ab-angle->ABCF D

Percentage Accurate: 82.1% → 99.7%

Time: 4.0s

Alternatives: 4

Speedup: 1.0×

• 28.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(a \cdot \left(-b\right)\right) \cdot \left(a \cdot b\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * -b) * (a * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Step-by-step derivation

   1. associate-*l* 72.8%

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

   2. unswap-sqr 99.7%

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

   3. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-out 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\left(a \cdot \left(-b\right)\right) \cdot \left(a \cdot b\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.9 \cdot 10^{-201}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(a \cdot b\right) \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 5.9e-201) (* a (* b (* a (- b)))) (* b (* (* a b) (- a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 5.9e-201) {
		tmp = a * (b * (a * -b));
	} else {
		tmp = b * ((a * b) * -a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 5.9d-201) then
        tmp = a * (b * (a * -b))
    else
        tmp = b * ((a * b) * -a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 5.9e-201) {
		tmp = a * (b * (a * -b));
	} else {
		tmp = b * ((a * b) * -a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 5.9e-201:
		tmp = a * (b * (a * -b))
	else:
		tmp = b * ((a * b) * -a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 5.9e-201)
		tmp = Float64(a * Float64(b * Float64(a * Float64(-b))));
	else
		tmp = Float64(b * Float64(Float64(a * b) * Float64(-a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 5.9e-201)
		tmp = a * (b * (a * -b));
	else
		tmp = b * ((a * b) * -a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 5.9e-201], N[(a * N[(b * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(a * b), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.89999999999999981e-201

   1. Initial program 74.6%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 67.8%

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

      2. swap-sqr 99.6%

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

      3. pow2 99.6%

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

      4. *-commutative 99.6%

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

      5. add-sqr-sqrt 14.7%

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

      6. associate-*r* 14.7%

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

      7. unpow-prod-down 14.8%

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

      8. pow2 14.8%

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

      9. add-sqr-sqrt 14.8%

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

   4. Applied egg-rr 14.8%

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

      1. unpow2 14.8%

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

      2. *-commutative 14.8%

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

      3. associate-*r* 14.8%

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

      4. associate-*r* 14.8%

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

      5. add-sqr-sqrt 96.7%

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

      6. *-commutative 96.7%

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

   6. Applied egg-rr 96.7%

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

   if 5.89999999999999981e-201 < a

   1. Initial program 86.8%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 94.6%

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

      2. *-commutative 94.6%

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

   4. Applied egg-rr 94.6%

      \[\leadsto -\color{blue}{\left(\left(a \cdot b\right) \cdot a\right)} \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.9 \cdot 10^{-201}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(a \cdot b\right) \cdot \left(-a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(\left(a \cdot b\right) \cdot \left(-a\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * ((a * b) * -a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* 91.6%

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

   2. *-commutative 91.6%

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

4. Applied egg-rr 91.6%

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

5. Final simplification 91.6%

   \[\leadsto b \cdot \left(\left(a \cdot b\right) \cdot \left(-a\right)\right) \]
6. Add Preprocessing

Alternative 4: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(\left(-b\right) \cdot \left(a \cdot a\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * (-b * (a * a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Add Preprocessing

3. Final simplification 79.7%

   \[\leadsto b \cdot \left(\left(-b\right) \cdot \left(a \cdot a\right)\right) \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))