Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.7s

Alternatives: 5

Speedup: 1.0×

• 61.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(b * b))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(b * b))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\left({b}^{4} + \mathsf{fma}\left(2, a \cdot \left(b \cdot \left(b \cdot a\right)\right), {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) + -1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+
  (+ (+ (pow b 4.0) (fma 2.0 (* a (* b (* b a))) (pow a 4.0))) (* 4.0 (* b b)))
  -1.0))

C

double code(double a, double b) {
	return ((pow(b, 4.0) + fma(2.0, (a * (b * (b * a))), pow(a, 4.0))) + (4.0 * (b * b))) + -1.0;
}

Julia

function code(a, b)
	return Float64(Float64(Float64((b ^ 4.0) + fma(2.0, Float64(a * Float64(b * Float64(b * a))), (a ^ 4.0))) + Float64(4.0 * Float64(b * b))) + -1.0)
end

Wolfram

code[a_, b_] := N[(N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(2.0 * N[(a * N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0 93.3%

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

   1. +-commutative 93.3%

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

   2. distribute-lft-in 88.2%

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

   3. associate-*r* 88.2%

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

   4. *-commutative 88.2%

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

   5. associate-*r* 88.2%

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

   6. pow-sqr 88.3%

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

   7. metadata-eval 88.3%

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

   8. fma-define 88.3%

      \[\leadsto \left(\left({b}^{4} + \color{blue}{\mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {a}^{4}\right)}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   9. unpow2 88.3%

      \[\leadsto \left(\left({b}^{4} + \mathsf{fma}\left(2, \color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}, {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   10. unpow2 88.3%

       \[\leadsto \left(\left({b}^{4} + \mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}, {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   11. swap-sqr 100.0%

       \[\leadsto \left(\left({b}^{4} + \mathsf{fma}\left(2, \color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}, {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   12. unpow2 100.0%

       \[\leadsto \left(\left({b}^{4} + \mathsf{fma}\left(2, \color{blue}{{\left(a \cdot b\right)}^{2}}, {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   13. *-commutative 100.0%

       \[\leadsto \left(\left({b}^{4} + \mathsf{fma}\left(2, {\color{blue}{\left(b \cdot a\right)}}^{2}, {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

5. Simplified 100.0%

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

   1. unpow2 100.0%

      \[\leadsto \left(\left({b}^{4} + \mathsf{fma}\left(2, \color{blue}{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}, {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   2. associate-*r* 100.0%

      \[\leadsto \left(\left({b}^{4} + \mathsf{fma}\left(2, \color{blue}{\left(\left(b \cdot a\right) \cdot b\right) \cdot a}, {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

7. Applied egg-rr 100.0%

   \[\leadsto \left(\left({b}^{4} + \mathsf{fma}\left(2, \color{blue}{\left(\left(b \cdot a\right) \cdot b\right) \cdot a}, {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

8. Final simplification 100.0%

   \[\leadsto \left(\left({b}^{4} + \mathsf{fma}\left(2, a \cdot \left(b \cdot \left(b \cdot a\right)\right), {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) + -1 \]
9. Add Preprocessing

Alternative 2: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+28)
   (+ (+ (pow a 4.0) (* 4.0 (* b b))) -1.0)
   (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+28) {
		tmp = (pow(a, 4.0) + (4.0 * (b * b))) + -1.0;
	} else {
		tmp = pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+28) then
        tmp = ((a ** 4.0d0) + (4.0d0 * (b * b))) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+28) {
		tmp = (Math.pow(a, 4.0) + (4.0 * (b * b))) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+28:
		tmp = (math.pow(a, 4.0) + (4.0 * (b * b))) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+28)
		tmp = Float64(Float64((a ^ 4.0) + Float64(4.0 * Float64(b * b))) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+28)
		tmp = ((a ^ 4.0) + (4.0 * (b * b))) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+28], N[(N[(N[Power[a, 4.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999957e28

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 99.4%

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

   if 4.99999999999999957e28 < (*.f64 b b)

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 93.1%

      \[\leadsto \left(\color{blue}{{b}^{4}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   4. Taylor expanded in b around inf 93.1%

      \[\leadsto \color{blue}{{b}^{4}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(4 \cdot \left(b \cdot b\right) + {\left(b \cdot b + a \cdot a\right)}^{2}\right) + -1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+ (+ (* 4.0 (* b b)) (pow (+ (* b b) (* a a)) 2.0)) -1.0))

C

double code(double a, double b) {
	return ((4.0 * (b * b)) + pow(((b * b) + (a * a)), 2.0)) + -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((4.0d0 * (b * b)) + (((b * b) + (a * a)) ** 2.0d0)) + (-1.0d0)
end function

Java

public static double code(double a, double b) {
	return ((4.0 * (b * b)) + Math.pow(((b * b) + (a * a)), 2.0)) + -1.0;
}

Python

def code(a, b):
	return ((4.0 * (b * b)) + math.pow(((b * b) + (a * a)), 2.0)) + -1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(4.0 * Float64(b * b)) + (Float64(Float64(b * b) + Float64(a * a)) ^ 2.0)) + -1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((4.0 * (b * b)) + (((b * b) + (a * a)) ^ 2.0)) + -1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(b \cdot b + a \cdot a\right)}^{2}\right) + -1 \]
4. Add Preprocessing

Alternative 4: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.8e+14) (+ (pow a 4.0) -1.0) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.8e+14) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.8d+14) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.8e+14) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.8e+14:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.8e+14)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.8e+14)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.8e+14], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8e14

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 89.2%

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

   4. Taylor expanded in a around inf 80.2%

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]

   if 3.8e14 < b

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 92.5%

      \[\leadsto \left(\color{blue}{{b}^{4}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   4. Taylor expanded in b around inf 92.5%

      \[\leadsto \color{blue}{{b}^{4}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {a}^{4} + -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ (pow a 4.0) -1.0))

C

double code(double a, double b) {
	return pow(a, 4.0) + -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a ** 4.0d0) + (-1.0d0)
end function

Java

public static double code(double a, double b) {
	return Math.pow(a, 4.0) + -1.0;
}

Python

def code(a, b):
	return math.pow(a, 4.0) + -1.0

Julia

function code(a, b)
	return Float64((a ^ 4.0) + -1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = (a ^ 4.0) + -1.0;
end

Wolfram

code[a_, b_] := N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around inf 82.8%

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

4. Taylor expanded in a around inf 68.4%

   \[\leadsto \color{blue}{{a}^{4}} - 1 \]

5. Final simplification 68.4%

   \[\leadsto {a}^{4} + -1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))