Expression, p14

Average Error: 0.0 → 0.2

Time: 4.5s

Precision: binary64

Cost: 448

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

\[ \begin{array}{c}[b, c, d] = \mathsf{sort}([b, c, d])\\ \end{array} \]

Math

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

↓

\[c \cdot a + a \cdot d \]

FPCore

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

↓

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

Python

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

↓

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

Target

Original	0.0
Target	0.0
Herbie	0.2

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in b around 0 0.2

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

3. Taylor expanded in c around 0 0.2

   \[\leadsto \color{blue}{c \cdot a + a \cdot d} \]

4. Final simplification 0.2

   \[\leadsto c \cdot a + a \cdot d \]

Alternatives

Alternative 1
Error	0.2
Cost	320

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

Alternative 2
Error	3.9
Cost	192

\[d \cdot a \]

Reproduce

herbie shell --seed 2023073 
(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)))