Expression 4, p15

Average Accuracy: 100.0% → 100.0%

Time: 4.8s

Precision: binary64

Cost: 704

\[\left(5 \leq a \land a \leq 10\right) \land \left(0 \leq b \land b \leq 0.001\right)\]

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a + b) * (a + b)
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * (a + (b + b))) + (b * b)
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 99.8%

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

   [Start] 100.0	\[ \left(a + b\right) \cdot \left(a + b\right) \]
   flip-+ [=>] 100.0	\[ \color{blue}{\frac{a \cdot a - b \cdot b}{a - b}} \cdot \left(a + b\right) \]
   associate-*l/ [=>] 99.8	\[ \color{blue}{\frac{\left(a \cdot a - b \cdot b\right) \cdot \left(a + b\right)}{a - b}} \]

3. Taylor expanded in a around 0 100.0%

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

4. Simplified 100.0%

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

   [Start] 100.0	\[ -1 \cdot \left({a}^{2} \cdot \left(1 + -1 \cdot \frac{b - -1 \cdot b}{b}\right)\right) + \left(a \cdot \left(b - -1 \cdot b\right) + {b}^{2}\right) \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	448

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

Alternative 2
Accuracy	100.0%
Cost	448

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

Alternative 3
Accuracy	98.6%
Cost	192

\[b \cdot b \]

Reproduce

herbie shell --seed 2023151 
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (and (<= 5.0 a) (<= a 10.0)) (and (<= 0.0 b) (<= b 0.001)))

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

  (* (+ a b) (+ a b)))