Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 5.1s

Alternatives: 7

Speedup: 1.0×

• 61.0% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(2.0 * N[(a * N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 92.9%

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

   1. +-commutative 92.9%

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

   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. fma-undefine 100.0%

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

   2. +-commutative 100.0%

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

7. Applied egg-rr 100.0%

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

   1. unpow2 100.0%

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

   2. associate-*r* 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 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 3: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq 3.7:\\ \;\;\;\;\left({b}^{4} + t\_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + t\_0\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= a 3.7) (+ (+ (pow b 4.0) t_0) -1.0) (+ (+ (pow a 4.0) t_0) -1.0))))

C

double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if (a <= 3.7) {
		tmp = (pow(b, 4.0) + t_0) + -1.0;
	} else {
		tmp = (pow(a, 4.0) + t_0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (b * b)
    if (a <= 3.7d0) then
        tmp = ((b ** 4.0d0) + t_0) + (-1.0d0)
    else
        tmp = ((a ** 4.0d0) + t_0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if (a <= 3.7) {
		tmp = (Math.pow(b, 4.0) + t_0) + -1.0;
	} else {
		tmp = (Math.pow(a, 4.0) + t_0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 4.0 * (b * b)
	tmp = 0
	if a <= 3.7:
		tmp = (math.pow(b, 4.0) + t_0) + -1.0
	else:
		tmp = (math.pow(a, 4.0) + t_0) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(4.0 * Float64(b * b))
	tmp = 0.0
	if (a <= 3.7)
		tmp = Float64(Float64((b ^ 4.0) + t_0) + -1.0);
	else
		tmp = Float64(Float64((a ^ 4.0) + t_0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 4.0 * (b * b);
	tmp = 0.0;
	if (a <= 3.7)
		tmp = ((b ^ 4.0) + t_0) + -1.0;
	else
		tmp = ((a ^ 4.0) + t_0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3.7], N[(N[(N[Power[b, 4.0], $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[Power[a, 4.0], $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.7000000000000002

   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 78.8%

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

   if 3.7000000000000002 < a

   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 98.3%

      \[\leadsto \left(\color{blue}{{a}^{4}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.7:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (a <= 4.0) {
		tmp = pow(b, 4.0) + -1.0;
	} else {
		tmp = (pow(a, 4.0) + (4.0 * (b * b))) + -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 (a <= 4.0d0) then
        tmp = (b ** 4.0d0) + (-1.0d0)
    else
        tmp = ((a ** 4.0d0) + (4.0d0 * (b * b))) + (-1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if a < 4

   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 91.0%

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

      1. +-commutative 91.0%

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

      2. distribute-lft-in 86.0%

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

      3. associate-*r* 86.0%

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

      4. *-commutative 86.0%

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

      5. associate-*r* 86.0%

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

      6. pow-sqr 86.0%

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

      7. metadata-eval 86.0%

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

      8. fma-define 86.0%

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

      9. unpow2 86.0%

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

      10. unpow2 86.0%

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

      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. Taylor expanded in b around inf 77.7%

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

   if 4 < a

   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 98.3%

      \[\leadsto \left(\color{blue}{{a}^{4}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

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

Alternative 5: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+21}:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.08e+21) (+ (pow b 4.0) -1.0) (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.08e+21) {
		tmp = pow(b, 4.0) + -1.0;
	} else {
		tmp = pow(a, 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 (a <= 1.08d+21) then
        tmp = (b ** 4.0d0) + (-1.0d0)
    else
        tmp = (a ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.08e+21) {
		tmp = Math.pow(b, 4.0) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.08e+21:
		tmp = math.pow(b, 4.0) + -1.0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.08e+21)
		tmp = Float64((b ^ 4.0) + -1.0);
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.08e+21)
		tmp = (b ^ 4.0) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.08e+21], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.08e21

   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 91.1%

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

      1. +-commutative 91.1%

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

      2. distribute-lft-in 86.2%

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

      3. associate-*r* 86.2%

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

      4. *-commutative 86.2%

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

      5. associate-*r* 86.2%

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

      6. pow-sqr 86.3%

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

      7. metadata-eval 86.3%

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

      8. fma-define 86.3%

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

      9. unpow2 86.3%

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

      10. unpow2 86.3%

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

      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. Taylor expanded in b around inf 77.2%

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

   if 1.08e21 < a

   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 99.9%

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

      1. +-commutative 99.9%

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

      2. distribute-lft-in 96.1%

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

      3. associate-*r* 96.1%

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

      4. *-commutative 96.1%

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

      5. associate-*r* 96.1%

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

      6. pow-sqr 96.2%

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

      7. metadata-eval 96.2%

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

      8. fma-define 96.2%

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

      9. unpow2 96.2%

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

      10. unpow2 96.2%

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

      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. Taylor expanded in b around 0 98.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+21}:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.5% 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 0 92.9%

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

   1. +-commutative 92.9%

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

   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. Taylor expanded in b around 0 72.7%

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

7. Final simplification 72.7%

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

Alternative 7: 25.4% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

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 69.3%

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

4. Taylor expanded in b around 0 28.9%

   \[\leadsto \color{blue}{-1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024097 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))