Bouland and Aaronson, Equation (24)

Percentage Accurate: 72.7% → 98.4%

Time: 7.0s

Alternatives: 9

Speedup: 8.5×

• 61.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 72.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.4% accurate, 0.2× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) (+ a 3.0))))
   (if (<=
        (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) t_0)))
        INFINITY)
     (fma 4.0 (fma a (- a (* a a)) t_0) (+ (pow (hypot a b) 4.0) -1.0))
     (+ -1.0 (* (* a (* a a)) (+ a -4.0))))))

C

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

Julia

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

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(4.0 * N[(a * N[(a - N[(a * a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 97.8%

   \[\begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a + -4\right)\\ \end{array} \]

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))))
   (if (<= t_0 INFINITY) (+ t_0 -1.0) (+ -1.0 (* (* a (* a a)) (+ a -4.0))))))

C

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

Java

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

Python

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

Julia

b = abs(b)
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(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(-1.0 + Float64(Float64(a * Float64(a * a)) * Float64(a + -4.0)));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: b should be positive before calling this function
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[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(-1.0 + N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 97.6%

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

Alternative 3: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0002:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0002)
   (+ -1.0 (* (+ a -4.0) (pow a 3.0)))
   (+ -1.0 (+ (pow b 4.0) (* b (* b 12.0))))))

C

b = abs(b);
double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0002) {
		tmp = -1.0 + ((a + -4.0) * pow(a, 3.0));
	} else {
		tmp = -1.0 + (pow(b, 4.0) + (b * (b * 12.0)));
	}
	return tmp;
}

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.0002d0) then
        tmp = (-1.0d0) + ((a + (-4.0d0)) * (a ** 3.0d0))
    else
        tmp = (-1.0d0) + ((b ** 4.0d0) + (b * (b * 12.0d0)))
    end if
    code = tmp
end function

Java

b = Math.abs(b);
public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0002) {
		tmp = -1.0 + ((a + -4.0) * Math.pow(a, 3.0));
	} else {
		tmp = -1.0 + (Math.pow(b, 4.0) + (b * (b * 12.0)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b):
	tmp = 0
	if (b * b) <= 0.0002:
		tmp = -1.0 + ((a + -4.0) * math.pow(a, 3.0))
	else:
		tmp = -1.0 + (math.pow(b, 4.0) + (b * (b * 12.0)))
	return tmp

Julia

b = abs(b)
function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0002)
		tmp = Float64(-1.0 + Float64(Float64(a + -4.0) * (a ^ 3.0)));
	else
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(b * Float64(b * 12.0))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.0002)
		tmp = -1.0 + ((a + -4.0) * (a ^ 3.0));
	else
		tmp = -1.0 + ((b ^ 4.0) + (b * (b * 12.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0002], N[(-1.0 + N[(N[(a + -4.0), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.5%

   \[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0002:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \end{array} \]

Alternative 4: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0002:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0002)
   (+ -1.0 (* (+ a -4.0) (pow a 3.0)))
   (+ -1.0 (pow b 4.0))))

C

b = abs(b);
double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0002) {
		tmp = -1.0 + ((a + -4.0) * pow(a, 3.0));
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.0002d0) then
        tmp = (-1.0d0) + ((a + (-4.0d0)) * (a ** 3.0d0))
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

b = Math.abs(b);
public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0002) {
		tmp = -1.0 + ((a + -4.0) * Math.pow(a, 3.0));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b):
	tmp = 0
	if (b * b) <= 0.0002:
		tmp = -1.0 + ((a + -4.0) * math.pow(a, 3.0))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

b = abs(b)
function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0002)
		tmp = Float64(-1.0 + Float64(Float64(a + -4.0) * (a ^ 3.0)));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.0002)
		tmp = -1.0 + ((a + -4.0) * (a ^ 3.0));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0002], N[(-1.0 + N[(N[(a + -4.0), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.0%

   \[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0002:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 5: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0002:\\ \;\;\;\;-1 + \left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0002)
   (+ -1.0 (* (* a (* a a)) (+ a -4.0)))
   (+ -1.0 (pow b 4.0))))

C

b = abs(b);
double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0002) {
		tmp = -1.0 + ((a * (a * a)) * (a + -4.0));
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.0002d0) then
        tmp = (-1.0d0) + ((a * (a * a)) * (a + (-4.0d0)))
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

b = Math.abs(b);
public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0002) {
		tmp = -1.0 + ((a * (a * a)) * (a + -4.0));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b):
	tmp = 0
	if (b * b) <= 0.0002:
		tmp = -1.0 + ((a * (a * a)) * (a + -4.0))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

b = abs(b)
function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0002)
		tmp = Float64(-1.0 + Float64(Float64(a * Float64(a * a)) * Float64(a + -4.0)));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.0002)
		tmp = -1.0 + ((a * (a * a)) * (a + -4.0));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0002], N[(-1.0 + N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.9%

   \[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0002:\\ \;\;\;\;-1 + \left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 6: 83.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+285}:\\ \;\;\;\;-1 + \left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+285)
   (+ -1.0 (* (* a (* a a)) (+ a -4.0)))
   (+ -1.0 (* b (* b 12.0)))))

C

b = abs(b);
double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+285) {
		tmp = -1.0 + ((a * (a * a)) * (a + -4.0));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+285) then
        tmp = (-1.0d0) + ((a * (a * a)) * (a + (-4.0d0)))
    else
        tmp = (-1.0d0) + (b * (b * 12.0d0))
    end if
    code = tmp
end function

Java

b = Math.abs(b);
public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+285) {
		tmp = -1.0 + ((a * (a * a)) * (a + -4.0));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b):
	tmp = 0
	if (b * b) <= 4e+285:
		tmp = -1.0 + ((a * (a * a)) * (a + -4.0))
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

b = abs(b)
function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+285)
		tmp = Float64(-1.0 + Float64(Float64(a * Float64(a * a)) * Float64(a + -4.0)));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+285)
		tmp = -1.0 + ((a * (a * a)) * (a + -4.0));
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+285], N[(-1.0 + N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 85.8%

   \[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+285}:\\ \;\;\;\;-1 + \left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 7: 72.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+65}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot -4\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+150}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b)
 :precision binary64
 (if (<= a -5.8e+65)
   (+ -1.0 (* (* a a) (+ 4.0 (* a -4.0))))
   (if (<= a 6.2e+150) (+ -1.0 (* b (* b 12.0))) (+ -1.0 (* (* a a) 4.0)))))

C

b = abs(b);
double code(double a, double b) {
	double tmp;
	if (a <= -5.8e+65) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * -4.0)));
	} else if (a <= 6.2e+150) {
		tmp = -1.0 + (b * (b * 12.0));
	} else {
		tmp = -1.0 + ((a * a) * 4.0);
	}
	return tmp;
}

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.8d+65)) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (a * (-4.0d0))))
    else if (a <= 6.2d+150) then
        tmp = (-1.0d0) + (b * (b * 12.0d0))
    else
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    end if
    code = tmp
end function

Java

b = Math.abs(b);
public static double code(double a, double b) {
	double tmp;
	if (a <= -5.8e+65) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * -4.0)));
	} else if (a <= 6.2e+150) {
		tmp = -1.0 + (b * (b * 12.0));
	} else {
		tmp = -1.0 + ((a * a) * 4.0);
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b):
	tmp = 0
	if a <= -5.8e+65:
		tmp = -1.0 + ((a * a) * (4.0 + (a * -4.0)))
	elif a <= 6.2e+150:
		tmp = -1.0 + (b * (b * 12.0))
	else:
		tmp = -1.0 + ((a * a) * 4.0)
	return tmp

Julia

b = abs(b)
function code(a, b)
	tmp = 0.0
	if (a <= -5.8e+65)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(a * -4.0))));
	elseif (a <= 6.2e+150)
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.8e+65)
		tmp = -1.0 + ((a * a) * (4.0 + (a * -4.0)));
	elseif (a <= 6.2e+150)
		tmp = -1.0 + (b * (b * 12.0));
	else
		tmp = -1.0 + ((a * a) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_] := If[LessEqual[a, -5.8e+65], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.2e+150], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 71.7%

   \[\begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+65}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot -4\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+150}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \end{array} \]

Alternative 8: 69.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+285}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+285) (+ -1.0 (* (* a a) 4.0)) (+ -1.0 (* b (* b 12.0)))))

C

b = abs(b);
double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+285) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+285) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = (-1.0d0) + (b * (b * 12.0d0))
    end if
    code = tmp
end function

Java

b = Math.abs(b);
public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+285) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b):
	tmp = 0
	if (b * b) <= 4e+285:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

b = abs(b)
function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+285)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+285)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+285], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 69.5%

   \[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+285}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 9: 51.7% accurate, 18.3× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b) :precision binary64 (+ -1.0 (* (* a a) 4.0)))

C

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

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + ((a * a) * 4.0d0)
end function

Java

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

Python

b = abs(b)
def code(a, b):
	return -1.0 + ((a * a) * 4.0)

Julia

b = abs(b)
function code(a, b)
	return Float64(-1.0 + Float64(Float64(a * a) * 4.0))
end

MATLAB

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

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_] := N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.3%

   \[-1 + \left(a \cdot a\right) \cdot 4 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))