ab-angle->ABCF D

Average Error: 16.3 → 0.3

Time: 3.8s

Precision: binary64

Cost: 512

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

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(a * Float64(-b)))
end

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 16.3

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

2. Taylor expanded in a around 0 22.0

   \[\leadsto -\color{blue}{{a}^{2} \cdot {b}^{2}} \]

3. Simplified 0.3

   \[\leadsto -\color{blue}{{\left(a \cdot b\right)}^{2}} \]
   Proof
   (neg.f64 (pow.f64 (*.f64 a b) 2)): 0 points increase in error, 0 points decrease in error
   (neg.f64 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 a b) (*.f64 a b)))): 0 points increase in error, 0 points decrease in error
   (neg.f64 (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 a a) (*.f64 b b)))): 2 points increase in error, 0 points decrease in error
   (neg.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 b b))): 0 points increase in error, 2 points decrease in error
   (neg.f64 (*.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	38.4
Cost	448

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

Alternative 2
Error	38.4
Cost	448

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

Reproduce

herbie shell --seed 2022340 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))