ab-angle->ABCF D

Percentage Accurate: 82.0% → 99.7%

Time: 6.8s

Alternatives: 8

Speedup: 1.0×

• 28.6% 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: 82.0% 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, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \left(a\_m \cdot b\right) \cdot \frac{1}{\frac{\frac{-1}{a\_m}}{b}} \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* (* a_m b) (/ 1.0 (/ (/ -1.0 a_m) b))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return (a_m * b) * (1.0 / ((-1.0 / a_m) / b));
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = (a_m * b) * (1.0d0 / (((-1.0d0) / a_m) / b))
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return (a_m * b) * (1.0 / ((-1.0 / a_m) / b));
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return (a_m * b) * (1.0 / ((-1.0 / a_m) / b))

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(Float64(a_m * b) * Float64(1.0 / Float64(Float64(-1.0 / a_m) / b)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (a_m * b) * (1.0 / ((-1.0 / a_m) / b));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m * b), $MachinePrecision] * N[(1.0 / N[(N[(-1.0 / a$95$m), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in a around 0 80.1%

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

   1. mul-1-neg 80.1%

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

   2. unpow2 80.1%

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

   3. unpow2 80.1%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 54.4%

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

   2. sqrt-prod 62.6%

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

   3. unpow2 62.6%

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

   4. add-sqr-sqrt 62.6%

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

   5. sqrt-prod 56.5%

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

   6. sqr-neg 56.5%

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

   7. sqrt-unprod 25.0%

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

   8. add-sqr-sqrt 25.0%

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

   9. metadata-eval 25.0%

      \[\leadsto \sqrt{-{\left(a \cdot b\right)}^{\color{blue}{\left(4 - 2\right)}}} \cdot \left(-a \cdot b\right) \]

   10. pow-div 0.8%

       \[\leadsto \sqrt{-\color{blue}{\frac{{\left(a \cdot b\right)}^{4}}{{\left(a \cdot b\right)}^{2}}}} \cdot \left(-a \cdot b\right) \]

   11. distribute-frac-neg2 0.8%

       \[\leadsto \sqrt{\color{blue}{\frac{{\left(a \cdot b\right)}^{4}}{-{\left(a \cdot b\right)}^{2}}}} \cdot \left(-a \cdot b\right) \]

   12. clear-num 0.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{-{\left(a \cdot b\right)}^{2}}{{\left(a \cdot b\right)}^{4}}}}} \cdot \left(-a \cdot b\right) \]

   13. sqrt-div 0.5%

       \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{-{\left(a \cdot b\right)}^{2}}{{\left(a \cdot b\right)}^{4}}}}} \cdot \left(-a \cdot b\right) \]

   14. metadata-eval 0.5%

       \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{-{\left(a \cdot b\right)}^{2}}{{\left(a \cdot b\right)}^{4}}}} \cdot \left(-a \cdot b\right) \]

   15. clear-num 0.5%

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{{\left(a \cdot b\right)}^{4}}{-{\left(a \cdot b\right)}^{2}}}}}} \cdot \left(-a \cdot b\right) \]

   16. distribute-frac-neg2 0.5%

       \[\leadsto \frac{1}{\sqrt{\frac{1}{\color{blue}{-\frac{{\left(a \cdot b\right)}^{4}}{{\left(a \cdot b\right)}^{2}}}}}} \cdot \left(-a \cdot b\right) \]

   17. pow-div 16.8%

       \[\leadsto \frac{1}{\sqrt{\frac{1}{-\color{blue}{{\left(a \cdot b\right)}^{\left(4 - 2\right)}}}}} \cdot \left(-a \cdot b\right) \]

   18. metadata-eval 16.8%

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

9. Applied egg-rr 62.6%

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

10. Taylor expanded in a around 0 99.7%

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

    1. associate-/r* 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

    \[\leadsto \left(a \cdot b\right) \cdot \frac{1}{\frac{\frac{-1}{a}}{b}} \]
14. Add Preprocessing

Alternative 2: 92.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot \left(b \cdot \left(a\_m \cdot a\_m\right)\right) \leq 5 \cdot 10^{-251}:\\ \;\;\;\;a\_m \cdot \left(b \cdot \left(a\_m \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(a\_m \cdot a\_m\right) \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b)
 :precision binary64
 (if (<= (* b (* b (* a_m a_m))) 5e-251)
   (* a_m (* b (* a_m (- b))))
   (* b (* (* a_m a_m) (- b)))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	double tmp;
	if ((b * (b * (a_m * a_m))) <= 5e-251) {
		tmp = a_m * (b * (a_m * -b));
	} else {
		tmp = b * ((a_m * a_m) * -b);
	}
	return tmp;
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * (b * (a_m * a_m))) <= 5d-251) then
        tmp = a_m * (b * (a_m * -b))
    else
        tmp = b * ((a_m * a_m) * -b)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	double tmp;
	if ((b * (b * (a_m * a_m))) <= 5e-251) {
		tmp = a_m * (b * (a_m * -b));
	} else {
		tmp = b * ((a_m * a_m) * -b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	tmp = 0
	if (b * (b * (a_m * a_m))) <= 5e-251:
		tmp = a_m * (b * (a_m * -b))
	else:
		tmp = b * ((a_m * a_m) * -b)
	return tmp

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	tmp = 0.0
	if (Float64(b * Float64(b * Float64(a_m * a_m))) <= 5e-251)
		tmp = Float64(a_m * Float64(b * Float64(a_m * Float64(-b))));
	else
		tmp = Float64(b * Float64(Float64(a_m * a_m) * Float64(-b)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if ((b * (b * (a_m * a_m))) <= 5e-251)
		tmp = a_m * (b * (a_m * -b));
	else
		tmp = b * ((a_m * a_m) * -b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := If[LessEqual[N[(b * N[(b * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-251], N[(a$95$m * N[(b * N[(a$95$m * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(a$95$m * a$95$m), $MachinePrecision] * (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 a a) b) b) < 5.0000000000000003e-251

   1. Initial program 79.9%

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

      1. associate-*l* 79.1%

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

      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* 97.7%

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

   3. Simplified 97.7%

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

   if 5.0000000000000003e-251 < (*.f64 (*.f64 (*.f64 a a) b) b)

   1. Initial program 86.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(b \cdot \left(a \cdot a\right)\right) \leq 5 \cdot 10^{-251}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(a \cdot a\right) \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \left(a\_m \cdot b\right) \cdot \frac{-1}{\frac{1}{a\_m \cdot b}} \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* (* a_m b) (/ -1.0 (/ 1.0 (* a_m b)))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return (a_m * b) * (-1.0 / (1.0 / (a_m * b)));
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = (a_m * b) * ((-1.0d0) / (1.0d0 / (a_m * b)))
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return (a_m * b) * (-1.0 / (1.0 / (a_m * b)));
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return (a_m * b) * (-1.0 / (1.0 / (a_m * b)))

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(Float64(a_m * b) * Float64(-1.0 / Float64(1.0 / Float64(a_m * b))))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (a_m * b) * (-1.0 / (1.0 / (a_m * b)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m * b), $MachinePrecision] * N[(-1.0 / N[(1.0 / N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in a around 0 80.1%

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

   1. mul-1-neg 80.1%

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

   2. unpow2 80.1%

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

   3. unpow2 80.1%

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

   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. Step-by-step derivation

   1. add-cbrt-cube 90.0%

      \[\leadsto \left(a \cdot b\right) \cdot \left(-\color{blue}{\sqrt[3]{\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) \cdot \left(a \cdot b\right)}}\right) \]

   2. pow3 90.0%

      \[\leadsto \left(a \cdot b\right) \cdot \left(-\sqrt[3]{\color{blue}{{\left(a \cdot b\right)}^{3}}}\right) \]

9. Applied egg-rr 90.0%

   \[\leadsto \left(a \cdot b\right) \cdot \left(-\color{blue}{\sqrt[3]{{\left(a \cdot b\right)}^{3}}}\right) \]
10. Step-by-step derivation

    1. rem-cbrt-cube 99.7%

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

    2. add-sqr-sqrt 56.0%

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

    3. sqrt-unprod 63.2%

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

    4. sqr-neg 63.2%

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

    5. unpow2 63.2%

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

    6. /-rgt-identity 63.2%

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

    7. clear-num 63.1%

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

    8. metadata-eval 63.1%

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

    9. add-sqr-sqrt 63.1%

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

    10. frac-times 63.1%

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

    11. sqrt-unprod 63.1%

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

    12. add-sqr-sqrt 63.1%

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

    13. frac-2neg 63.1%

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

    14. metadata-eval 63.1%

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

    15. sqrt-div 63.2%

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

    16. metadata-eval 63.2%

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

    17. sqrt-pow1 26.1%

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

    18. metadata-eval 26.1%

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

    19. pow1 26.1%

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

    20. distribute-neg-frac2 26.1%

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

    21. add-sqr-sqrt 19.0%

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

11. Applied egg-rr 99.7%

    \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{\frac{-1}{\frac{1}{a \cdot b}}} \]
12. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \left(a\_m \cdot b\right) \cdot \left(a\_m \cdot \left(-b\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* (* a_m b) (* a_m (- b))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return (a_m * b) * (a_m * -b);
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = (a_m * b) * (a_m * -b)
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return (a_m * b) * (a_m * -b);
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return (a_m * b) * (a_m * -b)

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(Float64(a_m * b) * Float64(a_m * Float64(-b)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (a_m * b) * (a_m * -b);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m * b), $MachinePrecision] * N[(a$95$m * (-b)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in a around 0 80.1%

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

   1. mul-1-neg 80.1%

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

   2. unpow2 80.1%

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

   3. unpow2 80.1%

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

   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 5: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ b \cdot \left(\left(a\_m \cdot a\_m\right) \cdot \left(-b\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* b (* (* a_m a_m) (- b))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return b * ((a_m * a_m) * -b);
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = b * ((a_m * a_m) * -b)
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return b * ((a_m * a_m) * -b);
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return b * ((a_m * a_m) * -b)

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(b * Float64(Float64(a_m * a_m) * Float64(-b)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = b * ((a_m * a_m) * -b);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(b * N[(N[(a$95$m * a$95$m), $MachinePrecision] * (-b)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Final simplification 84.2%

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

Alternative 6: 28.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ b \cdot \left(a\_m \cdot \left(a\_m \cdot b\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* b (* a_m (* a_m b))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return b * (a_m * (a_m * b));
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = b * (a_m * (a_m * b))
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return b * (a_m * (a_m * b));
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return b * (a_m * (a_m * b))

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(b * Float64(a_m * Float64(a_m * b)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = b * (a_m * (a_m * b));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(b * N[(a$95$m * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. distribute-rgt-neg-in 84.2%

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

   2. associate-*l* 92.9%

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

3. Simplified 92.9%

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

   1. neg-sub0 92.9%

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

   2. sub-neg 92.9%

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

   3. add-sqr-sqrt 44.3%

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

   4. sqrt-unprod 54.2%

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

   5. sqr-neg 54.2%

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

   6. sqrt-unprod 14.9%

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

   7. add-sqr-sqrt 26.2%

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

6. Applied egg-rr 26.2%

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

   1. +-lft-identity 26.2%

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

8. Simplified 26.2%

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

9. Final simplification 26.2%

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

Alternative 7: 28.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \left(a\_m \cdot b\right) \cdot \left(a\_m \cdot b\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* (* a_m b) (* a_m b)))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return (a_m * b) * (a_m * b);
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = (a_m * b) * (a_m * b)
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return (a_m * b) * (a_m * b);
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return (a_m * b) * (a_m * b)

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(Float64(a_m * b) * Float64(a_m * b))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (a_m * b) * (a_m * b);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m * b), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. add-sqr-sqrt 25.3%

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

   2. sqrt-unprod 26.2%

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

   3. sqr-neg 26.2%

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

   4. sqrt-unprod 26.2%

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

   5. add-sqr-sqrt 26.2%

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

   6. associate-*l* 25.9%

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

   7. swap-sqr 26.1%

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

4. Applied egg-rr 26.1%

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

Alternative 8: 28.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ a\_m \cdot \left(b \cdot \left(a\_m \cdot b\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* a_m (* b (* a_m b))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return a_m * (b * (a_m * b));
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = a_m * (b * (a_m * b))
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return a_m * (b * (a_m * b));
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return a_m * (b * (a_m * b))

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(a_m * Float64(b * Float64(a_m * b)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = a_m * (b * (a_m * b));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(a$95$m * N[(b * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. associate-*l* 80.1%

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

   2. associate-*r* 84.8%

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

   3. *-commutative 84.8%

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

   4. distribute-rgt-neg-in 84.8%

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

   5. distribute-rgt-neg-in 84.8%

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

   6. associate-*r* 96.2%

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

3. Simplified 96.2%

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

   1. neg-sub0 96.2%

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

   2. sub-neg 96.2%

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

   3. add-sqr-sqrt 47.6%

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

   4. sqrt-unprod 55.3%

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

   5. sqr-neg 55.3%

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

   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 26.1%

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

6. Applied egg-rr 26.1%

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

   1. +-lft-identity 26.1%

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

8. Simplified 26.1%

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

9. Final simplification 26.1%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))