[ \left(c + \frac{\mathsf{fma}\left(\color{blue}{a + d}, d + a, -b \cdot b\right)}{\left(a + d\right) - b}\right) \cdot 2 \]
   +-commutative [<=] 98.1%	\[ \left(c + \frac{\mathsf{fma}\left(a + d, \color{blue}{a + d}, -b \cdot b\right)}{\left(a + d\right) - b}\right) \cdot 2 \]

4. Simplified 95.0%

   \[\leadsto \left(c + \color{blue}{\frac{\left(d + \left(a + b\right)\right) \cdot \left(d - \left(b - a\right)\right)}{d - \left(b - a\right)}}\right) \cdot 2 \]
   Step-by-step derivation

   [Start] 98.1%	\[ \left(c + \frac{\mathsf{fma}\left(a + d, a + d, -b \cdot b\right)}{\left(a + d\right) - b}\right) \cdot 2 \]
   fma-neg [<=] 98.2%	\[ \left(c + \frac{\color{blue}{\left(a + d\right) \cdot \left(a + d\right) - b \cdot b}}{\left(a + d\right) - b}\right) \cdot 2 \]
   difference-of-squares [=>] 99.3%	\[ \left(c + \frac{\color{blue}{\left(\left(a + d\right) + b\right) \cdot \left(\left(a + d\right) - b\right)}}{\left(a + d\right) - b}\right) \cdot 2 \]
   +-commutative [=>] 99.3%	\[ \left(c + \frac{\left(\color{blue}{\left(d + a\right)} + b\right) \cdot \left(\left(a + d\right) - b\right)}{\left(a + d\right) - b}\right) \cdot 2 \]
   associate-+r+ [<=] 94.8%	\[ \left(c + \frac{\color{blue}{\left(d + \left(a + b\right)\right)} \cdot \left(\left(a + d\right) - b\right)}{\left(a + d\right) - b}\right) \cdot 2 \]
   associate--l+ [=>] 94.1%	\[ \left(c + \frac{\left(d + \left(a + b\right)\right) \cdot \color{blue}{\left(a + \left(d - b\right)\right)}}{\left(a + d\right) - b}\right) \cdot 2 \]
   +-commutative [=>] 94.1%	\[ \left(c + \frac{\left(d + \left(a + b\right)\right) \cdot \color{blue}{\left(\left(d - b\right) + a\right)}}{\left(a + d\right) - b}\right) \cdot 2 \]
   associate-+l- [=>] 93.8%	\[ \left(c + \frac{\left(d + \left(a + b\right)\right) \cdot \color{blue}{\left(d - \left(b - a\right)\right)}}{\left(a + d\right) - b}\right) \cdot 2 \]
   associate--l+ [=>] 94.5%	\[ \left(c + \frac{\left(d + \left(a + b\right)\right) \cdot \left(d - \left(b - a\right)\right)}{\color{blue}{a + \left(d - b\right)}}\right) \cdot 2 \]
   +-commutative [=>] 94.5%	\[ \left(c + \frac{\left(d + \left(a + b\right)\right) \cdot \left(d - \left(b - a\right)\right)}{\color{blue}{\left(d - b\right) + a}}\right) \cdot 2 \]
   associate-+l- [=>] 95.0%	\[ \left(c + \frac{\left(d + \left(a + b\right)\right) \cdot \left(d - \left(b - a\right)\right)}{\color{blue}{d - \left(b - a\right)}}\right) \cdot 2 \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(c + \color{blue}{\left(\left(a + d\right) + b\right)}\right) \cdot 2 \]
   Step-by-step derivation

   [Start] 95.0%	\[ \left(c + \frac{\left(d + \left(a + b\right)\right) \cdot \left(d - \left(b - a\right)\right)}{d - \left(b - a\right)}\right) \cdot 2 \]
   associate-/l* [=>] 95.0%	\[ \left(c + \color{blue}{\frac{d + \left(a + b\right)}{\frac{d - \left(b - a\right)}{d - \left(b - a\right)}}}\right) \cdot 2 \]
   *-inverses [=>] 95.0%	\[ \left(c + \frac{d + \left(a + b\right)}{\color{blue}{1}}\right) \cdot 2 \]
   flip3-+ [=>] 95.2%	\[ \left(c + \frac{\color{blue}{\frac{{d}^{3} + {\left(a + b\right)}^{3}}{d \cdot d + \left(\left(a + b\right) \cdot \left(a + b\right) - d \cdot \left(a + b\right)\right)}}}{1}\right) \cdot 2 \]
   associate-/l/ [=>] 95.2%	\[ \left(c + \color{blue}{\frac{{d}^{3} + {\left(a + b\right)}^{3}}{1 \cdot \left(d \cdot d + \left(\left(a + b\right) \cdot \left(a + b\right) - d \cdot \left(a + b\right)\right)\right)}}\right) \cdot 2 \]
   *-un-lft-identity [<=] 95.2%	\[ \left(c + \frac{{d}^{3} + {\left(a + b\right)}^{3}}{\color{blue}{d \cdot d + \left(\left(a + b\right) \cdot \left(a + b\right) - d \cdot \left(a + b\right)\right)}}\right) \cdot 2 \]
   flip3-+ [<=] 95.0%	\[ \left(c + \color{blue}{\left(d + \left(a + b\right)\right)}\right) \cdot 2 \]
   associate-+r+ [=>] 100.0%	\[ \left(c + \color{blue}{\left(\left(d + a\right) + b\right)}\right) \cdot 2 \]
   +-commutative [=>] 100.0%	\[ \left(c + \left(\color{blue}{\left(a + d\right)} + b\right)\right) \cdot 2 \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	576

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

Alternative 2
Accuracy	94.4%
Cost	576

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

Alternative 3
Accuracy	95.7%
Cost	576

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

Alternative 4
Accuracy	14.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq -13.67:\\ \;\;\;\;\left(a + d\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(c + b\right)\\ \end{array} \]

Alternative 5
Accuracy	12.4%
Cost	324

\[\begin{array}{l} \mathbf{if}\;a \leq -13.66:\\ \;\;\;\;b \cdot 2\\ \mathbf{else}:\\ \;\;\;\;c \cdot 2\\ \end{array} \]

Alternative 6
Accuracy	14.0%
Cost	320

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

Alternative 7
Accuracy	6.2%
Cost	192

\[b \cdot 2 \]

Reproduce

herbie shell --seed 2023178 
(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))