ab-angle->ABCF D

Average Error: 16.8 → 0.3

Time: 6.0s

Precision: binary64

Cost: 704

Math

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

↓

\[\frac{b \cdot \left(a \cdot 4\right)}{\frac{\frac{-4}{a}}{b}} \]

FPCore

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

↓

(FPCore (a b) :precision binary64 (/ (* b (* a 4.0)) (/ (/ -4.0 a) b)))

C

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

↓

double code(double a, double b) {
	return (b * (a * 4.0)) / ((-4.0 / 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 = (b * (a * 4.0d0)) / (((-4.0d0) / 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 (b * (a * 4.0)) / ((-4.0 / a) / b);
}

Python

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

↓

def code(a, b):
	return (b * (a * 4.0)) / ((-4.0 / a) / b)

Julia

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

↓

function code(a, b)
	return Float64(Float64(b * Float64(a * 4.0)) / Float64(Float64(-4.0 / a) / b))
end

MATLAB

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

↓

function tmp = code(a, b)
	tmp = (b * (a * 4.0)) / ((-4.0 / a) / b);
end

Wolfram

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

↓

code[a_, b_] := N[(N[(b * N[(a * 4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.8

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

2. Simplified 16.8

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

   [Start] 16.8	\[ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   rational.json-simplify-10 [=>] 16.8	\[ \color{blue}{\frac{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}{-1}} \]
   rational.json-simplify-49 [=>] 16.8	\[ \color{blue}{b \cdot \frac{\left(a \cdot a\right) \cdot b}{-1}} \]
   rational.json-simplify-2 [=>] 16.8	\[ b \cdot \frac{\color{blue}{b \cdot \left(a \cdot a\right)}}{-1} \]
   rational.json-simplify-49 [=>] 16.8	\[ b \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b}{-1}\right)} \]
   rational.json-simplify-11 [=>] 16.8	\[ b \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(-b\right)}\right) \]

3. Applied egg-rr 17.1

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

4. Applied egg-rr 16.8

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

5. Simplified 5.7

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

   [Start] 16.8	\[ b \cdot \left(2 \cdot \frac{-b}{\frac{\frac{2}{a}}{a}}\right) \]
   rational.json-simplify-5 [<=] 16.8	\[ b \cdot \left(2 \cdot \frac{-b}{\color{blue}{\frac{\frac{2}{a}}{a} - 0}}\right) \]
   rational.json-simplify-50 [<=] 16.8	\[ b \cdot \left(2 \cdot \color{blue}{\frac{b}{0 - \frac{\frac{2}{a}}{a}}}\right) \]
   rational.json-simplify-12 [<=] 16.8	\[ b \cdot \left(2 \cdot \frac{b}{\color{blue}{-\frac{\frac{2}{a}}{a}}}\right) \]
   rational.json-simplify-10 [=>] 16.8	\[ b \cdot \left(2 \cdot \frac{b}{\color{blue}{\frac{\frac{\frac{2}{a}}{a}}{-1}}}\right) \]
   rational.json-simplify-47 [=>] 17.1	\[ b \cdot \left(2 \cdot \frac{b}{\frac{\color{blue}{\frac{2}{a \cdot a}}}{-1}}\right) \]
   rational.json-simplify-44 [=>] 17.1	\[ b \cdot \left(2 \cdot \frac{b}{\color{blue}{\frac{\frac{2}{-1}}{a \cdot a}}}\right) \]
   metadata-eval [=>] 17.1	\[ b \cdot \left(2 \cdot \frac{b}{\frac{\color{blue}{-2}}{a \cdot a}}\right) \]
   rational.json-simplify-61 [<=] 16.8	\[ b \cdot \left(2 \cdot \color{blue}{\frac{a \cdot a}{\frac{-2}{b}}}\right) \]
   rational.json-simplify-49 [=>] 5.8	\[ b \cdot \left(2 \cdot \color{blue}{\left(a \cdot \frac{a}{\frac{-2}{b}}\right)}\right) \]
   rational.json-simplify-43 [=>] 5.7	\[ b \cdot \color{blue}{\left(a \cdot \left(\frac{a}{\frac{-2}{b}} \cdot 2\right)\right)} \]
   rational.json-simplify-61 [=>] 5.7	\[ b \cdot \left(a \cdot \left(\color{blue}{\frac{b}{\frac{-2}{a}}} \cdot 2\right)\right) \]

6. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\frac{b \cdot \left(a \cdot 4\right)}{\frac{\frac{-4}{a}}{b}}} \]

7. Final simplification 0.3

   \[\leadsto \frac{b \cdot \left(a \cdot 4\right)}{\frac{\frac{-4}{a}}{b}} \]

Alternatives

Alternative 1
Error	0.3
Cost	576

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

Alternative 2
Error	0.3
Cost	512

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

Reproduce

herbie shell --seed 2023064 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))