Expression, p14

Average Accuracy: 99.9% → 99.9%

Time: 6.1s

Precision: binary64

Cost: 576

\[\left(\left(\left(56789 \leq a \land a \leq 98765\right) \land \left(0 \leq b \land b \leq 1\right)\right) \land \left(0 \leq c \land c \leq 0.0016773\right)\right) \land \left(0 \leq d \land d \leq 0.0016773\right)\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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)
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 = (a * (c + b)) + (a * d)
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	99.9%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9	\[ a \cdot \left(\left(b + c\right) + d\right) \]
   +-commutative [=>] 99.9	\[ a \cdot \left(\color{blue}{\left(c + b\right)} + d\right) \]
   associate-+l+ [=>] 99.9	\[ a \cdot \color{blue}{\left(c + \left(b + d\right)\right)} \]

3. Taylor expanded in d around 0 99.9%

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

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	52.5%
Cost	720

\[\begin{array}{l} \mathbf{if}\;c \leq 5.9 \cdot 10^{-264}:\\ \;\;\;\;a \cdot d\\ \mathbf{elif}\;c \leq 1.18 \cdot 10^{-225}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{-167}:\\ \;\;\;\;a \cdot d\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{-108}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot c\\ \end{array} \]

Alternative 2
Accuracy	77.7%
Cost	717

\[\begin{array}{l} \mathbf{if}\;d \leq 3 \cdot 10^{-131} \lor \neg \left(d \leq 7.4 \cdot 10^{-106}\right) \land d \leq 2.2 \cdot 10^{-98}:\\ \;\;\;\;a \cdot \left(c + b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot d\\ \end{array} \]

Alternative 3
Accuracy	85.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-121}:\\ \;\;\;\;a \cdot \left(c + d\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(c + b\right)\\ \end{array} \]

Alternative 4
Accuracy	84.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-103}:\\ \;\;\;\;a \cdot \left(c + d\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b + d\right)\\ \end{array} \]

Alternative 5
Accuracy	99.9%
Cost	448

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

Alternative 6
Accuracy	52.8%
Cost	324

\[\begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-103}:\\ \;\;\;\;a \cdot c\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 7
Accuracy	35.7%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023141 
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (and (and (and (<= 56789.0 a) (<= a 98765.0)) (and (<= 0.0 b) (<= b 1.0))) (and (<= 0.0 c) (<= c 0.0016773))) (and (<= 0.0 d) (<= d 0.0016773)))

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

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