ab-angle->ABCF D

Percentage Accurate: 81.7% → 99.6%

Time: 5.7s

Alternatives: 6

Speedup: 1.0×

• 27.8% 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: 81.7% 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: 99.6% accurate, 1.0× speedup

Math

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

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 * b) * Float64(a * Float64(-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 83.1%

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

3. Taylor expanded in a around 0 77.4%

   \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot {b}^{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg 77.4%

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

   2. unpow2 77.4%

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

   3. unpow2 77.4%

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

   4. swap-sqr 99.7%

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

   5. unpow2 99.7%

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

5. Simplified 99.7%

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

   1. unpow2 99.7%

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

   2. distribute-rgt-neg-in 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 88.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(\left(a \cdot a\right) \cdot \left(-b\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-288}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* (* a a) (- b)))))
   (if (<= t_0 -1e-288) t_0 (* a (* b (* a (- b)))))))

C

double code(double a, double b) {
	double t_0 = b * ((a * a) * -b);
	double tmp;
	if (t_0 <= -1e-288) {
		tmp = t_0;
	} else {
		tmp = a * (b * (a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * ((a * a) * -b)
    if (t_0 <= (-1d-288)) then
        tmp = t_0
    else
        tmp = a * (b * (a * -b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * ((a * a) * -b);
	double tmp;
	if (t_0 <= -1e-288) {
		tmp = t_0;
	} else {
		tmp = a * (b * (a * -b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * ((a * a) * -b)
	tmp = 0
	if t_0 <= -1e-288:
		tmp = t_0
	else:
		tmp = a * (b * (a * -b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(Float64(a * a) * Float64(-b)))
	tmp = 0.0
	if (t_0 <= -1e-288)
		tmp = t_0;
	else
		tmp = Float64(a * Float64(b * Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * ((a * a) * -b);
	tmp = 0.0;
	if (t_0 <= -1e-288)
		tmp = t_0;
	else
		tmp = a * (b * (a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(N[(a * a), $MachinePrecision] * (-b)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-288], t$95$0, N[(a * N[(b * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 (*.f64 (*.f64 (*.f64 a a) b) b)) < -1.00000000000000006e-288

   1. Initial program 86.9%

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

   if -1.00000000000000006e-288 < (neg.f64 (*.f64 (*.f64 (*.f64 a a) b) b))

   1. Initial program 77.9%

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

      1. associate-*l* 76.0%

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

      2. associate-*r* 83.3%

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

      3. *-commutative 83.3%

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

      4. distribute-rgt-neg-in 83.3%

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

      5. distribute-rgt-neg-in 83.3%

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

      6. associate-*r* 95.8%

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

   3. Simplified 95.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(\left(a \cdot a\right) \cdot \left(-b\right)\right) \leq -1 \cdot 10^{-288}:\\ \;\;\;\;b \cdot \left(\left(a \cdot a\right) \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Final simplification 83.1%

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

Alternative 4: 28.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. distribute-rgt-neg-in 83.1%

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

   2. associate-*l* 94.5%

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

3. Simplified 94.5%

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

   1. neg-sub0 94.5%

      \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{\left(0 - b\right)} \]

   2. sub-neg 94.5%

      \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{\left(0 + \left(-b\right)\right)} \]

   3. add-sqr-sqrt 42.1%

      \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \left(0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}\right) \]

   4. sqrt-unprod 54.1%

      \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \left(0 + \color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}\right) \]

   5. sqr-neg 54.1%

      \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \left(0 + \sqrt{\color{blue}{b \cdot b}}\right) \]

   6. sqrt-unprod 16.5%

      \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \left(0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) \]

   7. add-sqr-sqrt 32.3%

      \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \left(0 + \color{blue}{b}\right) \]

6. Applied egg-rr 32.3%

   \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{\left(0 + b\right)} \]
7. Step-by-step derivation

   1. +-lft-identity 32.3%

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

8. Simplified 32.3%

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

9. Final simplification 32.3%

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

Alternative 5: 28.0% accurate, 1.1× speedup

Math

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

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 * 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 83.1%

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

   1. add-sqr-sqrt 31.3%

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

   2. sqrt-unprod 32.4%

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

   3. sqr-neg 32.4%

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

   4. sqrt-unprod 32.3%

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

   5. add-sqr-sqrt 32.3%

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

   6. associate-*l* 32.1%

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

   7. swap-sqr 32.2%

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

4. Applied egg-rr 32.2%

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

Alternative 6: 28.1% accurate, 1.1× speedup

Math

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

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(a * Float64(b * Float64(a * b)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. associate-*l* 77.4%

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

   2. associate-*r* 83.4%

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

   3. *-commutative 83.4%

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

   4. distribute-rgt-neg-in 83.4%

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

   5. distribute-rgt-neg-in 83.4%

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

   6. associate-*r* 95.0%

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

3. Simplified 95.0%

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

   1. neg-sub0 95.0%

      \[\leadsto a \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(0 - a\right)}\right)\right) \]

   2. sub-neg 95.0%

      \[\leadsto a \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(0 + \left(-a\right)\right)}\right)\right) \]

   3. add-sqr-sqrt 46.3%

      \[\leadsto a \cdot \left(b \cdot \left(b \cdot \left(0 + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)\right) \]

   4. sqrt-unprod 56.3%

      \[\leadsto a \cdot \left(b \cdot \left(b \cdot \left(0 + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)\right) \]

   5. sqr-neg 56.3%

      \[\leadsto a \cdot \left(b \cdot \left(b \cdot \left(0 + \sqrt{\color{blue}{a \cdot a}}\right)\right)\right) \]

   6. sqrt-prod 17.7%

      \[\leadsto a \cdot \left(b \cdot \left(b \cdot \left(0 + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)\right) \]

   7. add-sqr-sqrt 32.3%

      \[\leadsto a \cdot \left(b \cdot \left(b \cdot \left(0 + \color{blue}{a}\right)\right)\right) \]

6. Applied egg-rr 32.3%

   \[\leadsto a \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(0 + a\right)}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 32.3%

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

8. Simplified 32.3%

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

9. Final simplification 32.3%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))