Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.4% → 99.2%

Time: 6.8s

Alternatives: 6

Speedup: 1.2×

• 60.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a + \left(b \cdot b\right) \cdot 3\right)\right) + -1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+
  (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* a a) (* (* b b) 3.0))))
  -1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a * a) + ((b * b) * 3.0)))) + -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) + (4.0d0 * ((a * a) + ((b * b) * 3.0d0)))) + (-1.0d0)
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a * a) + ((b * b) * 3.0)))) + -1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a * a) + ((b * b) * 3.0)))) + -1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(a * a) + Float64(Float64(b * b) * 3.0)))) + -1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * ((a * a) + ((b * b) * 3.0)))) + -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[(4.0 * N[(N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in a around 0 87.7%

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

4. Taylor expanded in a around 0 99.1%

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

5. Final simplification 99.1%

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a + \left(b \cdot b\right) \cdot 3\right)\right) + -1 \]
6. Add Preprocessing

Alternative 2: 69.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.92 \lor \neg \left(a \leq 3.1 \cdot 10^{-17}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.92) (not (<= a 3.1e-17)))
   (pow a 4.0)
   (+ (* (* a a) 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.92) || !(a <= 3.1e-17)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = ((a * a) * 4.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.92d0)) .or. (.not. (a <= 3.1d-17))) then
        tmp = a ** 4.0d0
    else
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.92) || !(a <= 3.1e-17)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = ((a * a) * 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.92) or not (a <= 3.1e-17):
		tmp = math.pow(a, 4.0)
	else:
		tmp = ((a * a) * 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.92) || !(a <= 3.1e-17))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.92) || ~((a <= 3.1e-17)))
		tmp = a ^ 4.0;
	else
		tmp = ((a * a) * 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.92], N[Not[LessEqual[a, 3.1e-17]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9199999999999999 or 3.0999999999999998e-17 < a

   1. Initial program 49.2%

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

      1. associate--l+ 49.2%

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

      2. fma-define 49.2%

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

      3. sqr-neg 49.2%

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

      4. fma-define 49.2%

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

      5. distribute-rgt-in 49.2%

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

      6. sqr-neg 49.2%

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

      7. distribute-rgt-in 49.2%

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

      8. fma-define 49.2%

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

      9. sqr-neg 49.2%

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

   3. Simplified 51.2%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, b \cdot \left(b \cdot \left(a + 3\right)\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 86.8%

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

   if -1.9199999999999999 < a < 3.0999999999999998e-17

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0 99.2%

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

   4. Taylor expanded in a around 0 89.3%

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

      1. fma-define 89.3%

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

      2. unpow2 89.3%

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

      3. unpow2 89.3%

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

      4. swap-sqr 99.3%

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

      5. unpow2 99.3%

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

      6. *-commutative 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in b around 0 49.4%

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

      1. *-commutative 49.4%

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

   9. Simplified 49.4%

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

       1. pow2 49.4%

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

   11. Applied egg-rr 49.4%

       \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot 4 - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.92 \lor \neg \left(a \leq 3.1 \cdot 10^{-17}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3300000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3300000000.0) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3300000000.0) {
		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 <= 3300000000.0d0) 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 <= 3300000000.0) {
		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 <= 3300000000.0:
		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 (b <= 3300000000.0)
		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 <= 3300000000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3300000000.0], 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 b < 3.3e9

   1. Initial program 73.2%

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

   3. Taylor expanded in a around 0 89.9%

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

   4. Taylor expanded in a around inf 82.8%

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

   if 3.3e9 < b

   1. Initial program 64.5%

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

      1. associate--l+ 64.5%

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

      2. fma-define 64.5%

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

      3. sqr-neg 64.5%

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

      4. fma-define 64.5%

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

      5. distribute-rgt-in 64.5%

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

      6. sqr-neg 64.5%

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

      7. distribute-rgt-in 64.5%

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

      8. fma-define 64.5%

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

      9. sqr-neg 64.5%

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

   3. Simplified 67.6%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, b \cdot \left(b \cdot \left(a + 3\right)\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3300000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2000000000.0) (+ (* (* a a) 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2000000000.0) {
		tmp = ((a * 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 <= 2000000000.0d0) then
        tmp = ((a * 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 <= 2000000000.0) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2000000000.0:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2000000000.0)
		tmp = Float64(Float64(Float64(a * 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 <= 2000000000.0)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2000000000.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2e9

   1. Initial program 73.2%

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

   3. Taylor expanded in a around 0 89.9%

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

   4. Taylor expanded in a around 0 57.6%

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

      1. fma-define 57.6%

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

      2. unpow2 57.6%

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

      3. unpow2 57.6%

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

      4. swap-sqr 71.8%

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

      5. unpow2 71.8%

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

      6. *-commutative 71.8%

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

   6. Simplified 71.8%

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

   7. Taylor expanded in b around 0 61.1%

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

      1. *-commutative 61.1%

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

   9. Simplified 61.1%

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

       1. pow2 61.1%

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

   11. Applied egg-rr 61.1%

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

   if 2e9 < b

   1. Initial program 64.5%

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

      1. associate--l+ 64.5%

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

      2. fma-define 64.5%

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

      3. sqr-neg 64.5%

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

      4. fma-define 64.5%

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

      5. distribute-rgt-in 64.5%

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

      6. sqr-neg 64.5%

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

      7. distribute-rgt-in 64.5%

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

      8. fma-define 64.5%

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

      9. sqr-neg 64.5%

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

   3. Simplified 67.6%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, b \cdot \left(b \cdot \left(a + 3\right)\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.6% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in a around 0 87.7%

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

4. Taylor expanded in a around 0 60.6%

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

   1. fma-define 60.6%

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

   2. unpow2 60.6%

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

   3. unpow2 60.6%

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

   4. swap-sqr 74.3%

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

   5. unpow2 74.3%

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

   6. *-commutative 74.3%

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

6. Simplified 74.3%

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

7. Taylor expanded in b around 0 51.9%

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

   1. *-commutative 51.9%

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

9. Simplified 51.9%

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

    1. pow2 51.9%

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

11. Applied egg-rr 51.9%

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

12. Final simplification 51.9%

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

Alternative 6: 25.2% accurate, 128.0× 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 71.0%

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

3. Taylor expanded in a around 0 87.7%

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

4. Taylor expanded in a around inf 70.7%

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

5. Taylor expanded in a around 0 21.6%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))