ab-angle->ABCF D

Average Error: 15.9 → 0.3

Time: 5.7s

Precision: binary64

Cost: 576

\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]

Math

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

↓

\[\frac{a \cdot b}{\frac{\frac{-1}{b}}{a}} \]

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

MATLAB

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

↓

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

Derivation

1. Initial program 15.9

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

2. Simplified 5.4

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

   [Start] 15.9	\[ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   rational.json-simplify-10 [=>] 15.9	\[ \color{blue}{\frac{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}{-1}} \]
   rational.json-simplify-49 [=>] 15.9	\[ \color{blue}{b \cdot \frac{\left(a \cdot a\right) \cdot b}{-1}} \]
   rational.json-simplify-2 [=>] 15.9	\[ b \cdot \frac{\color{blue}{b \cdot \left(a \cdot a\right)}}{-1} \]
   rational.json-simplify-49 [=>] 15.9	\[ b \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b}{-1}\right)} \]
   rational.json-simplify-43 [<=] 15.9	\[ \color{blue}{\frac{b}{-1} \cdot \left(b \cdot \left(a \cdot a\right)\right)} \]
   rational.json-simplify-43 [=>] 5.1	\[ \frac{b}{-1} \cdot \color{blue}{\left(a \cdot \left(a \cdot b\right)\right)} \]
   rational.json-simplify-43 [=>] 5.4	\[ \color{blue}{a \cdot \left(\left(a \cdot b\right) \cdot \frac{b}{-1}\right)} \]
   rational.json-simplify-11 [=>] 5.4	\[ a \cdot \left(\left(a \cdot b\right) \cdot \color{blue}{\left(-b\right)}\right) \]
   rational.json-simplify-8 [=>] 5.4	\[ a \cdot \left(\left(a \cdot b\right) \cdot \color{blue}{\left(b \cdot -1\right)}\right) \]
   rational.json-simplify-43 [=>] 5.4	\[ a \cdot \color{blue}{\left(b \cdot \left(-1 \cdot \left(a \cdot b\right)\right)\right)} \]
   rational.json-simplify-2 [=>] 5.4	\[ a \cdot \left(b \cdot \left(-1 \cdot \color{blue}{\left(b \cdot a\right)}\right)\right) \]
   rational.json-simplify-43 [<=] 5.4	\[ a \cdot \left(b \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)}\right) \]
   rational.json-simplify-2 [<=] 5.4	\[ a \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(b \cdot -1\right)}\right)\right) \]
   rational.json-simplify-8 [<=] 5.4	\[ a \cdot \left(b \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right)\right) \]

3. Applied egg-rr 15.9

   \[\leadsto \color{blue}{b \cdot \left(\left(-b\right) \cdot \left(a \cdot a\right)\right) + 0} \]

4. Simplified 16.2

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

   [Start] 15.9	\[ b \cdot \left(\left(-b\right) \cdot \left(a \cdot a\right)\right) + 0 \]
   rational.json-simplify-4 [=>] 15.9	\[ \color{blue}{b \cdot \left(\left(-b\right) \cdot \left(a \cdot a\right)\right)} \]
   rational.json-simplify-43 [<=] 21.7	\[ \color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot \left(-b\right)\right)} \]
   rational.json-simplify-2 [=>] 21.7	\[ \color{blue}{\left(b \cdot \left(-b\right)\right) \cdot \left(a \cdot a\right)} \]
   rational.json-simplify-43 [<=] 16.2	\[ \color{blue}{a \cdot \left(\left(b \cdot \left(-b\right)\right) \cdot a\right)} \]
   rational.json-simplify-43 [<=] 16.2	\[ \color{blue}{a \cdot \left(a \cdot \left(b \cdot \left(-b\right)\right)\right)} \]

5. Applied egg-rr 16.6

   \[\leadsto a \cdot \color{blue}{\frac{-a}{\frac{1}{b \cdot b}}} \]

6. Simplified 5.4

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

   [Start] 16.6	\[ a \cdot \frac{-a}{\frac{1}{b \cdot b}} \]
   rational.json-simplify-61 [=>] 16.2	\[ a \cdot \color{blue}{\frac{b \cdot b}{\frac{1}{-a}}} \]
   rational.json-simplify-49 [=>] 5.4	\[ a \cdot \color{blue}{\left(b \cdot \frac{b}{\frac{1}{-a}}\right)} \]
   rational.json-simplify-10 [=>] 5.4	\[ a \cdot \left(b \cdot \frac{b}{\frac{1}{\color{blue}{\frac{a}{-1}}}}\right) \]
   rational.json-simplify-61 [=>] 5.4	\[ a \cdot \left(b \cdot \frac{b}{\color{blue}{\frac{-1}{\frac{a}{1}}}}\right) \]
   rational.json-simplify-7 [=>] 5.4	\[ a \cdot \left(b \cdot \frac{b}{\frac{-1}{\color{blue}{a}}}\right) \]
   rational.json-simplify-61 [=>] 5.4	\[ a \cdot \left(b \cdot \color{blue}{\frac{a}{\frac{-1}{b}}}\right) \]

7. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\frac{a \cdot b}{\frac{-1}{a \cdot b}}} \]

8. Applied egg-rr 0.3

   \[\leadsto \frac{a \cdot b}{\color{blue}{\frac{-1}{a \cdot b} + 0}} \]

9. Simplified 0.3

   \[\leadsto \frac{a \cdot b}{\color{blue}{\frac{\frac{-1}{b}}{a}}} \]
   Proof

   [Start] 0.3	\[ \frac{a \cdot b}{\frac{-1}{a \cdot b} + 0} \]
   rational.json-simplify-4 [=>] 0.3	\[ \frac{a \cdot b}{\color{blue}{\frac{-1}{a \cdot b}}} \]
   rational.json-simplify-46 [=>] 0.3	\[ \frac{a \cdot b}{\color{blue}{\frac{\frac{-1}{a}}{b}}} \]
   rational.json-simplify-44 [=>] 0.3	\[ \frac{a \cdot b}{\color{blue}{\frac{\frac{-1}{b}}{a}}} \]

10. Final simplification 0.3

    \[\leadsto \frac{a \cdot b}{\frac{\frac{-1}{b}}{a}} \]

Alternatives

Alternative 1
Error	0.3
Cost	512

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

Reproduce

herbie shell --seed 2023073 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))