ab-angle->ABCF D

Average Error: 16.3 → 0.3

Time: 3.9s

Precision: binary64

Cost: 512

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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) * (b * -a)
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 (a * b) * (b * -a);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 16.3

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

2. Simplified 21.6

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

3. Taylor expanded in a around 0 21.6

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

4. Simplified 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2022228 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))