ab-angle->ABCF D

Average Error: 15.9 → 0.3

Time: 2.4s

Precision: binary64

Cost: 6720

Math

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

↓

\[-{\left(a \cdot b\right)}^{2} \]

FPCore

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

↓

(FPCore (a b) :precision binary64 (- (pow (* a b) 2.0)))

C

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

↓

double code(double a, double b) {
	return -pow((a * b), 2.0);
}

Fortran

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

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -((a * b) ** 2.0d0)
end function

Java

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

↓

public static double code(double a, double b) {
	return -Math.pow((a * b), 2.0);
}

Python

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

↓

def code(a, b):
	return -math.pow((a * b), 2.0)

Julia

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

↓

function code(a, b)
	return Float64(-(Float64(a * b) ^ 2.0))
end

MATLAB

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

↓

function tmp = code(a, b)
	tmp = -((a * b) ^ 2.0);
end

Wolfram

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

↓

code[a_, b_] := (-N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision])

Derivation

1. Initial program 15.9

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

2. Simplified 16.0

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

   [Start] 15.9	\[ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   rational_best-simplify-9 [=>] 15.9	\[ \color{blue}{\frac{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}{-1}} \]
   rational_best-simplify-2 [=>] 15.9	\[ \frac{\color{blue}{b \cdot \left(\left(a \cdot a\right) \cdot b\right)}}{-1} \]
   rational_best-simplify-44 [=>] 21.4	\[ \frac{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}}{-1} \]
   rational_best-simplify-47 [=>] 21.4	\[ \color{blue}{\left(b \cdot b\right) \cdot \frac{a \cdot a}{-1}} \]
   rational_best-simplify-47 [=>] 21.4	\[ \left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot \frac{a}{-1}\right)} \]
   rational_best-simplify-44 [=>] 16.0	\[ \color{blue}{a \cdot \left(\left(b \cdot b\right) \cdot \frac{a}{-1}\right)} \]
   rational_best-simplify-8 [=>] 16.0	\[ a \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(-a\right)}\right) \]

3. Taylor expanded in a around 0 21.4

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

4. Simplified 0.3

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

   [Start] 21.4	\[ -1 \cdot \left({a}^{2} \cdot {b}^{2}\right) \]
   rational_best-simplify-2 [=>] 21.4	\[ \color{blue}{\left({a}^{2} \cdot {b}^{2}\right) \cdot -1} \]
   rational_best-simplify-12 [=>] 21.4	\[ \color{blue}{-{a}^{2} \cdot {b}^{2}} \]
   exponential-simplify-27 [=>] 0.3	\[ -\color{blue}{{\left(b \cdot a\right)}^{2}} \]
   rational_best-simplify-2 [=>] 0.3	\[ -{\color{blue}{\left(a \cdot b\right)}}^{2} \]

5. Final simplification 0.3

   \[\leadsto -{\left(a \cdot b\right)}^{2} \]

Alternatives

Alternative 1
Error	3.0
Cost	1032

\[\begin{array}{l} t_0 := a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\\ \mathbf{if}\;a \cdot a \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \cdot a \leq 2 \cdot 10^{+245}:\\ \;\;\;\;-\left(\left(a \cdot a\right) \cdot b\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	3.3
Cost	908

\[\begin{array}{l} t_0 := a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\\ t_1 := b \cdot \left(a \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{if}\;a \leq -3.6 \cdot 10^{+217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-304}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	15.9
Cost	512

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

Reproduce

herbie shell --seed 2023096 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))