Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.4s

Alternatives: 7

Speedup: 1.0×

• 62.4% 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

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. Step-by-step derivation

   1. associate--l+ 99.9%

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

   2. sqr-pow 99.9%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 58.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{-252}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-179}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 0.49:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.4e-252)
   (pow a 4.0)
   (if (<= b 1.7e-179)
     -1.0
     (if (<= b 2.5e-17) (pow a 4.0) (if (<= b 0.49) -1.0 (pow b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.4e-252) {
		tmp = pow(a, 4.0);
	} else if (b <= 1.7e-179) {
		tmp = -1.0;
	} else if (b <= 2.5e-17) {
		tmp = pow(a, 4.0);
	} else if (b <= 0.49) {
		tmp = -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 <= 8.4d-252) then
        tmp = a ** 4.0d0
    else if (b <= 1.7d-179) then
        tmp = -1.0d0
    else if (b <= 2.5d-17) then
        tmp = a ** 4.0d0
    else if (b <= 0.49d0) then
        tmp = -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 <= 8.4e-252) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 1.7e-179) {
		tmp = -1.0;
	} else if (b <= 2.5e-17) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 0.49) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8.4e-252:
		tmp = math.pow(a, 4.0)
	elif b <= 1.7e-179:
		tmp = -1.0
	elif b <= 2.5e-17:
		tmp = math.pow(a, 4.0)
	elif b <= 0.49:
		tmp = -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.4e-252)
		tmp = a ^ 4.0;
	elseif (b <= 1.7e-179)
		tmp = -1.0;
	elseif (b <= 2.5e-17)
		tmp = a ^ 4.0;
	elseif (b <= 0.49)
		tmp = -1.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.4e-252)
		tmp = a ^ 4.0;
	elseif (b <= 1.7e-179)
		tmp = -1.0;
	elseif (b <= 2.5e-17)
		tmp = a ^ 4.0;
	elseif (b <= 0.49)
		tmp = -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.4e-252], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 1.7e-179], -1.0, If[LessEqual[b, 2.5e-17], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 0.49], -1.0, N[Power[b, 4.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 8.4000000000000001e-252 or 1.6999999999999999e-179 < b < 2.4999999999999999e-17

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around inf 50.3%

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

   if 8.4000000000000001e-252 < b < 1.6999999999999999e-179 or 2.4999999999999999e-17 < b < 0.48999999999999999

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 95.1%

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

   6. Taylor expanded in a around 0 73.6%

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

   if 0.48999999999999999 < b

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around inf 88.2%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{-252}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-179}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 0.49:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.052:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+40}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.052)
   (+ (pow a 4.0) -1.0)
   (if (<= b 3.6e+40)
     (+ (+ (* 4.0 (* b b)) (pow b 4.0)) -1.0)
     (if (<= b 2.9e+49) (pow a 4.0) (pow b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.052) {
		tmp = pow(a, 4.0) + -1.0;
	} else if (b <= 3.6e+40) {
		tmp = ((4.0 * (b * b)) + pow(b, 4.0)) + -1.0;
	} else if (b <= 2.9e+49) {
		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 <= 0.052d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else if (b <= 3.6d+40) then
        tmp = ((4.0d0 * (b * b)) + (b ** 4.0d0)) + (-1.0d0)
    else if (b <= 2.9d+49) 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 <= 0.052) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else if (b <= 3.6e+40) {
		tmp = ((4.0 * (b * b)) + Math.pow(b, 4.0)) + -1.0;
	} else if (b <= 2.9e+49) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.052:
		tmp = math.pow(a, 4.0) + -1.0
	elif b <= 3.6e+40:
		tmp = ((4.0 * (b * b)) + math.pow(b, 4.0)) + -1.0
	elif b <= 2.9e+49:
		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 <= 0.052)
		tmp = Float64((a ^ 4.0) + -1.0);
	elseif (b <= 3.6e+40)
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + (b ^ 4.0)) + -1.0);
	elseif (b <= 2.9e+49)
		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 <= 0.052)
		tmp = (a ^ 4.0) + -1.0;
	elseif (b <= 3.6e+40)
		tmp = ((4.0 * (b * b)) + (b ^ 4.0)) + -1.0;
	elseif (b <= 2.9e+49)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.052], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[b, 3.6e+40], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[b, 2.9e+49], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < 0.0519999999999999976

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 79.0%

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

   if 0.0519999999999999976 < b < 3.59999999999999996e40

   1. Initial program 99.2%

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

      1. add-cbrt-cube 83.2%

         \[\leadsto \left(\color{blue}{\sqrt[3]{\left({\left(a \cdot a + b \cdot b\right)}^{2} \cdot {\left(a \cdot a + b \cdot b\right)}^{2}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{2}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

      2. pow3 83.2%

         \[\leadsto \left(\sqrt[3]{\color{blue}{{\left({\left(a \cdot a + b \cdot b\right)}^{2}\right)}^{3}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

      3. metadata-eval 83.2%

         \[\leadsto \left(\sqrt[3]{{\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{\left(\frac{4}{2}\right)}}\right)}^{3}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

      4. sqrt-pow2 83.6%

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

      5. hypot-udef 83.6%

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

      6. pow-pow 83.8%

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

      7. metadata-eval 83.8%

         \[\leadsto \left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{\color{blue}{12}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   4. Applied egg-rr 83.8%

      \[\leadsto \left(\color{blue}{\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{12}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   5. Taylor expanded in a around 0 67.7%

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

   if 3.59999999999999996e40 < b < 2.9e49

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sqr-pow 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

   if 2.9e49 < b

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around inf 97.3%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.052:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+40}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 610:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+40} \lor \neg \left(b \leq 3.4 \cdot 10^{+49}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 610.0)
   (+ (pow a 4.0) -1.0)
   (if (or (<= b 6.6e+40) (not (<= b 3.4e+49))) (pow b 4.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 610.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else if ((b <= 6.6e+40) || !(b <= 3.4e+49)) {
		tmp = pow(b, 4.0);
	} 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 (b <= 610.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else if ((b <= 6.6d+40) .or. (.not. (b <= 3.4d+49))) then
        tmp = b ** 4.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 610.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else if ((b <= 6.6e+40) || !(b <= 3.4e+49)) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 610.0:
		tmp = math.pow(a, 4.0) + -1.0
	elif (b <= 6.6e+40) or not (b <= 3.4e+49):
		tmp = math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 610.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	elseif ((b <= 6.6e+40) || !(b <= 3.4e+49))
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 610.0)
		tmp = (a ^ 4.0) + -1.0;
	elseif ((b <= 6.6e+40) || ~((b <= 3.4e+49)))
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 610.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], If[Or[LessEqual[b, 6.6e+40], N[Not[LessEqual[b, 3.4e+49]], $MachinePrecision]], N[Power[b, 4.0], $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 610

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 78.7%

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

   if 610 < b < 6.5999999999999997e40 or 3.4000000000000001e49 < b

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around inf 95.3%

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

   if 6.5999999999999997e40 < b < 3.4000000000000001e49

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sqr-pow 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 610:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+40} \lor \neg \left(b \leq 3.4 \cdot 10^{+49}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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]

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%

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

Alternative 6: 46.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.0027:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= a 0.0027) -1.0 (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.0027) {
		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 <= 0.0027d0) 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 <= 0.0027) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.0027:
		tmp = -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.0027)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 0.0027)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.0027], -1.0, N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 0.0027000000000000001

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 62.4%

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

   6. Taylor expanded in a around 0 30.5%

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

   if 0.0027000000000000001 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around inf 80.4%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 0.0027:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 24.2% 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. Step-by-step derivation

   1. associate--l+ 99.9%

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

   2. sqr-pow 99.9%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 67.0%

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

6. Taylor expanded in a around 0 23.3%

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

7. Final simplification 23.3%

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

Reproduce

herbie shell --seed 2024018 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))