ab-angle->ABCF D

Average Error: 16.0 → 0.3

Time: 5.9s

Precision: binary64

Cost: 640

Math

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

↓

\[-\left(a \cdot b\right) \cdot \frac{b}{\frac{1}{a}} \]

FPCore

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

↓

(FPCore (a b) :precision binary64 (- (* (* a b) (/ b (/ 1.0 a)))))

C

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

↓

double code(double a, double b) {
	return -((a * b) * (b / (1.0 / 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 / (1.0d0 / 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 / (1.0 / a)));
}

Python

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

↓

def code(a, b):
	return -((a * b) * (b / (1.0 / a)))

Julia

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

↓

function code(a, b)
	return Float64(-Float64(Float64(a * b) * Float64(b / Float64(1.0 / a))))
end

MATLAB

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

↓

function tmp = code(a, b)
	tmp = -((a * b) * (b / (1.0 / 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 / N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 16.0

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

2. Simplified 5.0

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

   [Start] 16.0	\[ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   rational.json-simplify-10 [=>] 16.0	\[ \color{blue}{\frac{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}{-1}} \]
   rational.json-simplify-49 [=>] 16.0	\[ \color{blue}{b \cdot \frac{\left(a \cdot a\right) \cdot b}{-1}} \]
   rational.json-simplify-2 [=>] 16.0	\[ b \cdot \frac{\color{blue}{b \cdot \left(a \cdot a\right)}}{-1} \]
   rational.json-simplify-49 [=>] 16.0	\[ b \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b}{-1}\right)} \]
   rational.json-simplify-43 [<=] 16.0	\[ \color{blue}{\frac{b}{-1} \cdot \left(b \cdot \left(a \cdot a\right)\right)} \]
   rational.json-simplify-43 [=>] 5.3	\[ \frac{b}{-1} \cdot \color{blue}{\left(a \cdot \left(a \cdot b\right)\right)} \]
   rational.json-simplify-43 [=>] 5.0	\[ \color{blue}{a \cdot \left(\left(a \cdot b\right) \cdot \frac{b}{-1}\right)} \]
   rational.json-simplify-11 [=>] 5.0	\[ a \cdot \left(\left(a \cdot b\right) \cdot \color{blue}{\left(-b\right)}\right) \]
   rational.json-simplify-8 [=>] 5.0	\[ a \cdot \left(\left(a \cdot b\right) \cdot \color{blue}{\left(b \cdot -1\right)}\right) \]
   rational.json-simplify-43 [=>] 5.0	\[ a \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \left(a \cdot b\right)\right)\right)} \]
   rational.json-simplify-2 [=>] 5.0	\[ a \cdot \left(b \cdot \left(-1 \cdot \color{blue}{\left(b \cdot a\right)}\right)\right) \]
   rational.json-simplify-43 [<=] 5.0	\[ a \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)}\right) \]
   rational.json-simplify-2 [<=] 5.0	\[ a \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(b \cdot -1\right)}\right)\right) \]
   rational.json-simplify-8 [<=] 5.0	\[ a \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right)\right) \]

3. Applied egg-rr 0.3

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

4. Applied egg-rr 0.7

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

5. Applied egg-rr 0.3

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

6. Final simplification 0.3

   \[\leadsto -\left(a \cdot b\right) \cdot \frac{b}{\frac{1}{a}} \]

Alternatives

Alternative 1
Error	0.3
Cost	512

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

Reproduce

herbie shell --seed 2023065 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))