Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 98.3%

Time: 6.3s

Alternatives: 6

Speedup: 1.1×

• 61.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 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 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 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 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))))
   (if (<= t_0 INFINITY) (+ t_0 -1.0) (+ (pow a 4.0) -1.0))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 95.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+17}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+69} \lor \neg \left(b \leq 5.2 \cdot 10^{+75}\right):\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.75e+17)
   (+ (* (pow a 3.0) (+ a 4.0)) -1.0)
   (if (or (<= b 6.5e+69) (not (<= b 5.2e+75)))
     (+ (pow b 4.0) -1.0)
     (+ (pow a 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.75e+17) {
		tmp = (pow(a, 3.0) * (a + 4.0)) + -1.0;
	} else if ((b <= 6.5e+69) || !(b <= 5.2e+75)) {
		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 (b <= 1.75d+17) then
        tmp = ((a ** 3.0d0) * (a + 4.0d0)) + (-1.0d0)
    else if ((b <= 6.5d+69) .or. (.not. (b <= 5.2d+75))) 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 (b <= 1.75e+17) {
		tmp = (Math.pow(a, 3.0) * (a + 4.0)) + -1.0;
	} else if ((b <= 6.5e+69) || !(b <= 5.2e+75)) {
		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 b <= 1.75e+17:
		tmp = (math.pow(a, 3.0) * (a + 4.0)) + -1.0
	elif (b <= 6.5e+69) or not (b <= 5.2e+75):
		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 (b <= 1.75e+17)
		tmp = Float64(Float64((a ^ 3.0) * Float64(a + 4.0)) + -1.0);
	elseif ((b <= 6.5e+69) || !(b <= 5.2e+75))
		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 (b <= 1.75e+17)
		tmp = ((a ^ 3.0) * (a + 4.0)) + -1.0;
	elseif ((b <= 6.5e+69) || ~((b <= 5.2e+75)))
		tmp = (b ^ 4.0) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.75e+17], N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[Or[LessEqual[b, 6.5e+69], N[Not[LessEqual[b, 5.2e+75]], $MachinePrecision]], 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 3 regimes

2. if b < 1.75e17

   1. Initial program 79.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 80.4%

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

      1. associate-*r/ 80.4%

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

      2. metadata-eval 80.4%

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

   5. Simplified 80.4%

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

   6. Taylor expanded in a around 0 80.4%

      \[\leadsto \color{blue}{{a}^{3} \cdot \left(4 + a\right)} - 1 \]

   if 1.75e17 < b < 6.5000000000000001e69 or 5.1999999999999997e75 < b

   1. Initial program 72.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 96.1%

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

   if 6.5000000000000001e69 < b < 5.1999999999999997e75

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+17}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+69} \lor \neg \left(b \leq 5.2 \cdot 10^{+75}\right):\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{+17}:\\ \;\;\;\;{a}^{4} \cdot \left(1 + \frac{4}{a}\right) + -1\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+70} \lor \neg \left(b \leq 4.1 \cdot 10^{+74}\right):\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.22e+17)
   (+ (* (pow a 4.0) (+ 1.0 (/ 4.0 a))) -1.0)
   (if (or (<= b 3.1e+70) (not (<= b 4.1e+74)))
     (+ (pow b 4.0) -1.0)
     (+ (pow a 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.22e+17) {
		tmp = (pow(a, 4.0) * (1.0 + (4.0 / a))) + -1.0;
	} else if ((b <= 3.1e+70) || !(b <= 4.1e+74)) {
		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 (b <= 1.22d+17) then
        tmp = ((a ** 4.0d0) * (1.0d0 + (4.0d0 / a))) + (-1.0d0)
    else if ((b <= 3.1d+70) .or. (.not. (b <= 4.1d+74))) 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 (b <= 1.22e+17) {
		tmp = (Math.pow(a, 4.0) * (1.0 + (4.0 / a))) + -1.0;
	} else if ((b <= 3.1e+70) || !(b <= 4.1e+74)) {
		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 b <= 1.22e+17:
		tmp = (math.pow(a, 4.0) * (1.0 + (4.0 / a))) + -1.0
	elif (b <= 3.1e+70) or not (b <= 4.1e+74):
		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 (b <= 1.22e+17)
		tmp = Float64(Float64((a ^ 4.0) * Float64(1.0 + Float64(4.0 / a))) + -1.0);
	elseif ((b <= 3.1e+70) || !(b <= 4.1e+74))
		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 (b <= 1.22e+17)
		tmp = ((a ^ 4.0) * (1.0 + (4.0 / a))) + -1.0;
	elseif ((b <= 3.1e+70) || ~((b <= 4.1e+74)))
		tmp = (b ^ 4.0) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.22e+17], N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[Or[LessEqual[b, 3.1e+70], N[Not[LessEqual[b, 4.1e+74]], $MachinePrecision]], 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 3 regimes

2. if b < 1.22e17

   1. Initial program 79.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 80.4%

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

      1. associate-*r/ 80.4%

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

      2. metadata-eval 80.4%

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

   5. Simplified 80.4%

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

   if 1.22e17 < b < 3.1000000000000003e70 or 4.1e74 < b

   1. Initial program 72.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 96.1%

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

   if 3.1000000000000003e70 < b < 4.1e74

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{+17}:\\ \;\;\;\;{a}^{4} \cdot \left(1 + \frac{4}{a}\right) + -1\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+70} \lor \neg \left(b \leq 4.1 \cdot 10^{+74}\right):\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -620000000000 \lor \neg \left(a \leq 2.75 \cdot 10^{+22}\right):\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -620000000000.0) (not (<= a 2.75e+22)))
   (+ (pow a 4.0) -1.0)
   (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -620000000000.0) || !(a <= 2.75e+22)) {
		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 ((a <= (-620000000000.0d0)) .or. (.not. (a <= 2.75d+22))) 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 ((a <= -620000000000.0) || !(a <= 2.75e+22)) {
		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 (a <= -620000000000.0) or not (a <= 2.75e+22):
		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 ((a <= -620000000000.0) || !(a <= 2.75e+22))
		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 ((a <= -620000000000.0) || ~((a <= 2.75e+22)))
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -620000000000.0], N[Not[LessEqual[a, 2.75e+22]], $MachinePrecision]], 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 a < -6.2e11 or 2.7500000000000001e22 < a

   1. Initial program 50.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 92.6%

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

   if -6.2e11 < a < 2.7500000000000001e22

   1. Initial program 99.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 95.2%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -620000000000 \lor \neg \left(a \leq 2.75 \cdot 10^{+22}\right):\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 1.3× 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 77.3%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around inf 71.1%

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

4. Final simplification 71.1%

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

Alternative 6: 24.9% accurate, 130.0× 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 77.3%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in b around inf 67.1%

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

4. Taylor expanded in b around 0 28.6%

   \[\leadsto \color{blue}{-1} \]

5. Final simplification 28.6%

   \[\leadsto -1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))