ab-angle->ABCF D

Average Error: 16.2 → 0.3

Time: 3.6s

Precision: binary64

Cost: 512

Math

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

↓

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

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(Float64(a * b) * Float64(a * 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 16.2

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

2. Simplified 5.3

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

   [Start] 16.2	\[ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 16.2	\[ -\color{blue}{b \cdot \left(\left(a \cdot a\right) \cdot b\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 16.2	\[ -b \cdot \color{blue}{\left(b \cdot \left(a \cdot a\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-89 [=>] 16.2	\[ -b \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{a}{a}\right)\right)}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 5.4	\[ -b \cdot \color{blue}{\left(a \cdot \left(b \cdot \left(a \cdot \frac{a}{a}\right)\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 5.3	\[ -\color{blue}{a \cdot \left(b \cdot \left(b \cdot \left(a \cdot \frac{a}{a}\right)\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 5.3	\[ -a \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(b \cdot \frac{a}{a}\right)\right)}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-6 [=>] 5.3	\[ -a \cdot \left(b \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

3. Applied egg-rr 5.3

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

4. Simplified 0.3

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

   [Start] 5.3	\[ -\left|\left|a\right| \cdot \left(\left(a \cdot b\right) \cdot \left(\frac{a}{a} \cdot \left(b \cdot \frac{a}{a}\right)\right)\right)\right| \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.3	\[ -\left|\color{blue}{\left(a \cdot b\right) \cdot \left(\left|a\right| \cdot \left(\frac{a}{a} \cdot \left(b \cdot \frac{a}{a}\right)\right)\right)}\right| \]
   rational_best_oopsla_all_46_json_45_simplify-72 [=>] 0.3	\[ -\color{blue}{\left|a \cdot b\right| \cdot \left|\left|a\right| \cdot \left(\frac{a}{a} \cdot \left(b \cdot \frac{a}{a}\right)\right)\right|} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ -\left|a \cdot b\right| \cdot \left|\left|a\right| \cdot \color{blue}{\left(\left(b \cdot \frac{a}{a}\right) \cdot \frac{a}{a}\right)}\right| \]
   rational_best_oopsla_all_46_json_45_simplify-30 [=>] 0.3	\[ -\left|a \cdot b\right| \cdot \left|\left|a\right| \cdot \left(\left(b \cdot \frac{a}{a}\right) \cdot \color{blue}{\frac{\left|a\right|}{\left|a\right|}}\right)\right| \]
   rational_best_oopsla_all_46_json_45_simplify-6 [=>] 0.3	\[ -\left|a \cdot b\right| \cdot \left|\color{blue}{\left|a\right| \cdot \left(b \cdot \frac{a}{a}\right)}\right| \]
   rational_best_oopsla_all_46_json_45_simplify-30 [=>] 0.3	\[ -\left|a \cdot b\right| \cdot \left|\left|a\right| \cdot \left(b \cdot \color{blue}{\frac{\left|a\right|}{\left|a\right|}}\right)\right| \]
   rational_best_oopsla_all_46_json_45_simplify-6 [=>] 0.3	\[ -\left|a \cdot b\right| \cdot \left|\color{blue}{\left|a\right| \cdot b}\right| \]
   rational_best_oopsla_all_46_json_45_simplify-80 [<=] 0.3	\[ -\left|a \cdot b\right| \cdot \color{blue}{\left(\left|\left|a\right|\right| \cdot \left|b\right|\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-31 [<=] 0.3	\[ -\left|a \cdot b\right| \cdot \left(\color{blue}{\left|a\right|} \cdot \left|b\right|\right) \]
   rational_best_oopsla_all_46_json_45_simplify-72 [<=] 0.3	\[ -\left|a \cdot b\right| \cdot \color{blue}{\left|a \cdot b\right|} \]
   rational_best_oopsla_all_46_json_45_simplify-80 [=>] 0.3	\[ -\color{blue}{\left|\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right|} \]
   rational_best_oopsla_all_46_json_45_simplify-68 [<=] 0.3	\[ -\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	5.3
Cost	512

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

Reproduce

herbie shell --seed 2023090 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))