ab-angle->ABCF D

Average Error: 16.0 → 0.3

Time: 3.5s

Precision: binary64

Cost: 512

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 16.0

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

2. Simplified 16.4

   \[\leadsto \color{blue}{a \cdot \left(\left(b \cdot b\right) \cdot \left(-a\right)\right)} \]
   Proof
   (*.f64 a (*.f64 (*.f64 b b) (neg.f64 a))): 0 points increase in error, 0 points decrease in error
   (*.f64 a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 b b) a)))): 0 points increase in error, 0 points decrease in error
   (*.f64 a (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 a (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (*.f64 a (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a b) b)))): 11 points increase in error, 54 points decrease in error
   (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a (*.f64 (*.f64 a b) b)))): 0 points increase in error, 0 points decrease in error
   (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a (*.f64 a b)) b))): 41 points increase in error, 48 points decrease in error
   (neg.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a a) b)) b)): 55 points increase in error, 14 points decrease in error

3. Applied egg-rr 0.3

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

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	38.4
Cost	448

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

Reproduce

herbie shell --seed 2022338 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))