ab-angle->ABCF D

Percentage Accurate: 81.4% → 99.7%

Time: 4.5s

Precision: binary64

Cost: 6720

• Unsound rule application detected in e-graph. Results may not be sound.

• 28.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[-{\left(a \cdot b\right)}^{2} \]

FPCore

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

↓

(FPCore (a b) :precision binary64 (- (pow (* a b) 2.0)))

C

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

↓

double code(double a, double b) {
	return -pow((a * b), 2.0);
}

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 = -((a * b) ** 2.0d0)
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 -Math.pow((a * b), 2.0);
}

Python

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

↓

def code(a, b):
	return -math.pow((a * b), 2.0)

Julia

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

↓

function code(a, b)
	return Float64(-(Float64(a * b) ^ 2.0))
end

MATLAB

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

↓

function tmp = code(a, b)
	tmp = -((a * b) ^ 2.0);
end

Wolfram

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

↓

code[a_, b_] := (-N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision])

Derivation

1. Initial program 83.4%

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

2. Taylor expanded in a around 0 77.0%

   \[\leadsto -\color{blue}{{a}^{2} \cdot {b}^{2}} \]

3. Simplified 99.7%

   \[\leadsto -\color{blue}{{\left(a \cdot b\right)}^{2}} \]
   Step-by-step derivation

   [Start] 77.0	\[ -{a}^{2} \cdot {b}^{2} \]
   unpow2 [=>] 77.0	\[ -\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2} \]
   unpow2 [=>] 77.0	\[ -\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} \]
   swap-sqr [<=] 99.7	\[ -\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   unpow2 [<=] 99.7	\[ -\color{blue}{{\left(a \cdot b\right)}^{2}} \]

4. Final simplification 99.7%

   \[\leadsto -{\left(a \cdot b\right)}^{2} \]

Alternatives

Alternative 1
Accuracy	75.3%
Cost	512

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

Alternative 2
Accuracy	99.7%
Cost	512

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

Alternative 3
Accuracy	27.9%
Cost	448

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

Reproduce

herbie shell --seed 2023159 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))