ab-angle->ABCF D

Percentage Accurate: 82.4% → 99.5%

Time: 4.4s

Alternatives: 5

Speedup: 1.0×

• 27.7% 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: 82.4% 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: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ [a, b_m] = \mathsf{sort}([a, b_m])\\ \\ \begin{array}{l} t_0 := a \cdot \sqrt{b\_m}\\ t\_0 \cdot \left(b\_m \cdot \left(-t\_0\right)\right) \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
NOTE: a and b_m should be sorted in increasing order before calling this function.
(FPCore (a b_m)
 :precision binary64
 (let* ((t_0 (* a (sqrt b_m)))) (* t_0 (* b_m (- t_0)))))

C

b_m = fabs(b);
assert(a < b_m);
double code(double a, double b_m) {
	double t_0 = a * sqrt(b_m);
	return t_0 * (b_m * -t_0);
}

Fortran

b_m = abs(b)
NOTE: a and b_m should be sorted in increasing order before calling this function.
real(8) function code(a, b_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: t_0
    t_0 = a * sqrt(b_m)
    code = t_0 * (b_m * -t_0)
end function

Java

b_m = Math.abs(b);
assert a < b_m;
public static double code(double a, double b_m) {
	double t_0 = a * Math.sqrt(b_m);
	return t_0 * (b_m * -t_0);
}

Python

b_m = math.fabs(b)
[a, b_m] = sort([a, b_m])
def code(a, b_m):
	t_0 = a * math.sqrt(b_m)
	return t_0 * (b_m * -t_0)

Julia

b_m = abs(b)
a, b_m = sort([a, b_m])
function code(a, b_m)
	t_0 = Float64(a * sqrt(b_m))
	return Float64(t_0 * Float64(b_m * Float64(-t_0)))
end

MATLAB

b_m = abs(b);
a, b_m = num2cell(sort([a, b_m])){:}
function tmp = code(a, b_m)
	t_0 = a * sqrt(b_m);
	tmp = t_0 * (b_m * -t_0);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
NOTE: a and b_m should be sorted in increasing order before calling this function.
code[a_, b$95$m_] := Block[{t$95$0 = N[(a * N[Sqrt[b$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(b$95$m * (-t$95$0)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.7%

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

   1. add-cube-cbrt 79.4%

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

   2. pow3 79.4%

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

   3. associate-*l* 76.7%

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

   4. swap-sqr 99.1%

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

   5. cbrt-unprod 98.7%

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

   6. pow2 98.7%

      \[\leadsto -{\color{blue}{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2}\right)}}^{3} \]

4. Applied egg-rr 98.7%

   \[\leadsto -\color{blue}{{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2}\right)}^{3}} \]
5. Step-by-step derivation

   1. unpow3 98.7%

      \[\leadsto -\color{blue}{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{a \cdot b}\right)}^{2}\right) \cdot {\left(\sqrt[3]{a \cdot b}\right)}^{2}} \]

   2. pow2 98.7%

      \[\leadsto -\color{blue}{{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2}\right)}^{2}} \cdot {\left(\sqrt[3]{a \cdot b}\right)}^{2} \]

   3. unpow-prod-down 98.7%

      \[\leadsto -\color{blue}{{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2} \cdot \sqrt[3]{a \cdot b}\right)}^{2}} \]

   4. unpow2 98.7%

      \[\leadsto -{\left(\color{blue}{\left(\sqrt[3]{a \cdot b} \cdot \sqrt[3]{a \cdot b}\right)} \cdot \sqrt[3]{a \cdot b}\right)}^{2} \]

   5. add-cube-cbrt 99.7%

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

   6. unpow-prod-down 77.0%

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

   7. unpow2 77.0%

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

   8. associate-*r* 79.7%

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

   9. add-sqr-sqrt 45.3%

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

   10. pow2 45.3%

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

   11. pow-prod-down 53.5%

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

   12. *-commutative 53.5%

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

   13. unpow2 53.5%

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

   14. associate-*r* 56.4%

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

6. Applied egg-rr 56.4%

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

7. Final simplification 56.4%

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

Alternative 2: 50.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ [a, b_m] = \mathsf{sort}([a, b_m])\\ \\ \begin{array}{l} t_0 := b\_m \cdot \sqrt{a}\\ -t\_0 \cdot \left(a \cdot t\_0\right) \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
NOTE: a and b_m should be sorted in increasing order before calling this function.
(FPCore (a b_m)
 :precision binary64
 (let* ((t_0 (* b_m (sqrt a)))) (- (* t_0 (* a t_0)))))

C

b_m = fabs(b);
assert(a < b_m);
double code(double a, double b_m) {
	double t_0 = b_m * sqrt(a);
	return -(t_0 * (a * t_0));
}

Fortran

b_m = abs(b)
NOTE: a and b_m should be sorted in increasing order before calling this function.
real(8) function code(a, b_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: t_0
    t_0 = b_m * sqrt(a)
    code = -(t_0 * (a * t_0))
end function

Java

b_m = Math.abs(b);
assert a < b_m;
public static double code(double a, double b_m) {
	double t_0 = b_m * Math.sqrt(a);
	return -(t_0 * (a * t_0));
}

Python

b_m = math.fabs(b)
[a, b_m] = sort([a, b_m])
def code(a, b_m):
	t_0 = b_m * math.sqrt(a)
	return -(t_0 * (a * t_0))

Julia

b_m = abs(b)
a, b_m = sort([a, b_m])
function code(a, b_m)
	t_0 = Float64(b_m * sqrt(a))
	return Float64(-Float64(t_0 * Float64(a * t_0)))
end

MATLAB

b_m = abs(b);
a, b_m = num2cell(sort([a, b_m])){:}
function tmp = code(a, b_m)
	t_0 = b_m * sqrt(a);
	tmp = -(t_0 * (a * t_0));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
NOTE: a and b_m should be sorted in increasing order before calling this function.
code[a_, b$95$m_] := Block[{t$95$0 = N[(b$95$m * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]}, (-N[(t$95$0 * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Initial program 79.7%

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

   1. add-cube-cbrt 79.4%

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

   2. pow3 79.4%

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

   3. associate-*l* 76.7%

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

   4. swap-sqr 99.1%

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

   5. cbrt-unprod 98.7%

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

   6. pow2 98.7%

      \[\leadsto -{\color{blue}{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2}\right)}}^{3} \]

4. Applied egg-rr 98.7%

   \[\leadsto -\color{blue}{{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2}\right)}^{3}} \]
5. Step-by-step derivation

   1. unpow3 98.7%

      \[\leadsto -\color{blue}{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{a \cdot b}\right)}^{2}\right) \cdot {\left(\sqrt[3]{a \cdot b}\right)}^{2}} \]

   2. pow2 98.7%

      \[\leadsto -\color{blue}{{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2}\right)}^{2}} \cdot {\left(\sqrt[3]{a \cdot b}\right)}^{2} \]

   3. unpow-prod-down 98.7%

      \[\leadsto -\color{blue}{{\left({\left(\sqrt[3]{a \cdot b}\right)}^{2} \cdot \sqrt[3]{a \cdot b}\right)}^{2}} \]

   4. unpow2 98.7%

      \[\leadsto -{\left(\color{blue}{\left(\sqrt[3]{a \cdot b} \cdot \sqrt[3]{a \cdot b}\right)} \cdot \sqrt[3]{a \cdot b}\right)}^{2} \]

   5. add-cube-cbrt 99.7%

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

   6. unpow-prod-down 77.0%

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

   7. unpow2 77.0%

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

   8. add-sqr-sqrt 77.0%

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

   9. sqrt-unprod 68.6%

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

   10. sqr-neg 68.6%

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

   11. sqrt-unprod 16.7%

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

   12. add-sqr-sqrt 27.9%

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

   13. associate-*r* 28.1%

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

   14. *-commutative 28.1%

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

   15. add-sqr-sqrt 25.1%

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

   16. associate-*l* 25.1%

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

6. Applied egg-rr 52.6%

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

7. Final simplification 52.6%

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

Alternative 3: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ [a, b_m] = \mathsf{sort}([a, b_m])\\ \\ \begin{array}{l} t_0 := b\_m \cdot \left(a \cdot a\right)\\ \mathbf{if}\;b\_m \cdot t\_0 \leq 10^{-12}:\\ \;\;\;\;a \cdot \left(b\_m \cdot \left(b\_m \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\_m\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
NOTE: a and b_m should be sorted in increasing order before calling this function.
(FPCore (a b_m)
 :precision binary64
 (let* ((t_0 (* b_m (* a a))))
   (if (<= (* b_m t_0) 1e-12) (* a (* b_m (* b_m (- a)))) (* (- b_m) t_0))))

C

b_m = fabs(b);
assert(a < b_m);
double code(double a, double b_m) {
	double t_0 = b_m * (a * a);
	double tmp;
	if ((b_m * t_0) <= 1e-12) {
		tmp = a * (b_m * (b_m * -a));
	} else {
		tmp = -b_m * t_0;
	}
	return tmp;
}

Fortran

b_m = abs(b)
NOTE: a and b_m should be sorted in increasing order before calling this function.
real(8) function code(a, b_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b_m * (a * a)
    if ((b_m * t_0) <= 1d-12) then
        tmp = a * (b_m * (b_m * -a))
    else
        tmp = -b_m * t_0
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
assert a < b_m;
public static double code(double a, double b_m) {
	double t_0 = b_m * (a * a);
	double tmp;
	if ((b_m * t_0) <= 1e-12) {
		tmp = a * (b_m * (b_m * -a));
	} else {
		tmp = -b_m * t_0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
[a, b_m] = sort([a, b_m])
def code(a, b_m):
	t_0 = b_m * (a * a)
	tmp = 0
	if (b_m * t_0) <= 1e-12:
		tmp = a * (b_m * (b_m * -a))
	else:
		tmp = -b_m * t_0
	return tmp

Julia

b_m = abs(b)
a, b_m = sort([a, b_m])
function code(a, b_m)
	t_0 = Float64(b_m * Float64(a * a))
	tmp = 0.0
	if (Float64(b_m * t_0) <= 1e-12)
		tmp = Float64(a * Float64(b_m * Float64(b_m * Float64(-a))));
	else
		tmp = Float64(Float64(-b_m) * t_0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
a, b_m = num2cell(sort([a, b_m])){:}
function tmp_2 = code(a, b_m)
	t_0 = b_m * (a * a);
	tmp = 0.0;
	if ((b_m * t_0) <= 1e-12)
		tmp = a * (b_m * (b_m * -a));
	else
		tmp = -b_m * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
NOTE: a and b_m should be sorted in increasing order before calling this function.
code[a_, b$95$m_] := Block[{t$95$0 = N[(b$95$m * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b$95$m * t$95$0), $MachinePrecision], 1e-12], N[(a * N[(b$95$m * N[(b$95$m * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b$95$m) * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 a a) b) b) < 9.9999999999999998e-13

   1. Initial program 80.2%

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

      1. associate-*l* 77.4%

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

      2. associate-*r* 83.5%

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

      3. *-commutative 83.5%

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

      4. distribute-rgt-neg-in 83.5%

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

      5. distribute-rgt-neg-in 83.5%

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

      6. associate-*r* 95.1%

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

   3. Simplified 95.1%

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

   if 9.9999999999999998e-13 < (*.f64 (*.f64 (*.f64 a a) b) b)

   1. Initial program 79.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(b \cdot \left(a \cdot a\right)\right) \leq 10^{-12}:\\ \;\;\;\;a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(b \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ [a, b_m] = \mathsf{sort}([a, b_m])\\ \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.25 \cdot 10^{+267}:\\ \;\;\;\;a \cdot \left(b\_m \cdot \left(b\_m \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b\_m \cdot \left(-a \cdot \left(b\_m \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
NOTE: a and b_m should be sorted in increasing order before calling this function.
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 3.25e+267)
   (* a (* b_m (* b_m (- a))))
   (* b_m (- (* a (* b_m a))))))

C

b_m = fabs(b);
assert(a < b_m);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 3.25e+267) {
		tmp = a * (b_m * (b_m * -a));
	} else {
		tmp = b_m * -(a * (b_m * a));
	}
	return tmp;
}

Fortran

b_m = abs(b)
NOTE: a and b_m should be sorted in increasing order before calling this function.
real(8) function code(a, b_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if (b_m <= 3.25d+267) then
        tmp = a * (b_m * (b_m * -a))
    else
        tmp = b_m * -(a * (b_m * a))
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
assert a < b_m;
public static double code(double a, double b_m) {
	double tmp;
	if (b_m <= 3.25e+267) {
		tmp = a * (b_m * (b_m * -a));
	} else {
		tmp = b_m * -(a * (b_m * a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
[a, b_m] = sort([a, b_m])
def code(a, b_m):
	tmp = 0
	if b_m <= 3.25e+267:
		tmp = a * (b_m * (b_m * -a))
	else:
		tmp = b_m * -(a * (b_m * a))
	return tmp

Julia

b_m = abs(b)
a, b_m = sort([a, b_m])
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 3.25e+267)
		tmp = Float64(a * Float64(b_m * Float64(b_m * Float64(-a))));
	else
		tmp = Float64(b_m * Float64(-Float64(a * Float64(b_m * a))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a, b_m = num2cell(sort([a, b_m])){:}
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if (b_m <= 3.25e+267)
		tmp = a * (b_m * (b_m * -a));
	else
		tmp = b_m * -(a * (b_m * a));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
NOTE: a and b_m should be sorted in increasing order before calling this function.
code[a_, b$95$m_] := If[LessEqual[b$95$m, 3.25e+267], N[(a * N[(b$95$m * N[(b$95$m * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b$95$m * (-N[(a * N[(b$95$m * a), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.24999999999999991e267

   1. Initial program 79.9%

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

      1. associate-*l* 77.1%

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

      2. associate-*r* 84.0%

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

      3. *-commutative 84.0%

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

      4. distribute-rgt-neg-in 84.0%

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

      5. distribute-rgt-neg-in 84.0%

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

      6. associate-*r* 94.9%

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

   3. Simplified 94.9%

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

   if 3.24999999999999991e267 < b

   1. Initial program 75.7%

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

      1. distribute-rgt-neg-in 75.7%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.25 \cdot 10^{+267}:\\ \;\;\;\;a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-a \cdot \left(b \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Final simplification 79.7%

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

Reproduce

herbie shell --seed 2024044 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))