ab-angle->ABCF D

Percentage Accurate: 82.7% → 99.7%

Time: 12.5s

Alternatives: 10

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: 82.7% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	return (a / (-1.0 / b)) * (a * b)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. distribute-rgt-neg-in 81.1%

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

   2. associate-*l* 92.7%

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

3. Simplified 92.7%

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

   1. add-sqr-sqrt_binary64 52.9%

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

5. Applied rewrite-once 52.9%

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

   1. rem-square-sqrt 92.7%

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

   2. associate-*r* 81.1%

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

   3. associate-*l* 73.0%

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

   4. /-rgt-identity 73.0%

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

   5. associate-/l* 73.0%

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

   6. associate-*l/ 79.5%

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

7. Applied egg-rr 79.5%

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

   1. associate-/l* 79.5%

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

   2. div-inv 79.5%

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

   3. associate-*r/ 79.5%

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

   4. div-inv 79.5%

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

   5. *-commutative 79.5%

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

   6. times-frac 91.9%

      \[\leadsto \frac{a \cdot 1}{\color{blue}{\frac{1}{-b} \cdot \frac{\frac{1}{a}}{b}}} \]

   7. times-frac 99.6%

      \[\leadsto \color{blue}{\frac{a}{\frac{1}{-b}} \cdot \frac{1}{\frac{\frac{1}{a}}{b}}} \]

   8. associate-/r* 99.6%

      \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \frac{1}{\color{blue}{\frac{1}{a \cdot b}}} \]

   9. clear-num 99.7%

      \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \color{blue}{\frac{a \cdot b}{1}} \]

   10. /-rgt-identity 99.7%

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

   11. neg-mul-1 99.7%

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

   12. associate-/r* 99.7%

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

   13. metadata-eval 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} a = |a|\\ [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+154) (* a (* a (- (* b b)))) (* b (* b (* a (- a))))))

C

a = abs(a);
assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+154) {
		tmp = a * (a * -(b * b));
	} else {
		tmp = b * (b * (a * -a));
	}
	return tmp;
}

Fortran

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.35d+154) then
        tmp = a * (a * -(b * b))
    else
        tmp = b * (b * (a * -a))
    end if
    code = tmp
end function

Java

a = Math.abs(a);
assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+154) {
		tmp = a * (a * -(b * b));
	} else {
		tmp = b * (b * (a * -a));
	}
	return tmp;
}

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.35e+154:
		tmp = a * (a * -(b * b))
	else:
		tmp = b * (b * (a * -a))
	return tmp

Julia

a = abs(a)
a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+154)
		tmp = Float64(a * Float64(a * Float64(-Float64(b * b))));
	else
		tmp = Float64(b * Float64(b * Float64(a * Float64(-a))));
	end
	return tmp
end

MATLAB

a = abs(a)
a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e+154)
		tmp = a * (a * -(b * b));
	else
		tmp = b * (b * (a * -a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1.35e+154], N[(a * N[(a * (-N[(b * b), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35000000000000003e154

   1. Initial program 82.8%

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

      1. associate-*l* 76.9%

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

      2. distribute-rgt-neg-in 76.9%

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

      3. associate-*r* 84.4%

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

   3. Simplified 84.4%

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

   if 1.35000000000000003e154 < b

   1. Initial program 71.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right)\\ \end{array} \]

Alternative 3: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} a = |a|\\ [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(a \cdot b\right) \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+154) (* a (* a (- (* b b)))) (* b (* (* a b) (- a)))))

C

a = abs(a);
assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+154) {
		tmp = a * (a * -(b * b));
	} else {
		tmp = b * ((a * b) * -a);
	}
	return tmp;
}

Fortran

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.35d+154) then
        tmp = a * (a * -(b * b))
    else
        tmp = b * ((a * b) * -a)
    end if
    code = tmp
end function

Java

a = Math.abs(a);
assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+154) {
		tmp = a * (a * -(b * b));
	} else {
		tmp = b * ((a * b) * -a);
	}
	return tmp;
}

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.35e+154:
		tmp = a * (a * -(b * b))
	else:
		tmp = b * ((a * b) * -a)
	return tmp

Julia

a = abs(a)
a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+154)
		tmp = Float64(a * Float64(a * Float64(-Float64(b * b))));
	else
		tmp = Float64(b * Float64(Float64(a * b) * Float64(-a)));
	end
	return tmp
end

MATLAB

a = abs(a)
a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e+154)
		tmp = a * (a * -(b * b));
	else
		tmp = b * ((a * b) * -a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1.35e+154], N[(a * N[(a * (-N[(b * b), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(a * b), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35000000000000003e154

   1. Initial program 82.8%

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

      1. associate-*l* 76.9%

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

      2. distribute-rgt-neg-in 76.9%

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

      3. associate-*r* 84.4%

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

   3. Simplified 84.4%

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

   if 1.35000000000000003e154 < b

   1. Initial program 71.2%

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

      1. distribute-rgt-neg-in 71.2%

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

      2. associate-*l* 92.0%

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

   3. Simplified 92.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(a \cdot b\right) \cdot \left(-a\right)\right)\\ \end{array} \]

Alternative 4: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	return b * (b * (a * -a))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]

2. Final simplification 81.1%

   \[\leadsto b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right) \]

Alternative 5: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	return -((a * b) * (a * b))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. distribute-rgt-neg-in 81.1%

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

   2. associate-*l* 92.7%

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

3. Simplified 92.7%

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

   1. add-sqr-sqrt_binary64 52.9%

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

5. Applied rewrite-once 52.9%

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

   1. rem-square-sqrt 92.7%

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

   2. associate-*r* 81.1%

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

   3. associate-*l* 73.0%

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

   4. /-rgt-identity 73.0%

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

   5. associate-/l* 73.0%

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

   6. associate-*l/ 79.5%

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

7. Applied egg-rr 79.5%

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

   1. clear-num 79.5%

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

   2. associate-/r/ 79.5%

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

   3. remove-double-div 79.5%

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

   4. distribute-rgt-neg-out 79.5%

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

   5. distribute-rgt-neg-out 79.5%

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

   6. distribute-lft-neg-out 79.5%

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

   7. associate-*l* 73.0%

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

   8. associate-*r* 81.1%

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

   9. *-commutative 81.1%

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

   10. *-commutative 81.1%

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

   11. associate-*l* 92.7%

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

   12. associate-*r* 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

    \[\leadsto -\left(a \cdot b\right) \cdot \left(a \cdot b\right) \]

Alternative 6: 40.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} a = |a|\\ [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-7}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= b 8e-7) 0.0 (* a (- a))))

C

a = abs(a);
assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 8e-7) {
		tmp = 0.0;
	} else {
		tmp = a * -a;
	}
	return tmp;
}

Fortran

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 8d-7) then
        tmp = 0.0d0
    else
        tmp = a * -a
    end if
    code = tmp
end function

Java

a = Math.abs(a);
assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 8e-7) {
		tmp = 0.0;
	} else {
		tmp = a * -a;
	}
	return tmp;
}

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 8e-7:
		tmp = 0.0
	else:
		tmp = a * -a
	return tmp

Julia

a = abs(a)
a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 8e-7)
		tmp = 0.0;
	else
		tmp = Float64(a * Float64(-a));
	end
	return tmp
end

MATLAB

a = abs(a)
a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8e-7)
		tmp = 0.0;
	else
		tmp = a * -a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 8e-7], 0.0, N[(a * (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.9999999999999996e-7

   1. Initial program 82.1%

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

      1. distribute-rgt-neg-in 82.1%

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

      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. Step-by-step derivation

      1. add-sqr-sqrt_binary64 40.4%

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

   5. Applied rewrite-once 40.4%

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

      1. rem-square-sqrt 92.9%

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

      2. associate-*r* 82.1%

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

      3. associate-*l* 75.4%

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

      4. /-rgt-identity 75.4%

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

      5. associate-/l* 75.4%

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

      6. associate-*l/ 82.3%

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

   7. Applied egg-rr 82.3%

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

      1. associate-/l* 82.4%

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

      2. div-inv 82.3%

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

      3. associate-*r/ 82.4%

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

      4. div-inv 82.4%

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

      5. *-commutative 82.4%

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

      6. times-frac 91.9%

         \[\leadsto \frac{a \cdot 1}{\color{blue}{\frac{1}{-b} \cdot \frac{\frac{1}{a}}{b}}} \]

      7. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{a}{\frac{1}{-b}} \cdot \frac{1}{\frac{\frac{1}{a}}{b}}} \]

      8. associate-/r* 99.7%

         \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \frac{1}{\color{blue}{\frac{1}{a \cdot b}}} \]

      9. clear-num 99.7%

         \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \color{blue}{\frac{a \cdot b}{1}} \]

      10. /-rgt-identity 99.7%

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

      11. neg-mul-1 99.7%

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

      12. associate-/r* 99.7%

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

      13. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   11. Applied egg-rr 39.0%

       \[\leadsto \color{blue}{0} \]

   if 7.9999999999999996e-7 < b

   1. Initial program 78.2%

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

      1. associate-*l* 65.5%

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

      2. distribute-rgt-neg-in 65.5%

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

      3. associate-*r* 70.6%

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

   3. Simplified 70.6%

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

   5. Applied egg-rr 34.9%

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

      1. +-rgt-identity 34.9%

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

   7. Simplified 34.9%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-7}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-a\right)\\ \end{array} \]

Alternative 7: 47.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	return (a * b) * -a

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. distribute-rgt-neg-in 81.1%

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

   2. associate-*l* 92.7%

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

3. Simplified 92.7%

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

5. Applied egg-rr 31.6%

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

   1. +-rgt-identity 31.6%

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

7. Simplified 31.6%

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

8. Taylor expanded in a around 0 31.6%

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

   1. mul-1-neg 31.6%

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

   2. unpow2 31.6%

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

   3. distribute-rgt-neg-out 31.6%

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

   4. associate-*l* 31.6%

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

10. Simplified 31.6%

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

11. Final simplification 31.6%

    \[\leadsto \left(a \cdot b\right) \cdot \left(-a\right) \]

Alternative 8: 4.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} a = |a|\\ [a, b] = \mathsf{sort}([a, b])\\ \\ -9 \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 -9.0)

C

a = abs(a);
assert(a < b);
double code(double a, double b) {
	return -9.0;
}

Fortran

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -9.0d0
end function

Java

a = Math.abs(a);
assert a < b;
public static double code(double a, double b) {
	return -9.0;
}

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	return -9.0

Julia

a = abs(a)
a, b = sort([a, b])
function code(a, b)
	return -9.0
end

MATLAB

a = abs(a)
a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = -9.0;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := -9.0

Derivation

1. Initial program 81.1%

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

   1. distribute-rgt-neg-in 81.1%

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

   2. associate-*l* 92.7%

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

3. Simplified 92.7%

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

   1. add-sqr-sqrt_binary64 52.9%

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

5. Applied rewrite-once 52.9%

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

   1. rem-square-sqrt 92.7%

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

   2. associate-*r* 81.1%

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

   3. associate-*l* 73.0%

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

   4. /-rgt-identity 73.0%

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

   5. associate-/l* 73.0%

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

   6. associate-*l/ 79.5%

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

7. Applied egg-rr 79.5%

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

   1. associate-/l* 79.5%

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

   2. div-inv 79.5%

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

   3. associate-*r/ 79.5%

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

   4. div-inv 79.5%

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

   5. *-commutative 79.5%

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

   6. times-frac 91.9%

      \[\leadsto \frac{a \cdot 1}{\color{blue}{\frac{1}{-b} \cdot \frac{\frac{1}{a}}{b}}} \]

   7. times-frac 99.6%

      \[\leadsto \color{blue}{\frac{a}{\frac{1}{-b}} \cdot \frac{1}{\frac{\frac{1}{a}}{b}}} \]

   8. associate-/r* 99.6%

      \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \frac{1}{\color{blue}{\frac{1}{a \cdot b}}} \]

   9. clear-num 99.7%

      \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \color{blue}{\frac{a \cdot b}{1}} \]

   10. /-rgt-identity 99.7%

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

   11. neg-mul-1 99.7%

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

   12. associate-/r* 99.7%

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

   13. metadata-eval 99.7%

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

9. Applied egg-rr 99.7%

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

11. Applied egg-rr 4.0%

    \[\leadsto \color{blue}{-9} \]

12. Final simplification 4.0%

    \[\leadsto -9 \]

Alternative 9: 4.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} a = |a|\\ [a, b] = \mathsf{sort}([a, b])\\ \\ -0.125 \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 -0.125)

C

a = abs(a);
assert(a < b);
double code(double a, double b) {
	return -0.125;
}

Fortran

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -0.125d0
end function

Java

a = Math.abs(a);
assert a < b;
public static double code(double a, double b) {
	return -0.125;
}

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	return -0.125

Julia

a = abs(a)
a, b = sort([a, b])
function code(a, b)
	return -0.125
end

MATLAB

a = abs(a)
a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = -0.125;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := -0.125

Derivation

1. Initial program 81.1%

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

   1. distribute-rgt-neg-in 81.1%

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

   2. associate-*l* 92.7%

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

3. Simplified 92.7%

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

   1. add-sqr-sqrt_binary64 52.9%

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

5. Applied rewrite-once 52.9%

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

   1. rem-square-sqrt 92.7%

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

   2. associate-*r* 81.1%

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

   3. associate-*l* 73.0%

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

   4. /-rgt-identity 73.0%

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

   5. associate-/l* 73.0%

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

   6. associate-*l/ 79.5%

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

7. Applied egg-rr 79.5%

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

   1. associate-/l* 79.5%

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

   2. div-inv 79.5%

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

   3. associate-*r/ 79.5%

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

   4. div-inv 79.5%

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

   5. *-commutative 79.5%

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

   6. times-frac 91.9%

      \[\leadsto \frac{a \cdot 1}{\color{blue}{\frac{1}{-b} \cdot \frac{\frac{1}{a}}{b}}} \]

   7. times-frac 99.6%

      \[\leadsto \color{blue}{\frac{a}{\frac{1}{-b}} \cdot \frac{1}{\frac{\frac{1}{a}}{b}}} \]

   8. associate-/r* 99.6%

      \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \frac{1}{\color{blue}{\frac{1}{a \cdot b}}} \]

   9. clear-num 99.7%

      \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \color{blue}{\frac{a \cdot b}{1}} \]

   10. /-rgt-identity 99.7%

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

   11. neg-mul-1 99.7%

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

   12. associate-/r* 99.7%

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

   13. metadata-eval 99.7%

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

9. Applied egg-rr 99.7%

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

11. Applied egg-rr 4.0%

    \[\leadsto \color{blue}{-0.125} \]

12. Final simplification 4.0%

    \[\leadsto -0.125 \]

Alternative 10: 29.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} a = |a|\\ [a, b] = \mathsf{sort}([a, b])\\ \\ 0 \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 0.0)

C

a = abs(a);
assert(a < b);
double code(double a, double b) {
	return 0.0;
}

Fortran

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.0d0
end function

Java

a = Math.abs(a);
assert a < b;
public static double code(double a, double b) {
	return 0.0;
}

Python

a = abs(a)
[a, b] = sort([a, b])
def code(a, b):
	return 0.0

Julia

a = abs(a)
a, b = sort([a, b])
function code(a, b)
	return 0.0
end

MATLAB

a = abs(a)
a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = 0.0;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := 0.0

Derivation

1. Initial program 81.1%

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

   1. distribute-rgt-neg-in 81.1%

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

   2. associate-*l* 92.7%

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

3. Simplified 92.7%

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

   1. add-sqr-sqrt_binary64 52.9%

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

5. Applied rewrite-once 52.9%

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

   1. rem-square-sqrt 92.7%

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

   2. associate-*r* 81.1%

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

   3. associate-*l* 73.0%

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

   4. /-rgt-identity 73.0%

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

   5. associate-/l* 73.0%

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

   6. associate-*l/ 79.5%

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

7. Applied egg-rr 79.5%

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

   1. associate-/l* 79.5%

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

   2. div-inv 79.5%

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

   3. associate-*r/ 79.5%

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

   4. div-inv 79.5%

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

   5. *-commutative 79.5%

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

   6. times-frac 91.9%

      \[\leadsto \frac{a \cdot 1}{\color{blue}{\frac{1}{-b} \cdot \frac{\frac{1}{a}}{b}}} \]

   7. times-frac 99.6%

      \[\leadsto \color{blue}{\frac{a}{\frac{1}{-b}} \cdot \frac{1}{\frac{\frac{1}{a}}{b}}} \]

   8. associate-/r* 99.6%

      \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \frac{1}{\color{blue}{\frac{1}{a \cdot b}}} \]

   9. clear-num 99.7%

      \[\leadsto \frac{a}{\frac{1}{-b}} \cdot \color{blue}{\frac{a \cdot b}{1}} \]

   10. /-rgt-identity 99.7%

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

   11. neg-mul-1 99.7%

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

   12. associate-/r* 99.7%

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

   13. metadata-eval 99.7%

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

9. Applied egg-rr 99.7%

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

11. Applied egg-rr 33.2%

    \[\leadsto \color{blue}{0} \]

12. Final simplification 33.2%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))