ab-angle->ABCF D

Percentage Accurate: 82.5% → 99.6%

Time: 5.8s

Alternatives: 5

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: 82.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Taylor expanded in a around 0 77.0%

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

   1. unpow2 77.0%

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

   2. unpow2 77.0%

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

   3. swap-sqr 99.7%

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

   4. 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%

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

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%

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

Alternative 2: 28.3% 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 82.4%

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

   1. distribute-rgt-neg-in 82.4%

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

   2. associate-*l* 93.2%

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

3. Simplified 93.2%

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

   1. neg-sub0 93.2%

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

   2. sub-neg 93.2%

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

   3. add-sqr-sqrt 40.4%

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

   4. sqrt-unprod 50.2%

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

   5. sqr-neg 50.2%

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

   6. sqrt-unprod 15.7%

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

   7. add-sqr-sqrt 28.7%

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

6. Applied egg-rr 28.7%

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

   1. +-lft-identity 28.7%

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

8. Simplified 28.7%

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

9. Final simplification 28.7%

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

Alternative 3: 28.3% 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 82.4%

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

   1. add-sqr-sqrt 28.1%

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

   2. sqrt-unprod 28.8%

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

   3. sqr-neg 28.8%

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

   4. sqrt-unprod 28.7%

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

   5. add-sqr-sqrt 28.7%

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

   6. associate-*l* 28.5%

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

   7. swap-sqr 28.7%

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

4. Applied egg-rr 28.7%

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

Alternative 4: 28.3% 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 82.4%

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

   1. associate-*l* 77.0%

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

   2. associate-*r* 82.9%

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

   3. *-commutative 82.9%

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

   4. distribute-rgt-neg-in 82.9%

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

   5. distribute-rgt-neg-in 82.9%

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

   6. associate-*r* 93.7%

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

3. Simplified 93.7%

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

   1. neg-sub0 93.7%

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

   2. sub-neg 93.7%

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

   3. add-sqr-sqrt 48.1%

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

   4. sqrt-unprod 57.9%

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

   5. sqr-neg 57.9%

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

   6. sqrt-prod 14.8%

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

   7. add-sqr-sqrt 28.7%

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

6. Applied egg-rr 28.7%

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

   1. +-lft-identity 28.7%

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

8. Simplified 28.7%

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

9. Final simplification 28.7%

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

Alternative 5: 28.3% accurate, 1.1× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. associate-*l* 77.0%

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

   2. associate-*r* 82.9%

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

   3. *-commutative 82.9%

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

   4. distribute-rgt-neg-in 82.9%

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

   5. distribute-rgt-neg-in 82.9%

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

   6. associate-*r* 93.7%

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

3. Simplified 93.7%

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

5. Taylor expanded in b around 0 82.9%

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

   1. neg-mul-1 82.9%

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

   2. distribute-rgt-neg-in 82.9%

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

7. Simplified 82.9%

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

   1. add-sqr-sqrt 15.2%

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

   2. sqrt-unprod 28.3%

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

   3. sqr-neg 28.3%

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

   4. sqrt-unprod 28.7%

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

   5. add-sqr-sqrt 28.7%

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

   6. unpow2 28.7%

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

9. Applied egg-rr 28.7%

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

Reproduce

herbie shell --seed 2024121 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))