Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.4s

Alternatives: 7

Speedup: 1.0×

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

Specification

Math

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 3: 47.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-275}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-127}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-119}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-79}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-49}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 2500000:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6e-275)
   -1.0
   (if (<= b 5.5e-127)
     (pow a 4.0)
     (if (<= b 6e-119)
       -1.0
       (if (<= b 8.2e-79)
         (pow a 4.0)
         (if (<= b 2.2e-49)
           -1.0
           (if (<= b 2500000.0) (pow a 4.0) (pow b 4.0))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.6e-275) {
		tmp = -1.0;
	} else if (b <= 5.5e-127) {
		tmp = pow(a, 4.0);
	} else if (b <= 6e-119) {
		tmp = -1.0;
	} else if (b <= 8.2e-79) {
		tmp = pow(a, 4.0);
	} else if (b <= 2.2e-49) {
		tmp = -1.0;
	} else if (b <= 2500000.0) {
		tmp = pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.6d-275) then
        tmp = -1.0d0
    else if (b <= 5.5d-127) then
        tmp = a ** 4.0d0
    else if (b <= 6d-119) then
        tmp = -1.0d0
    else if (b <= 8.2d-79) then
        tmp = a ** 4.0d0
    else if (b <= 2.2d-49) then
        tmp = -1.0d0
    else if (b <= 2500000.0d0) then
        tmp = a ** 4.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.6e-275) {
		tmp = -1.0;
	} else if (b <= 5.5e-127) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 6e-119) {
		tmp = -1.0;
	} else if (b <= 8.2e-79) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 2.2e-49) {
		tmp = -1.0;
	} else if (b <= 2500000.0) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.6e-275:
		tmp = -1.0
	elif b <= 5.5e-127:
		tmp = math.pow(a, 4.0)
	elif b <= 6e-119:
		tmp = -1.0
	elif b <= 8.2e-79:
		tmp = math.pow(a, 4.0)
	elif b <= 2.2e-49:
		tmp = -1.0
	elif b <= 2500000.0:
		tmp = math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.6e-275)
		tmp = -1.0;
	elseif (b <= 5.5e-127)
		tmp = a ^ 4.0;
	elseif (b <= 6e-119)
		tmp = -1.0;
	elseif (b <= 8.2e-79)
		tmp = a ^ 4.0;
	elseif (b <= 2.2e-49)
		tmp = -1.0;
	elseif (b <= 2500000.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.6e-275)
		tmp = -1.0;
	elseif (b <= 5.5e-127)
		tmp = a ^ 4.0;
	elseif (b <= 6e-119)
		tmp = -1.0;
	elseif (b <= 8.2e-79)
		tmp = a ^ 4.0;
	elseif (b <= 2.2e-49)
		tmp = -1.0;
	elseif (b <= 2500000.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.6e-275], -1.0, If[LessEqual[b, 5.5e-127], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 6e-119], -1.0, If[LessEqual[b, 8.2e-79], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 2.2e-49], -1.0, If[LessEqual[b, 2500000.0], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]]]]]

Derivation

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1. Add Preprocessing

Alternative 4: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(a \cdot \left(b \cdot \left(a \cdot b\right)\right)\right) + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1)
   (- (pow a 4.0) 1.0)
   (+ (* 2.0 (* a (* b (* a b)))) (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1) {
		tmp = pow(a, 4.0) - 1.0;
	} else {
		tmp = (2.0 * (a * (b * (a * b)))) + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.1d0) then
        tmp = (a ** 4.0d0) - 1.0d0
    else
        tmp = (2.0d0 * (a * (b * (a * b)))) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.1) {
		tmp = Math.pow(a, 4.0) - 1.0;
	} else {
		tmp = (2.0 * (a * (b * (a * b)))) + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.1:
		tmp = math.pow(a, 4.0) - 1.0
	else:
		tmp = (2.0 * (a * (b * (a * b)))) + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64((a ^ 4.0) - 1.0);
	else
		tmp = Float64(Float64(2.0 * Float64(a * Float64(b * Float64(a * b)))) + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.1)
		tmp = (a ^ 4.0) - 1.0;
	else
		tmp = (2.0 * (a * (b * (a * b)))) + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1], N[(N[Power[a, 4.0], $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(2.0 * N[(a * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 5: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 760000:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 760000.0) (- (pow a 4.0) 1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 760000.0) {
		tmp = pow(a, 4.0) - 1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 760000.0d0) then
        tmp = (a ** 4.0d0) - 1.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 760000.0) {
		tmp = Math.pow(a, 4.0) - 1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 760000.0:
		tmp = math.pow(a, 4.0) - 1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 760000.0)
		tmp = Float64((a ^ 4.0) - 1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 760000.0)
		tmp = (a ^ 4.0) - 1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 760000.0], N[(N[Power[a, 4.0], $MachinePrecision] - 1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 6: 47.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.44:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= a 0.44) -1.0 (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.44) {
		tmp = -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 0.44d0) then
        tmp = -1.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 0.44) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.44:
		tmp = -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.44)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 0.44)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.44], -1.0, N[Power[a, 4.0], $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 7: 25.6% 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

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1. Add Preprocessing

Reproduce

herbie shell --seed 2023342 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))