ab-angle->ABCF D

Average Error: 16.0 → 0.3

Time: 34.2s

Precision: binary64

Cost: 1344

Math

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

↓

\[\frac{\left(b \cdot a\right) \cdot \left(a \cdot \left(b \cdot -3\right)\right)}{4} - \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{4} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (- (/ (* (* b a) (* a (* b -3.0))) 4.0) (/ (* (* b a) (* b a)) 4.0)))

C

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

↓

double code(double a, double b) {
	return (((b * a) * (a * (b * -3.0))) / 4.0) - (((b * a) * (b * a)) / 4.0);
}

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) * (a * (b * (-3.0d0)))) / 4.0d0) - (((b * a) * (b * a)) / 4.0d0)
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) * (a * (b * -3.0))) / 4.0) - (((b * a) * (b * a)) / 4.0);
}

Python

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

↓

def code(a, b):
	return (((b * a) * (a * (b * -3.0))) / 4.0) - (((b * a) * (b * a)) / 4.0)

Julia

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

↓

function code(a, b)
	return Float64(Float64(Float64(Float64(b * a) * Float64(a * Float64(b * -3.0))) / 4.0) - Float64(Float64(Float64(b * a) * Float64(b * a)) / 4.0))
end

MATLAB

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

↓

function tmp = code(a, b)
	tmp = (((b * a) * (a * (b * -3.0))) / 4.0) - (((b * a) * (b * a)) / 4.0);
end

Wolfram

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

↓

code[a_, b_] := N[(N[(N[(N[(b * a), $MachinePrecision] * N[(a * N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] - N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.0

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

2. Simplified 5.3

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

   [Start] 16.0	\[ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   rational_best-simplify-11 [=>] 16.0	\[ \color{blue}{\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot -1} \]
   rational_best-simplify-1 [=>] 16.0	\[ \color{blue}{-1 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)} \]
   rational_best-simplify-50 [=>] 16.0	\[ \color{blue}{b \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot -1\right)} \]
   rational_best-simplify-1 [=>] 16.0	\[ b \cdot \color{blue}{\left(-1 \cdot \left(\left(a \cdot a\right) \cdot b\right)\right)} \]
   rational_best-simplify-1 [=>] 16.0	\[ b \cdot \left(-1 \cdot \color{blue}{\left(b \cdot \left(a \cdot a\right)\right)}\right) \]
   rational_best-simplify-50 [=>] 5.3	\[ b \cdot \left(-1 \cdot \color{blue}{\left(a \cdot \left(a \cdot b\right)\right)}\right) \]
   rational_best-simplify-1 [=>] 5.3	\[ b \cdot \left(-1 \cdot \color{blue}{\left(\left(a \cdot b\right) \cdot a\right)}\right) \]
   rational_best-simplify-50 [=>] 5.3	\[ b \cdot \color{blue}{\left(a \cdot \left(\left(a \cdot b\right) \cdot -1\right)\right)} \]
   rational_best-simplify-1 [=>] 5.3	\[ b \cdot \left(a \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right) \]
   rational_best-simplify-1 [=>] 5.3	\[ b \cdot \left(a \cdot \left(-1 \cdot \color{blue}{\left(b \cdot a\right)}\right)\right) \]
   rational_best-simplify-50 [=>] 5.3	\[ b \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(b \cdot -1\right)\right)}\right) \]
   rational_best-simplify-10 [=>] 5.3	\[ b \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right)\right) \]

3. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\frac{\left(b \cdot a\right) \cdot \left(a \cdot \left(b \cdot -3\right)\right)}{4} - \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{4}} \]

4. Final simplification 0.3

   \[\leadsto \frac{\left(b \cdot a\right) \cdot \left(a \cdot \left(b \cdot -3\right)\right)}{4} - \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{4} \]

Alternatives

Alternative 1
Error	0.4
Cost	960

\[\frac{\left(a \cdot b\right) \cdot \left(a \cdot \left(-3 \cdot b\right) - a \cdot b\right)}{4} \]

Alternative 2
Error	21.9
Cost	512

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

Alternative 3
Error	0.3
Cost	512

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

Reproduce

herbie shell --seed 2023099 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))