ab-angle->ABCF D

Percentage Accurate: 82.1% → 99.6%

Time: 5.0s

Alternatives: 5

Speedup: 1.0×

• 28.3% 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: 99.6% 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 82.8%

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

   1. add-sqr-sqrt 51.8%

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

   2. pow2 51.8%

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

   3. sqrt-prod 40.8%

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

   4. sqrt-prod 23.3%

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

   5. add-sqr-sqrt 48.7%

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

4. Applied egg-rr 48.7%

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

   1. add-sqr-sqrt 48.7%

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

   2. distribute-rgt-neg-in 48.7%

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

   3. sqrt-prod 48.7%

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

   4. sqrt-pow1 30.2%

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

   5. metadata-eval 30.2%

      \[\leadsto \left({\left(a \cdot \sqrt{b}\right)}^{\color{blue}{1}} \cdot \sqrt{b}\right) \cdot \left(-\sqrt{{\left(a \cdot \sqrt{b}\right)}^{2} \cdot b}\right) \]

   6. pow1 30.2%

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

   7. associate-*r* 30.2%

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

   8. add-sqr-sqrt 30.3%

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

   9. sqrt-prod 30.3%

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

   10. sqrt-pow1 50.5%

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

   11. metadata-eval 50.5%

       \[\leadsto \left(a \cdot b\right) \cdot \left(-{\left(a \cdot \sqrt{b}\right)}^{\color{blue}{1}} \cdot \sqrt{b}\right) \]

   12. pow1 50.5%

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

   13. associate-*r* 50.5%

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

   14. add-sqr-sqrt 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot \left(b \cdot \left(a \cdot a\right)\right) \leq 2 \cdot 10^{-109}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(-a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b (* b (* a a))) 2e-109)
   (* a (* b (* a (- b))))
   (* b (* b (- (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * (b * (a * a))) <= 2e-109) {
		tmp = a * (b * (a * -b));
	} else {
		tmp = b * (b * -(a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * (b * (a * a))) <= 2d-109) then
        tmp = a * (b * (a * -b))
    else
        tmp = b * (b * -(a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * (b * (a * a))) <= 2e-109) {
		tmp = a * (b * (a * -b));
	} else {
		tmp = b * (b * -(a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * (b * (a * a))) <= 2e-109:
		tmp = a * (b * (a * -b))
	else:
		tmp = b * (b * -(a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * Float64(b * Float64(a * a))) <= 2e-109)
		tmp = Float64(a * Float64(b * Float64(a * Float64(-b))));
	else
		tmp = Float64(b * Float64(b * Float64(-Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * (b * (a * a))) <= 2e-109)
		tmp = a * (b * (a * -b));
	else
		tmp = b * (b * -(a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-109], N[(a * N[(b * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * (-N[(a * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 a a) b) b) < 2e-109

   1. Initial program 82.9%

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

      1. associate-*l* 71.7%

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

      2. associate-*r* 76.3%

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

      3. *-commutative 76.3%

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

      4. distribute-rgt-neg-in 76.3%

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

      5. distribute-rgt-neg-in 76.3%

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

      6. associate-*r* 88.3%

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

   3. Simplified 88.3%

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

   if 2e-109 < (*.f64 (*.f64 (*.f64 a a) b) b)

   1. Initial program 82.8%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(b \cdot \left(a \cdot a\right)\right) \leq 2 \cdot 10^{-109}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(-a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 1.0× speedup

Math

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

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(b * Float64(-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 82.8%

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

3. Final simplification 82.8%

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

Alternative 4: 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 82.8%

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

   1. add-sqr-sqrt 27.3%

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

   2. sqrt-unprod 28.5%

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

   3. sqr-neg 28.5%

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

   4. sqrt-unprod 28.3%

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

   5. add-sqr-sqrt 28.3%

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

   6. associate-*l* 28.2%

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

   7. swap-sqr 28.3%

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

4. Applied egg-rr 28.3%

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

Alternative 5: 28.4% 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 82.8%

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

   1. add-sqr-sqrt 51.8%

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

   2. pow2 51.8%

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

   3. sqrt-prod 40.8%

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

   4. sqrt-prod 23.3%

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

   5. add-sqr-sqrt 48.7%

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

4. Applied egg-rr 48.7%

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

   1. distribute-rgt-neg-in 48.7%

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

   2. unpow2 48.7%

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

   3. swap-sqr 40.8%

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

   4. unpow2 40.8%

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

   5. add-sqr-sqrt 82.8%

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

   6. associate-*l* 74.8%

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

   7. distribute-rgt-neg-in 74.8%

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

   8. unpow2 74.8%

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

   9. unpow2 74.8%

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

   10. associate-*l* 79.4%

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

6. Applied egg-rr 79.4%

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

   1. add-sqr-sqrt 15.1%

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

   2. sqrt-unprod 27.0%

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

   3. sqr-neg 27.0%

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

   4. sqrt-unprod 28.4%

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

   5. add-sqr-sqrt 28.4%

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

   6. unpow2 28.4%

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

8. Applied egg-rr 28.4%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))