Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 6.8s

Alternatives: 14

Speedup: 1.0×

• 60.0% 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 99.9%

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

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

   4. sqr-pow 99.8%

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

   5. 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: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0)
   (+ (pow a 4.0) -1.0)
   (+ (pow b 4.0) (* (* b b) (+ 4.0 (* (* a a) 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 400000000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + ((b * b) * (4.0d0 + ((a * a) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 400000000.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + Float64(Float64(b * b) * Float64(4.0 + Float64(Float64(a * a) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 400000000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   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 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. 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. Taylor expanded in b around 0 100.0%

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

   if 4e8 < (*.f64 b 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%

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

      4. sqr-pow 99.9%

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

      5. 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 99.9%

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

   4. Taylor expanded in b around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. unpow2 98.1%

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

   6. Simplified 98.1%

      \[\leadsto \color{blue}{{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + b \cdot \left(b \cdot \left(\left(a \cdot a\right) \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0)
   (+ (pow a 4.0) -1.0)
   (+ (pow b 4.0) (* b (* b (* (* a a) 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0) + (b * (b * ((a * a) * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 400000000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (b * (b * ((a * a) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + (b * (b * ((a * a) * 2.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 400000000.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + (b * (b * ((a * a) * 2.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + Float64(b * Float64(b * Float64(Float64(a * a) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 400000000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + (b * (b * ((a * a) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   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 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. 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. Taylor expanded in b around 0 100.0%

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

   if 4e8 < (*.f64 b 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%

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

      4. sqr-pow 99.9%

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

      5. 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 99.9%

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

   4. Taylor expanded in b around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in a around inf 86.5%

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

      1. associate-*r* 86.5%

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

      2. unpow2 86.5%

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

      3. *-commutative 86.5%

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

      4. unpow2 86.5%

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

      5. associate-*l* 98.0%

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

   9. Simplified 98.0%

      \[\leadsto {b}^{4} + \color{blue}{b \cdot \left(b \cdot \left(2 \cdot \left(a \cdot a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + b \cdot \left(b \cdot \left(\left(a \cdot a\right) \cdot 2\right)\right)\\ \end{array} \]

Alternative 4: 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. Final simplification 99.9%

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

Alternative 5: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0)
   (+ (pow a 4.0) -1.0)
   (+ (pow b 4.0) (* b (* b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0) + (b * (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 * b) <= 400000000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (b * (b * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + (b * (b * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 400000000.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + (b * (b * 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + Float64(b * Float64(b * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 400000000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + (b * (b * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   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 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. 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. Taylor expanded in b around 0 100.0%

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

   if 4e8 < (*.f64 b 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%

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

      4. sqr-pow 99.9%

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

      5. 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 99.9%

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

   4. Taylor expanded in b around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in a around 0 93.0%

      \[\leadsto {b}^{4} + \color{blue}{4 \cdot {b}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 93.0%

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

      2. associate-*r* 93.0%

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

   9. Simplified 93.0%

      \[\leadsto {b}^{4} + \color{blue}{\left(4 \cdot b\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 6: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0) (+ (pow a 4.0) -1.0) (* b (* b (fma b b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = b * (b * fma(b, b, 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(b * Float64(b * fma(b, b, 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   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 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. 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. Taylor expanded in b around 0 100.0%

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

   if 4e8 < (*.f64 b 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%

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

      4. sqr-pow 99.9%

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

      5. 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 99.9%

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

   4. Taylor expanded in b around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in a around 0 93.0%

      \[\leadsto {b}^{4} + \color{blue}{4 \cdot {b}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 93.0%

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

      2. associate-*r* 93.0%

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

   9. Simplified 93.0%

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

       1. sqr-pow 93.0%

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

       2. fma-def 93.0%

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

       3. metadata-eval 93.0%

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

       4. pow2 93.0%

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

       5. metadata-eval 93.0%

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

       6. pow2 93.0%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot b}, \left(4 \cdot b\right) \cdot b\right) \]

       7. *-commutative 93.0%

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \color{blue}{b \cdot \left(4 \cdot b\right)}\right) \]

       8. *-commutative 93.0%

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, b \cdot \color{blue}{\left(b \cdot 4\right)}\right) \]

   11. Applied egg-rr 93.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot b, b \cdot \left(b \cdot 4\right)\right)} \]

   12. Taylor expanded in b around 0 93.0%

       \[\leadsto \color{blue}{{b}^{4} + 4 \cdot {b}^{2}} \]
   13. Step-by-step derivation

       1. metadata-eval 93.0%

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

       2. pow-plus 93.0%

          \[\leadsto \color{blue}{{b}^{3} \cdot b} + 4 \cdot {b}^{2} \]

       3. unpow3 93.0%

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

       4. associate-*r* 93.0%

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

       5. unpow2 93.0%

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

       6. distribute-rgt-in 93.0%

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

       7. fma-udef 93.0%

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, 4\right)} \]

       8. associate-*l* 93.0%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} \]

   14. Simplified 93.0%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \]

Alternative 7: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0)
   (+ (pow a 4.0) -1.0)
   (+ (* 4.0 (* b b)) (* (* b b) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 400000000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (4.0d0 * (b * b)) + ((b * b) * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 400000000.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(4.0 * Float64(b * b)) + Float64(Float64(b * b) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 400000000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   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 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. 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. Taylor expanded in b around 0 100.0%

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

   if 4e8 < (*.f64 b 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%

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

      4. sqr-pow 99.9%

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

      5. 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 99.9%

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

   4. Taylor expanded in b around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in a around 0 93.0%

      \[\leadsto {b}^{4} + \color{blue}{4 \cdot {b}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 93.0%

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

      2. associate-*r* 93.0%

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

   9. Simplified 93.0%

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

       1. add-cube-cbrt 92.7%

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

       2. unpow-prod-down 92.6%

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

       3. fma-def 92.6%

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

       4. pow2 92.6%

          \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\sqrt[3]{b}\right)}^{2}\right)}}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, \left(4 \cdot b\right) \cdot b\right) \]

       5. *-commutative 92.6%

          \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, \color{blue}{b \cdot \left(4 \cdot b\right)}\right) \]

       6. *-commutative 92.6%

          \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, b \cdot \color{blue}{\left(b \cdot 4\right)}\right) \]

   11. Applied egg-rr 92.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, b \cdot \left(b \cdot 4\right)\right)} \]
   12. Step-by-step derivation

       1. fma-udef 92.6%

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

       2. +-commutative 92.6%

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

       3. associate-*r* 92.6%

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

       4. *-commutative 92.6%

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

       5. sqr-pow 92.6%

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

       6. metadata-eval 92.6%

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

       7. pow-sqr 92.6%

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

       8. unpow2 92.6%

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

       9. metadata-eval 92.6%

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

       10. associate-*l* 92.6%

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

       11. cube-unmult 92.6%

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

   13. Simplified 92.8%

       \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right) + {\left(b \cdot \sqrt[3]{b}\right)}^{3}} \]
   14. Step-by-step derivation

       1. unpow3 92.8%

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

       2. *-commutative 92.8%

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

       3. associate-*r* 92.8%

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

       4. swap-sqr 92.8%

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

       5. associate-*r* 92.9%

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

       6. add-cube-cbrt 93.0%

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

       7. associate-*r* 93.0%

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

   15. Applied egg-rr 93.0%

       \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 8: 66.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2.5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \cdot a \leq 3.5 \cdot 10^{-211}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \cdot a \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2.5e-309)
   -1.0
   (if (<= (* a a) 3.5e-211)
     (* 4.0 (* b b))
     (if (<= (* a a) 1.0) -1.0 (* (* a a) (+ (* a a) 1.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2.5e-309) {
		tmp = -1.0;
	} else if ((a * a) <= 3.5e-211) {
		tmp = 4.0 * (b * b);
	} else if ((a * a) <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * ((a * a) + 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 * a) <= 2.5d-309) then
        tmp = -1.0d0
    else if ((a * a) <= 3.5d-211) then
        tmp = 4.0d0 * (b * b)
    else if ((a * a) <= 1.0d0) then
        tmp = -1.0d0
    else
        tmp = (a * a) * ((a * a) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2.5e-309) {
		tmp = -1.0;
	} else if ((a * a) <= 3.5e-211) {
		tmp = 4.0 * (b * b);
	} else if ((a * a) <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * ((a * a) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 2.5e-309:
		tmp = -1.0
	elif (a * a) <= 3.5e-211:
		tmp = 4.0 * (b * b)
	elif (a * a) <= 1.0:
		tmp = -1.0
	else:
		tmp = (a * a) * ((a * a) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 2.5e-309)
		tmp = -1.0;
	elseif (Float64(a * a) <= 3.5e-211)
		tmp = Float64(4.0 * Float64(b * b));
	elseif (Float64(a * a) <= 1.0)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * a) * Float64(Float64(a * a) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 2.5e-309)
		tmp = -1.0;
	elseif ((a * a) <= 3.5e-211)
		tmp = 4.0 * (b * b);
	elseif ((a * a) <= 1.0)
		tmp = -1.0;
	else
		tmp = (a * a) * ((a * a) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 2.5e-309], -1.0, If[LessEqual[N[(a * a), $MachinePrecision], 3.5e-211], N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * a), $MachinePrecision], 1.0], -1.0, N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a a) < 2.5000000000000022e-309 or 3.5e-211 < (*.f64 a a) < 1

   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%

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

      4. sqr-pow 99.9%

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

      5. 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. Taylor expanded in b around 0 52.4%

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

   5. Taylor expanded in a around 0 51.5%

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

   if 2.5000000000000022e-309 < (*.f64 a a) < 3.5e-211

   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%

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

      4. sqr-pow 100.0%

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

      5. associate-*r* 100.0%

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

   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. Taylor expanded in b around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

      3. unpow2 72.7%

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

   6. Simplified 72.7%

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

   7. Taylor expanded in a around 0 72.7%

      \[\leadsto {b}^{4} + \color{blue}{4 \cdot {b}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 72.7%

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

      2. associate-*r* 72.7%

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

   9. Simplified 72.7%

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

   10. Taylor expanded in b around 0 47.2%

       \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 47.2%

          \[\leadsto 4 \cdot \color{blue}{\left(b \cdot b\right)} \]

   12. Simplified 47.2%

       \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right)} \]

   if 1 < (*.f64 a 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%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.8%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 92.2%

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   5. Step-by-step derivation

      1. metadata-eval 92.2%

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

      2. pow-sqr 92.0%

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} - 1 \]

      3. pow2 92.0%

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

      4. pow2 92.0%

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

      5. difference-of-sqr-1 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in a around inf 91.3%

      \[\leadsto \left(a \cdot a + 1\right) \cdot \color{blue}{{a}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 91.3%

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

   9. Simplified 91.3%

      \[\leadsto \left(a \cdot a + 1\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2.5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \cdot a \leq 3.5 \cdot 10^{-211}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \cdot a \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + 1\right)\\ \end{array} \]

Alternative 9: 94.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(\left(a + -1\right) + a \cdot \left(a + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0)
   (* (+ (* a a) 1.0) (+ (+ a -1.0) (* a (+ a -1.0))))
   (+ (* 4.0 (* b b)) (* (* b b) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = ((a * a) + 1.0) * ((a + -1.0) + (a * (a + -1.0)));
	} else {
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 400000000.0d0) then
        tmp = ((a * a) + 1.0d0) * ((a + (-1.0d0)) + (a * (a + (-1.0d0))))
    else
        tmp = (4.0d0 * (b * b)) + ((b * b) * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = ((a * a) + 1.0) * ((a + -1.0) + (a * (a + -1.0)));
	} else {
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 400000000.0:
		tmp = ((a * a) + 1.0) * ((a + -1.0) + (a * (a + -1.0)))
	else:
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a + -1.0) + Float64(a * Float64(a + -1.0))));
	else
		tmp = Float64(Float64(4.0 * Float64(b * b)) + Float64(Float64(b * b) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 400000000.0)
		tmp = ((a * a) + 1.0) * ((a + -1.0) + (a * (a + -1.0)));
	else
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a + -1.0), $MachinePrecision] + N[(a * N[(a + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   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 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. 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. Taylor expanded in b around 0 100.0%

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   5. Step-by-step derivation

      1. metadata-eval 100.0%

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

      2. pow-sqr 99.8%

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} - 1 \]

      3. pow2 99.8%

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

      4. pow2 99.8%

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

      5. difference-of-sqr-1 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. difference-of-sqr-1 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. *-un-lft-identity 99.8%

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

   10. Applied egg-rr 99.8%

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

   if 4e8 < (*.f64 b 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%

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

      4. sqr-pow 99.9%

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

      5. 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 99.9%

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

   4. Taylor expanded in b around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in a around 0 93.0%

      \[\leadsto {b}^{4} + \color{blue}{4 \cdot {b}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 93.0%

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

      2. associate-*r* 93.0%

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

   9. Simplified 93.0%

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

       1. add-cube-cbrt 92.7%

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

       2. unpow-prod-down 92.6%

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

       3. fma-def 92.6%

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

       4. pow2 92.6%

          \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\sqrt[3]{b}\right)}^{2}\right)}}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, \left(4 \cdot b\right) \cdot b\right) \]

       5. *-commutative 92.6%

          \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, \color{blue}{b \cdot \left(4 \cdot b\right)}\right) \]

       6. *-commutative 92.6%

          \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, b \cdot \color{blue}{\left(b \cdot 4\right)}\right) \]

   11. Applied egg-rr 92.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, b \cdot \left(b \cdot 4\right)\right)} \]
   12. Step-by-step derivation

       1. fma-udef 92.6%

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

       2. +-commutative 92.6%

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

       3. associate-*r* 92.6%

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

       4. *-commutative 92.6%

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

       5. sqr-pow 92.6%

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

       6. metadata-eval 92.6%

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

       7. pow-sqr 92.6%

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

       8. unpow2 92.6%

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

       9. metadata-eval 92.6%

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

       10. associate-*l* 92.6%

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

       11. cube-unmult 92.6%

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

   13. Simplified 92.8%

       \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right) + {\left(b \cdot \sqrt[3]{b}\right)}^{3}} \]
   14. Step-by-step derivation

       1. unpow3 92.8%

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

       2. *-commutative 92.8%

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

       3. associate-*r* 92.8%

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

       4. swap-sqr 92.8%

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

       5. associate-*r* 92.9%

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

       6. add-cube-cbrt 93.0%

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

       7. associate-*r* 93.0%

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

   15. Applied egg-rr 93.0%

       \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(\left(a + -1\right) + a \cdot \left(a + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 10: 84.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.5 \cdot 10^{+302}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(\left(a + -1\right) \cdot \left(a + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4.5e+302)
   (* (+ (* a a) 1.0) (* (+ a -1.0) (+ a 1.0)))
   (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.5e+302) {
		tmp = ((a * a) + 1.0) * ((a + -1.0) * (a + 1.0));
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4.5d+302) then
        tmp = ((a * a) + 1.0d0) * ((a + (-1.0d0)) * (a + 1.0d0))
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.5e+302) {
		tmp = ((a * a) + 1.0) * ((a + -1.0) * (a + 1.0));
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4.5e+302:
		tmp = ((a * a) + 1.0) * ((a + -1.0) * (a + 1.0))
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4.5e+302)
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a + -1.0) * Float64(a + 1.0)));
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4.5e+302)
		tmp = ((a * a) + 1.0) * ((a + -1.0) * (a + 1.0));
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4.5e+302], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a + -1.0), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.5000000000000002e302

   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%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.8%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 80.6%

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   5. Step-by-step derivation

      1. metadata-eval 80.6%

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

      2. pow-sqr 80.5%

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} - 1 \]

      3. pow2 80.5%

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

      4. pow2 80.5%

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

      5. difference-of-sqr-1 80.5%

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

   6. Applied egg-rr 80.5%

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

      1. difference-of-sqr-1 80.5%

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

      2. sub-neg 80.5%

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

      3. metadata-eval 80.5%

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

   8. Applied egg-rr 80.5%

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

   if 4.5000000000000002e302 < (*.f64 b 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%

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

      4. sqr-pow 100.0%

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

      5. associate-*r* 100.0%

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

   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. Taylor expanded in b around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

      \[\leadsto {b}^{4} + \color{blue}{4 \cdot {b}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 97.6%

       \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 97.6%

          \[\leadsto 4 \cdot \color{blue}{\left(b \cdot b\right)} \]

   12. Simplified 97.6%

       \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.5 \cdot 10^{+302}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(\left(a + -1\right) \cdot \left(a + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 11: 94.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 75000000000:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(\left(a + -1\right) \cdot \left(a + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 75000000000.0)
   (* (+ (* a a) 1.0) (* (+ a -1.0) (+ a 1.0)))
   (+ (* 4.0 (* b b)) (* (* b b) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 75000000000.0) {
		tmp = ((a * a) + 1.0) * ((a + -1.0) * (a + 1.0));
	} else {
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 75000000000.0d0) then
        tmp = ((a * a) + 1.0d0) * ((a + (-1.0d0)) * (a + 1.0d0))
    else
        tmp = (4.0d0 * (b * b)) + ((b * b) * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 75000000000.0) {
		tmp = ((a * a) + 1.0) * ((a + -1.0) * (a + 1.0));
	} else {
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 75000000000.0:
		tmp = ((a * a) + 1.0) * ((a + -1.0) * (a + 1.0))
	else:
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 75000000000.0)
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a + -1.0) * Float64(a + 1.0)));
	else
		tmp = Float64(Float64(4.0 * Float64(b * b)) + Float64(Float64(b * b) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 75000000000.0)
		tmp = ((a * a) + 1.0) * ((a + -1.0) * (a + 1.0));
	else
		tmp = (4.0 * (b * b)) + ((b * b) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 75000000000.0], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a + -1.0), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 7.5e10

   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 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. 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. Taylor expanded in b around 0 100.0%

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   5. Step-by-step derivation

      1. metadata-eval 100.0%

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

      2. pow-sqr 99.8%

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} - 1 \]

      3. pow2 99.8%

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

      4. pow2 99.8%

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

      5. difference-of-sqr-1 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. difference-of-sqr-1 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

   if 7.5e10 < (*.f64 b 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%

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

      4. sqr-pow 99.9%

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

      5. 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 99.9%

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

   4. Taylor expanded in b around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in a around 0 93.0%

      \[\leadsto {b}^{4} + \color{blue}{4 \cdot {b}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 93.0%

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

      2. associate-*r* 93.0%

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

   9. Simplified 93.0%

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

       1. add-cube-cbrt 92.7%

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

       2. unpow-prod-down 92.6%

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

       3. fma-def 92.6%

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

       4. pow2 92.6%

          \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\sqrt[3]{b}\right)}^{2}\right)}}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, \left(4 \cdot b\right) \cdot b\right) \]

       5. *-commutative 92.6%

          \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, \color{blue}{b \cdot \left(4 \cdot b\right)}\right) \]

       6. *-commutative 92.6%

          \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, b \cdot \color{blue}{\left(b \cdot 4\right)}\right) \]

   11. Applied egg-rr 92.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{4}, {\left(\sqrt[3]{b}\right)}^{4}, b \cdot \left(b \cdot 4\right)\right)} \]
   12. Step-by-step derivation

       1. fma-udef 92.6%

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

       2. +-commutative 92.6%

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

       3. associate-*r* 92.6%

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

       4. *-commutative 92.6%

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

       5. sqr-pow 92.6%

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

       6. metadata-eval 92.6%

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

       7. pow-sqr 92.6%

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

       8. unpow2 92.6%

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

       9. metadata-eval 92.6%

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

       10. associate-*l* 92.6%

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

       11. cube-unmult 92.6%

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

   13. Simplified 92.8%

       \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right) + {\left(b \cdot \sqrt[3]{b}\right)}^{3}} \]
   14. Step-by-step derivation

       1. unpow3 92.8%

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

       2. *-commutative 92.8%

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

       3. associate-*r* 92.8%

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

       4. swap-sqr 92.8%

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

       5. associate-*r* 92.9%

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

       6. add-cube-cbrt 93.0%

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

       7. associate-*r* 93.0%

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

   15. Applied egg-rr 93.0%

       \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 75000000000:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(\left(a + -1\right) \cdot \left(a + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 12: 84.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.5 \cdot 10^{+302}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4.5e+302)
   (* (+ (* a a) 1.0) (+ (* a a) -1.0))
   (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.5e+302) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4.5d+302) then
        tmp = ((a * a) + 1.0d0) * ((a * a) + (-1.0d0))
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.5e+302) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4.5e+302:
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0)
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4.5e+302)
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a * a) + -1.0));
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4.5e+302)
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4.5e+302], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.5000000000000002e302

   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%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.8%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 80.6%

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   5. Step-by-step derivation

      1. metadata-eval 80.6%

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

      2. pow-sqr 80.5%

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} - 1 \]

      3. pow2 80.5%

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

      4. pow2 80.5%

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

      5. difference-of-sqr-1 80.5%

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

   6. Applied egg-rr 80.5%

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

   if 4.5000000000000002e302 < (*.f64 b 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%

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

      4. sqr-pow 100.0%

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

      5. associate-*r* 100.0%

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

   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. Taylor expanded in b around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

      \[\leadsto {b}^{4} + \color{blue}{4 \cdot {b}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 97.6%

       \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 97.6%

          \[\leadsto 4 \cdot \color{blue}{\left(b \cdot b\right)} \]

   12. Simplified 97.6%

       \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.5 \cdot 10^{+302}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 13: 50.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 7.5 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 7.5e-28) -1.0 (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 7.5e-28) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 7.5d-28) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 7.5e-28) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 7.5e-28:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 7.5e-28)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 7.5e-28)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 7.5e-28], -1.0, N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 7.5000000000000003e-28

   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%

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

      4. sqr-pow 99.8%

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

      5. 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. Taylor expanded in b around 0 100.0%

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

   5. Taylor expanded in a around 0 44.0%

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

   if 7.5000000000000003e-28 < (*.f64 b 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%

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

      4. sqr-pow 99.9%

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

      5. 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 99.9%

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

   4. Taylor expanded in b around inf 97.4%

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

      1. *-commutative 97.4%

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

      2. unpow2 97.4%

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

      3. unpow2 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in a around 0 90.9%

      \[\leadsto {b}^{4} + \color{blue}{4 \cdot {b}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 90.9%

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

      2. associate-*r* 90.9%

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

   9. Simplified 90.9%

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

   10. Taylor expanded in b around 0 56.0%

       \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 56.0%

          \[\leadsto 4 \cdot \color{blue}{\left(b \cdot b\right)} \]

   12. Simplified 56.0%

       \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 7.5 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 14: 25.3% 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. 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%

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

   4. sqr-pow 99.8%

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

   5. 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. Taylor expanded in b around 0 70.7%

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

5. Taylor expanded in a around 0 22.9%

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

6. Final simplification 22.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023256 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))