Expression, p6

Average Error: 3.6 → 0

Time: 7.6s

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

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

↓

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

FPCore

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

↓

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

Python

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

↓

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

Target

Original	3.6
Target	3.8
Herbie	0

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

Derivation

1. Initial program 3.6

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

2. Applied egg-rr 3.9

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

3. Simplified 3.6

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

   [Start] 3.9	\[ \left(\left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)}\right) \cdot 2 \]
   *-commutative [<=] 3.9	\[ \color{blue}{\left(\frac{1}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot \left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right)\right)} \cdot 2 \]
   associate-*l/ [=>] 3.9	\[ \color{blue}{\frac{1 \cdot \left({a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}\right)}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)}} \cdot 2 \]
   *-lft-identity [=>] 3.9	\[ \frac{\color{blue}{{a}^{3} + {\left(b + \left(c + d\right)\right)}^{3}}}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   +-commutative [=>] 3.9	\[ \frac{{a}^{3} + {\color{blue}{\left(\left(c + d\right) + b\right)}}^{3}}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   associate-+r+ [<=] 3.6	\[ \frac{{a}^{3} + {\color{blue}{\left(c + \left(d + b\right)\right)}}^{3}}{\mathsf{fma}\left(a, a, \left(b + \left(c + d\right)\right) \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   +-commutative [=>] 3.6	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{\mathsf{fma}\left(a, a, \color{blue}{\left(\left(c + d\right) + b\right)} \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   associate-+r+ [<=] 3.6	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{\mathsf{fma}\left(a, a, \color{blue}{\left(c + \left(d + b\right)\right)} \cdot \left(b + \left(\left(c + d\right) - a\right)\right)\right)} \cdot 2 \]
   associate-+r- [=>] 3.6	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{\mathsf{fma}\left(a, a, \left(c + \left(d + b\right)\right) \cdot \color{blue}{\left(\left(b + \left(c + d\right)\right) - a\right)}\right)} \cdot 2 \]
   +-commutative [=>] 3.6	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{\mathsf{fma}\left(a, a, \left(c + \left(d + b\right)\right) \cdot \left(\color{blue}{\left(\left(c + d\right) + b\right)} - a\right)\right)} \cdot 2 \]
   associate--l+ [=>] 3.6	\[ \frac{{a}^{3} + {\left(c + \left(d + b\right)\right)}^{3}}{\mathsf{fma}\left(a, a, \left(c + \left(d + b\right)\right) \cdot \color{blue}{\left(\left(c + d\right) + \left(b - a\right)\right)}\right)} \cdot 2 \]

4. Taylor expanded in a around 0 2.7

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

5. Simplified 0

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

   [Start] 2.7	\[ \left(c + \left(a + \left(d + b\right)\right)\right) \cdot 2 \]
   +-commutative [=>] 2.7	\[ \left(c + \color{blue}{\left(\left(d + b\right) + a\right)}\right) \cdot 2 \]
   +-commutative [=>] 2.7	\[ \left(c + \left(\color{blue}{\left(b + d\right)} + a\right)\right) \cdot 2 \]
   associate-+l+ [=>] 0.0	\[ \left(c + \color{blue}{\left(b + \left(d + a\right)\right)}\right) \cdot 2 \]
   +-commutative [<=] 0.0	\[ \left(c + \left(b + \color{blue}{\left(a + d\right)}\right)\right) \cdot 2 \]
   associate-+l+ [<=] 0	\[ \color{blue}{\left(\left(c + b\right) + \left(a + d\right)\right)} \cdot 2 \]
   +-commutative [=>] 0	\[ \left(\left(c + b\right) + \color{blue}{\left(d + a\right)}\right) \cdot 2 \]

6. Final simplification 0

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

Alternatives

Alternative 1
Error	3.6
Cost	576

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

Alternative 2
Error	2.7
Cost	576

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

Alternative 3
Error	55.2
Cost	320

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

Alternative 4
Error	60.0
Cost	192

\[b \cdot 2 \]

Alternative 5
Error	56.6
Cost	192

\[c \cdot 2 \]

Reproduce

herbie shell --seed 2023041 
(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))