Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.8s

Alternatives: 6

Speedup: 1.0×

• 61.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× 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 99.8%

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

   1. associate--l+ 99.8%

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

   2. sqr-pow 99.8%

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

   3. sqr-pow 99.8%

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

   4. unpow2 99.8%

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

   5. unpow1 99.8%

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

   6. sqr-pow 99.8%

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

   7. associate-*r* 99.9%

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

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. Final simplification 100.0%

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

Alternative 2: 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 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 34.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot {b}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.48) -1.0 (* a (pow b 3.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = a * pow(b, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.48d0) then
        tmp = -1.0d0
    else
        tmp = a * (b ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = a * Math.pow(b, 3.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.48:
		tmp = -1.0
	else:
		tmp = a * math.pow(b, 3.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.48)
		tmp = -1.0;
	else
		tmp = Float64(a * (b ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.48)
		tmp = -1.0;
	else
		tmp = a * (b ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.48], -1.0, N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.47999999999999998

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 36.5%

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

   5. Taylor expanded in b around 0 36.3%

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

   if 0.47999999999999998 < 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. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 90.8%

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

   5. Taylor expanded in a around inf 30.2%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot {b}^{3}\\ \end{array} \]

Alternative 4: 71.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+68}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot {b}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.4e+68) (+ (pow a 4.0) -1.0) (* a (pow b 3.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.4e+68) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = a * pow(b, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.4d+68) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = a * (b ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.4e+68) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = a * Math.pow(b, 3.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.4e+68:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = a * math.pow(b, 3.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.4e+68)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(a * (b ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.4e+68)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = a * (b ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.4e+68], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.40000000000000015e68

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 77.0%

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

   if 3.40000000000000015e68 < b

   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. unpow2 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. associate-*l* 100.0%

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

      6. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in a around inf 38.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+68}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot {b}^{3}\\ \end{array} \]

Alternative 5: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.25 \cdot 10^{+33}:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.25e+33) (+ (pow b 4.0) -1.0) (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.25e+33) {
		tmp = pow(b, 4.0) + -1.0;
	} else {
		tmp = pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 2.25d+33) then
        tmp = (b ** 4.0d0) + (-1.0d0)
    else
        tmp = (a ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 2.25e+33) {
		tmp = Math.pow(b, 4.0) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2.25e+33:
		tmp = math.pow(b, 4.0) + -1.0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.25e+33)
		tmp = Float64((b ^ 4.0) + -1.0);
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 2.25e+33)
		tmp = (b ^ 4.0) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.25e+33], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.25e33

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 95.1%

      \[\leadsto \color{blue}{\left({b}^{2} \cdot \left(4 + \left(0.5 \cdot {a}^{2} + a \cdot \left(a + 0.5 \cdot a\right)\right)\right) + \left({b}^{4} \cdot \left(0.25 + 0.5 \cdot \frac{a + 0.5 \cdot a}{a}\right) + {a}^{4}\right)\right) - 1} \]

   5. Taylor expanded in a around inf 88.9%

      \[\leadsto \left(\color{blue}{2 \cdot \left({a}^{2} \cdot {b}^{2}\right)} + \left({b}^{4} \cdot \left(0.25 + 0.5 \cdot \frac{a + 0.5 \cdot a}{a}\right) + {a}^{4}\right)\right) - 1 \]
   6. Step-by-step derivation

      1. associate-*r* 88.9%

         \[\leadsto \left(\color{blue}{\left(2 \cdot {a}^{2}\right) \cdot {b}^{2}} + \left({b}^{4} \cdot \left(0.25 + 0.5 \cdot \frac{a + 0.5 \cdot a}{a}\right) + {a}^{4}\right)\right) - 1 \]

      2. *-commutative 88.9%

         \[\leadsto \left(\color{blue}{\left({a}^{2} \cdot 2\right)} \cdot {b}^{2} + \left({b}^{4} \cdot \left(0.25 + 0.5 \cdot \frac{a + 0.5 \cdot a}{a}\right) + {a}^{4}\right)\right) - 1 \]

      3. associate-*l* 88.9%

         \[\leadsto \left(\color{blue}{{a}^{2} \cdot \left(2 \cdot {b}^{2}\right)} + \left({b}^{4} \cdot \left(0.25 + 0.5 \cdot \frac{a + 0.5 \cdot a}{a}\right) + {a}^{4}\right)\right) - 1 \]

   7. Simplified 88.9%

      \[\leadsto \left(\color{blue}{{a}^{2} \cdot \left(2 \cdot {b}^{2}\right)} + \left({b}^{4} \cdot \left(0.25 + 0.5 \cdot \frac{a + 0.5 \cdot a}{a}\right) + {a}^{4}\right)\right) - 1 \]

   8. Taylor expanded in a around 0 79.3%

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

   if 2.25e33 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 96.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.25 \cdot 10^{+33}:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

Alternative 6: 25.4% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. unpow2 99.8%

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

   3. unpow1 99.8%

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

   4. sqr-pow 99.8%

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

   5. associate-*l* 99.9%

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

   6. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in a around 0 49.5%

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

5. Taylor expanded in b around 0 27.8%

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

6. Final simplification 27.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023339 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))