Expression, p6

Average Accuracy: 94.3% → 100.0%

Time: 13.6s

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

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

↓

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

FPCore

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

↓

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

Python

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

↓

def code(a, b, c, d):
	return (b + (c + (a + d))) * 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(b + Float64(c + Float64(a + d))) * 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 = (b + (c + (a + d))) * 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[(b + N[(c + N[(a + d), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in a around 0 95.7%

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

3. Applied egg-rr 97.2%

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

   [Start] 95.7	\[ \left(c + \left(a + \left(d + b\right)\right)\right) \cdot 2 \]
   associate-+r+ [=>] 100.0	\[ \left(c + \color{blue}{\left(\left(a + d\right) + b\right)}\right) \cdot 2 \]
   flip3-+ [=>] 97.2	\[ \left(c + \color{blue}{\frac{{\left(a + d\right)}^{3} + {b}^{3}}{\left(a + d\right) \cdot \left(a + d\right) + \left(b \cdot b - \left(a + d\right) \cdot b\right)}}\right) \cdot 2 \]
   +-commutative [=>] 97.2	\[ \left(c + \frac{\color{blue}{{b}^{3} + {\left(a + d\right)}^{3}}}{\left(a + d\right) \cdot \left(a + d\right) + \left(b \cdot b - \left(a + d\right) \cdot b\right)}\right) \cdot 2 \]
   *-commutative [=>] 97.2	\[ \left(c + \frac{{b}^{3} + {\left(a + d\right)}^{3}}{\left(a + d\right) \cdot \left(a + d\right) + \left(b \cdot b - \color{blue}{b \cdot \left(a + d\right)}\right)}\right) \cdot 2 \]

4. Simplified 97.6%

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

   [Start] 97.2	\[ \left(c + \frac{{b}^{3} + {\left(a + d\right)}^{3}}{\left(a + d\right) \cdot \left(a + d\right) + \left(b \cdot b - b \cdot \left(a + d\right)\right)}\right) \cdot 2 \]
   +-commutative [=>] 97.2	\[ \left(c + \frac{{b}^{3} + {\color{blue}{\left(d + a\right)}}^{3}}{\left(a + d\right) \cdot \left(a + d\right) + \left(b \cdot b - b \cdot \left(a + d\right)\right)}\right) \cdot 2 \]
   fma-def [=>] 97.2	\[ \left(c + \frac{{b}^{3} + {\left(d + a\right)}^{3}}{\color{blue}{\mathsf{fma}\left(a + d, a + d, b \cdot b - b \cdot \left(a + d\right)\right)}}\right) \cdot 2 \]
   +-commutative [=>] 97.2	\[ \left(c + \frac{{b}^{3} + {\left(d + a\right)}^{3}}{\mathsf{fma}\left(\color{blue}{d + a}, a + d, b \cdot b - b \cdot \left(a + d\right)\right)}\right) \cdot 2 \]
   +-commutative [=>] 97.2	\[ \left(c + \frac{{b}^{3} + {\left(d + a\right)}^{3}}{\mathsf{fma}\left(d + a, \color{blue}{d + a}, b \cdot b - b \cdot \left(a + d\right)\right)}\right) \cdot 2 \]
   distribute-lft-out-- [=>] 97.6	\[ \left(c + \frac{{b}^{3} + {\left(d + a\right)}^{3}}{\mathsf{fma}\left(d + a, d + a, \color{blue}{b \cdot \left(b - \left(a + d\right)\right)}\right)}\right) \cdot 2 \]
   +-commutative [=>] 97.6	\[ \left(c + \frac{{b}^{3} + {\left(d + a\right)}^{3}}{\mathsf{fma}\left(d + a, d + a, b \cdot \left(b - \color{blue}{\left(d + a\right)}\right)\right)}\right) \cdot 2 \]

5. Taylor expanded in b around inf 100.0%

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

6. Simplified 100.0%

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

   [Start] 100.0	\[ \left(\left(c + b\right) - -1 \cdot \left(a + d\right)\right) \cdot 2 \]
   +-commutative [=>] 100.0	\[ \left(\color{blue}{\left(b + c\right)} - -1 \cdot \left(a + d\right)\right) \cdot 2 \]
   +-commutative [=>] 100.0	\[ \left(\left(b + c\right) - -1 \cdot \color{blue}{\left(d + a\right)}\right) \cdot 2 \]
   associate--l+ [=>] 100.0	\[ \color{blue}{\left(b + \left(c - -1 \cdot \left(d + a\right)\right)\right)} \cdot 2 \]
   mul-1-neg [=>] 100.0	\[ \left(b + \left(c - \color{blue}{\left(-\left(d + a\right)\right)}\right)\right) \cdot 2 \]
   +-commutative [<=] 100.0	\[ \left(b + \left(c - \left(-\color{blue}{\left(a + d\right)}\right)\right)\right) \cdot 2 \]

7. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	94.3%
Cost	576

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

Alternative 2
Accuracy	95.1%
Cost	576

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

Alternative 3
Accuracy	95.7%
Cost	576

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

Alternative 4
Accuracy	13.8%
Cost	320

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

Alternative 5
Accuracy	6.2%
Cost	192

\[b \cdot 2 \]

Alternative 6
Accuracy	11.6%
Cost	192

\[c \cdot 2 \]

Reproduce

herbie shell --seed 2023137 
(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))