ab-angle->ABCF D

Average Error: 16.3 → 0.3

Time: 5.9s

Precision: binary64

Cost: 576

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

def code(a, b):
	return (b * a) / ((-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(b * a) / 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 = (b * a) / ((-1.0 / 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[(N[(-1.0 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.3

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

2. Simplified 16.3

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

   [Start] 16.3	\[ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   rational.json-simplify-10 [=>] 16.3	\[ \color{blue}{\frac{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}{-1}} \]
   rational.json-simplify-49 [=>] 16.3	\[ \color{blue}{b \cdot \frac{\left(a \cdot a\right) \cdot b}{-1}} \]
   rational.json-simplify-2 [=>] 16.3	\[ b \cdot \frac{\color{blue}{b \cdot \left(a \cdot a\right)}}{-1} \]
   rational.json-simplify-49 [=>] 16.3	\[ b \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b}{-1}\right)} \]
   rational.json-simplify-11 [=>] 16.3	\[ b \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(-b\right)}\right) \]

3. Applied egg-rr 16.5

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

4. Applied egg-rr 5.5

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

5. Simplified 5.5

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

   [Start] 5.5	\[ b \cdot \left(a \cdot \frac{a + a}{\frac{-2}{b}}\right) \]
   rational.json-simplify-61 [=>] 5.5	\[ b \cdot \left(a \cdot \color{blue}{\frac{b}{\frac{-2}{a + a}}}\right) \]
   metadata-eval [<=] 5.5	\[ b \cdot \left(a \cdot \frac{b}{\frac{\color{blue}{-1 + -1}}{a + a}}\right) \]
   rational.json-simplify-35 [<=] 5.5	\[ b \cdot \left(a \cdot \frac{b}{\color{blue}{\frac{-1}{a}}}\right) \]
   rational.json-simplify-61 [<=] 5.5	\[ b \cdot \left(a \cdot \color{blue}{\frac{a}{\frac{-1}{b}}}\right) \]

6. Applied egg-rr 0.3

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

7. Taylor expanded in b around 0 0.3

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

8. Simplified 0.3

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

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

9. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	576

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

Alternative 2
Error	0.3
Cost	512

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

Reproduce

herbie shell --seed 2023074 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))