Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 8

Speedup: 1.0×

• 61.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + ((b * b) * 4.0)) + -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) + ((b * b) * 4.0d0)) + (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + ((b * b) * 4.0)) + -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[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 94.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54)
   (+ (+ (* (* b b) 4.0) (pow a 4.0)) -1.0)
   (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = (((b * b) * 4.0) + pow(a, 4.0)) + -1.0;
	} else {
		tmp = pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = (((b * b) * 4.0d0) + (a ** 4.0d0)) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = (((b * b) * 4.0) + Math.pow(a, 4.0)) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = (((b * b) * 4.0) + math.pow(a, 4.0)) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + (a ^ 4.0)) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = (((b * b) * 4.0) + (a ^ 4.0)) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf 98.6%

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

   if 4.0000000000000003e54 < (*.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. Add Preprocessing

   3. Taylor expanded in a around 0 87.7%

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

      1. fma-define 87.7%

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

      2. unpow2 87.7%

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

      3. unpow2 87.7%

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

      4. swap-sqr 100.0%

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

      5. unpow2 100.0%

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

   5. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in b around inf 96.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + {a}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54) (+ (pow a 4.0) -1.0) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf 98.6%

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

   4. Taylor expanded in b around 0 98.0%

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

   if 4.0000000000000003e54 < (*.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. Add Preprocessing

   3. Taylor expanded in a around 0 87.7%

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

      1. fma-define 87.7%

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

      2. unpow2 87.7%

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

      3. unpow2 87.7%

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

      4. swap-sqr 100.0%

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

      5. unpow2 100.0%

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

   5. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in b around inf 96.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf 98.6%

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

   4. Taylor expanded in b around 0 98.0%

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

   if 4.0000000000000003e54 < (*.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. Add Preprocessing

   3. Taylor expanded in a around 0 96.6%

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

   4. Taylor expanded in b around inf 96.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 600000:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 600000.0) (* (+ 1.0 (* b 2.0)) (+ (* b 2.0) -1.0)) (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 600000.0) {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -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 <= 600000.0d0) then
        tmp = (1.0d0 + (b * 2.0d0)) * ((b * 2.0d0) + (-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 <= 600000.0) {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 600000.0:
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 600000.0)
		tmp = Float64(Float64(1.0 + Float64(b * 2.0)) * Float64(Float64(b * 2.0) + -1.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 600000.0)
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 600000.0], N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6e5

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf 81.6%

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

   4. Taylor expanded in a around 0 54.3%

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

      1. add-sqr-sqrt 54.3%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2}} \cdot \sqrt{4 \cdot {b}^{2}}} - 1 \]

      2. difference-of-sqr-1 54.3%

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

      3. *-commutative 54.3%

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

      4. sqrt-prod 54.3%

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

      5. sqrt-pow1 43.2%

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

      6. metadata-eval 43.2%

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

      7. pow1 43.2%

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

      8. metadata-eval 43.2%

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

      9. *-commutative 43.2%

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

      10. sqrt-prod 43.2%

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

      11. sqrt-pow1 54.3%

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

      12. metadata-eval 54.3%

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

      13. pow1 54.3%

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

      14. metadata-eval 54.3%

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

   6. Applied egg-rr 54.3%

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

   if 6e5 < 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. Add Preprocessing

   3. Taylor expanded in a around inf 92.7%

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

   4. Taylor expanded in a around inf 88.2%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 600000:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0)

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(b * 2.0), $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. Add Preprocessing

3. Taylor expanded in a around inf 84.4%

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

4. Taylor expanded in a around 0 48.5%

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

   1. add-sqr-sqrt 48.5%

      \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2}} \cdot \sqrt{4 \cdot {b}^{2}}} - 1 \]

   2. difference-of-sqr-1 48.5%

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

   3. *-commutative 48.5%

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

   4. sqrt-prod 48.5%

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

   5. sqrt-pow1 36.2%

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

   6. metadata-eval 36.2%

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

   7. pow1 36.2%

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

   8. metadata-eval 36.2%

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

   9. *-commutative 36.2%

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

   10. sqrt-prod 36.2%

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

   11. sqrt-pow1 48.5%

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

   12. metadata-eval 48.5%

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

   13. pow1 48.5%

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

   14. metadata-eval 48.5%

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

6. Applied egg-rr 48.5%

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

7. Final simplification 48.5%

   \[\leadsto \left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right) \]
8. Add Preprocessing

Alternative 7: 52.0% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around inf 84.4%

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

4. Taylor expanded in a around 0 48.5%

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

   1. pow2 48.5%

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

6. Applied egg-rr 48.5%

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

7. Final simplification 48.5%

   \[\leadsto \left(b \cdot b\right) \cdot 4 + -1 \]
8. Add Preprocessing

Alternative 8: 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. Add Preprocessing

3. Taylor expanded in a around 0 67.7%

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

4. Taylor expanded in b around 0 20.9%

   \[\leadsto \color{blue}{-1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))