Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.8s

Alternatives: 11

Speedup: 1.0×

• 61.1% 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 99.9%

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

   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. 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0001:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0001)
   (+ (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) <= 0.0001) {
		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) <= 0.0001d0) 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) <= 0.0001) {
		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) <= 0.0001:
		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) <= 0.0001)
		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) <= 0.0001)
		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], 0.0001], 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) < 1.00000000000000005e-4

   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 99.4%

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

   if 1.00000000000000005e-4 < (*.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 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 95.0%

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

      1. *-commutative 95.0%

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

      2. unpow2 95.0%

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

      3. unpow2 95.0%

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

   6. Simplified 95.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0001:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 3: 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 4: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+54}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.7e-5)
   (+ (+ (* 4.0 (* b b)) (pow b 4.0)) -1.0)
   (if (<= a 2.8e+36)
     (+ (pow a 4.0) -1.0)
     (if (<= a 2e+54) (pow b 4.0) (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.7e-5) {
		tmp = ((4.0 * (b * b)) + pow(b, 4.0)) + -1.0;
	} else if (a <= 2.8e+36) {
		tmp = pow(a, 4.0) + -1.0;
	} else if (a <= 2e+54) {
		tmp = pow(b, 4.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 2.7d-5) then
        tmp = ((4.0d0 * (b * b)) + (b ** 4.0d0)) + (-1.0d0)
    else if (a <= 2.8d+36) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else if (a <= 2d+54) then
        tmp = b ** 4.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 2.7e-5) {
		tmp = ((4.0 * (b * b)) + Math.pow(b, 4.0)) + -1.0;
	} else if (a <= 2.8e+36) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else if (a <= 2e+54) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2.7e-5:
		tmp = ((4.0 * (b * b)) + math.pow(b, 4.0)) + -1.0
	elif a <= 2.8e+36:
		tmp = math.pow(a, 4.0) + -1.0
	elif a <= 2e+54:
		tmp = math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.7e-5)
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + (b ^ 4.0)) + -1.0);
	elseif (a <= 2.8e+36)
		tmp = Float64((a ^ 4.0) + -1.0);
	elseif (a <= 2e+54)
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 2.7e-5)
		tmp = ((4.0 * (b * b)) + (b ^ 4.0)) + -1.0;
	elseif (a <= 2.8e+36)
		tmp = (a ^ 4.0) + -1.0;
	elseif (a <= 2e+54)
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.7e-5], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.8e+36], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2e+54], N[Power[b, 4.0], $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < 2.6999999999999999e-5

   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. Taylor expanded in a around 0 84.1%

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

      1. +-commutative 84.1%

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

      2. fma-def 84.1%

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

      3. unpow2 84.1%

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

   4. Simplified 84.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. fma-udef 84.1%

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

      2. +-commutative 84.1%

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

   6. Applied egg-rr 84.1%

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

   if 2.6999999999999999e-5 < a < 2.8000000000000001e36

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. unpow2 99.5%

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

      3. unpow1 99.5%

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

      4. sqr-pow 99.5%

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

      5. associate-*r* 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in b around 0 76.2%

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

   if 2.8000000000000001e36 < a < 2.0000000000000002e54

   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%

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

   if 2.0000000000000002e54 < a

   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 a around inf 100.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+54}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.45:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{+36}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+55}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.45)
   (+ (* (* b b) (+ 4.0 (* b b))) -1.0)
   (if (<= a 3.2e+36)
     (pow a 4.0)
     (if (<= a 2.8e+55) (+ (* (* b b) (* b b)) -1.0) (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.45) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else if (a <= 3.2e+36) {
		tmp = pow(a, 4.0);
	} else if (a <= 2.8e+55) {
		tmp = ((b * b) * (b * b)) + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 1.45d0) then
        tmp = ((b * b) * (4.0d0 + (b * b))) + (-1.0d0)
    else if (a <= 3.2d+36) then
        tmp = a ** 4.0d0
    else if (a <= 2.8d+55) then
        tmp = ((b * b) * (b * b)) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.45) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else if (a <= 3.2e+36) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 2.8e+55) {
		tmp = ((b * b) * (b * b)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.45:
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0
	elif a <= 3.2e+36:
		tmp = math.pow(a, 4.0)
	elif a <= 2.8e+55:
		tmp = ((b * b) * (b * b)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.45)
		tmp = Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))) + -1.0);
	elseif (a <= 3.2e+36)
		tmp = a ^ 4.0;
	elseif (a <= 2.8e+55)
		tmp = Float64(Float64(Float64(b * b) * Float64(b * b)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.45)
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	elseif (a <= 3.2e+36)
		tmp = a ^ 4.0;
	elseif (a <= 2.8e+55)
		tmp = ((b * b) * (b * b)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.45], N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 3.2e+36], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 2.8e+55], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.44999999999999996

   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. Taylor expanded in a around 0 84.0%

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

      1. +-commutative 84.0%

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

      2. fma-def 84.0%

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

      3. unpow2 84.0%

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

   4. Simplified 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. fma-udef 84.0%

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

      2. metadata-eval 84.0%

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

      3. pow-prod-up 83.9%

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

      4. pow-prod-down 83.9%

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

      5. pow2 83.9%

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

      6. distribute-rgt-out 83.9%

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

   6. Applied egg-rr 83.9%

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

   if 1.44999999999999996 < a < 3.1999999999999999e36 or 2.8000000000000001e55 < a

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. 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 a around inf 94.9%

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

   if 3.1999999999999999e36 < a < 2.8000000000000001e55

   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. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. unpow2 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow-prod-up 100.0%

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

      4. pow-prod-down 100.0%

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

      5. pow2 100.0%

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

      6. distribute-rgt-out 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in b around inf 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.45:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{+36}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+55}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.9:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+36}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+55}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.9)
   (+ (* (* b b) (+ 4.0 (* b b))) -1.0)
   (if (<= a 2.1e+36)
     (pow a 4.0)
     (if (<= a 1.1e+55) (pow b 4.0) (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.9) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else if (a <= 2.1e+36) {
		tmp = pow(a, 4.0);
	} else if (a <= 1.1e+55) {
		tmp = pow(b, 4.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 1.9d0) then
        tmp = ((b * b) * (4.0d0 + (b * b))) + (-1.0d0)
    else if (a <= 2.1d+36) then
        tmp = a ** 4.0d0
    else if (a <= 1.1d+55) then
        tmp = b ** 4.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.9) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else if (a <= 2.1e+36) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 1.1e+55) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.9:
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0
	elif a <= 2.1e+36:
		tmp = math.pow(a, 4.0)
	elif a <= 1.1e+55:
		tmp = math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.9)
		tmp = Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))) + -1.0);
	elseif (a <= 2.1e+36)
		tmp = a ^ 4.0;
	elseif (a <= 1.1e+55)
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.9)
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	elseif (a <= 2.1e+36)
		tmp = a ^ 4.0;
	elseif (a <= 1.1e+55)
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.9], N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.1e+36], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 1.1e+55], N[Power[b, 4.0], $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.8999999999999999

   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. Taylor expanded in a around 0 84.0%

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

      1. +-commutative 84.0%

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

      2. fma-def 84.0%

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

      3. unpow2 84.0%

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

   4. Simplified 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. fma-udef 84.0%

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

      2. metadata-eval 84.0%

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

      3. pow-prod-up 83.9%

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

      4. pow-prod-down 83.9%

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

      5. pow2 83.9%

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

      6. distribute-rgt-out 83.9%

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

   6. Applied egg-rr 83.9%

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

   if 1.8999999999999999 < a < 2.10000000000000004e36 or 1.10000000000000005e55 < a

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. 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 a around inf 94.9%

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

   if 2.10000000000000004e36 < a < 1.10000000000000005e55

   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%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.9:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+36}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+55}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{+36}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+54}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.7e-5)
   (+ (* (* b b) (+ 4.0 (* b b))) -1.0)
   (if (<= a 2.15e+36)
     (+ (pow a 4.0) -1.0)
     (if (<= a 2.2e+54) (pow b 4.0) (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.7e-5) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else if (a <= 2.15e+36) {
		tmp = pow(a, 4.0) + -1.0;
	} else if (a <= 2.2e+54) {
		tmp = pow(b, 4.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 2.7d-5) then
        tmp = ((b * b) * (4.0d0 + (b * b))) + (-1.0d0)
    else if (a <= 2.15d+36) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else if (a <= 2.2d+54) then
        tmp = b ** 4.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 2.7e-5) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else if (a <= 2.15e+36) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else if (a <= 2.2e+54) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2.7e-5:
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0
	elif a <= 2.15e+36:
		tmp = math.pow(a, 4.0) + -1.0
	elif a <= 2.2e+54:
		tmp = math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.7e-5)
		tmp = Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))) + -1.0);
	elseif (a <= 2.15e+36)
		tmp = Float64((a ^ 4.0) + -1.0);
	elseif (a <= 2.2e+54)
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 2.7e-5)
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	elseif (a <= 2.15e+36)
		tmp = (a ^ 4.0) + -1.0;
	elseif (a <= 2.2e+54)
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.7e-5], N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.15e+36], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.2e+54], N[Power[b, 4.0], $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < 2.6999999999999999e-5

   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. Taylor expanded in a around 0 84.1%

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

      1. +-commutative 84.1%

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

      2. fma-def 84.1%

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

      3. unpow2 84.1%

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

   4. Simplified 84.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. fma-udef 84.1%

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

      2. metadata-eval 84.1%

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

      3. pow-prod-up 84.1%

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

      4. pow-prod-down 84.1%

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

      5. pow2 84.1%

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

      6. distribute-rgt-out 84.1%

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

   6. Applied egg-rr 84.1%

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

   if 2.6999999999999999e-5 < a < 2.15000000000000002e36

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. unpow2 99.5%

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

      3. unpow1 99.5%

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

      4. sqr-pow 99.5%

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

      5. associate-*r* 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in b around 0 76.2%

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

   if 2.15000000000000002e36 < a < 2.1999999999999999e54

   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%

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

   if 2.1999999999999999e54 < a

   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 a around inf 100.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{+36}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+54}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 8: 69.6% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $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. Taylor expanded in a around 0 73.5%

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

   1. +-commutative 73.5%

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

   2. fma-def 73.5%

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

   3. unpow2 73.5%

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

4. Simplified 73.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right)} - 1 \]
5. Step-by-step derivation

   1. fma-udef 73.5%

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

   2. metadata-eval 73.5%

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

   3. pow-prod-up 73.4%

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

   4. pow-prod-down 73.4%

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

   5. pow2 73.4%

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

   6. distribute-rgt-out 73.4%

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

6. Applied egg-rr 73.4%

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

7. Final simplification 73.4%

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

Alternative 9: 69.1% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $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. Taylor expanded in a around 0 73.5%

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

   1. +-commutative 73.5%

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

   2. fma-def 73.5%

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

   3. unpow2 73.5%

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

4. Simplified 73.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right)} - 1 \]
5. Step-by-step derivation

   1. fma-udef 73.5%

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

   2. metadata-eval 73.5%

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

   3. pow-prod-up 73.4%

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

   4. pow-prod-down 73.4%

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

   5. pow2 73.4%

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

   6. distribute-rgt-out 73.4%

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

6. Applied egg-rr 73.4%

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

7. Taylor expanded in b around inf 71.7%

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

   1. unpow2 71.7%

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

9. Simplified 71.7%

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

10. Final simplification 71.7%

    \[\leadsto \left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1 \]

Alternative 10: 50.8% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(4.0 * N[(b * b), $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. Taylor expanded in a around 0 73.5%

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

   1. +-commutative 73.5%

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

   2. fma-def 73.5%

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

   3. unpow2 73.5%

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

4. Simplified 73.5%

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

5. Taylor expanded in b around 0 50.4%

   \[\leadsto \color{blue}{4 \cdot {b}^{2}} - 1 \]
6. Step-by-step derivation

   1. unpow2 50.4%

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

7. Simplified 50.4%

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

8. Final simplification 50.4%

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

Alternative 11: 24.9% 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 99.9%

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

   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 68.6%

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

5. Taylor expanded in a around 0 27.9%

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

6. Final simplification 27.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023261 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))