Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.2% → 97.6%

Time: 10.4s

Alternatives: 8

Speedup: 1.2×

• 60.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(b \cdot \left(b \cdot \left(a \cdot -3\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+150)
   (+ (+ (pow (+ (* b b) (* a a)) 2.0) (* 4.0 (* b (* b (* a -3.0))))) -1.0)
   (+ (* (* b b) (+ (* b b) 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+150) {
		tmp = (pow(((b * b) + (a * a)), 2.0) + (4.0 * (b * (b * (a * -3.0))))) + -1.0;
	} else {
		tmp = ((b * b) * ((b * 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) <= 5d+150) then
        tmp = ((((b * b) + (a * a)) ** 2.0d0) + (4.0d0 * (b * (b * (a * (-3.0d0)))))) + (-1.0d0)
    else
        tmp = ((b * b) * ((b * 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) <= 5e+150) {
		tmp = (Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (b * (b * (a * -3.0))))) + -1.0;
	} else {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+150:
		tmp = (math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (b * (b * (a * -3.0))))) + -1.0
	else:
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+150)
		tmp = Float64(Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(b * Float64(b * Float64(a * -3.0))))) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+150)
		tmp = ((((b * b) + (a * a)) ^ 2.0) + (4.0 * (b * (b * (a * -3.0))))) + -1.0;
	else
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+150], N[(N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * N[(b * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000009e150

   1. Initial program 82.9%

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

   2. Taylor expanded in a around inf 82.4%

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

      1. *-commutative 82.4%

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

      2. *-commutative 82.4%

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

      3. unpow2 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in a around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

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

      3. *-commutative 98.5%

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

      4. associate-*r* 98.5%

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

      5. associate-*l* 98.5%

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

   7. Simplified 98.5%

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

   if 5.00000000000000009e150 < (*.f64 b b)

   1. Initial program 62.9%

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

   2. Taylor expanded in a around 0 61.9%

      \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   3. Step-by-step derivation

      1. associate-+r+ 61.9%

         \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

      2. +-commutative 61.9%

         \[\leadsto \color{blue}{\left({b}^{4} + \left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right)\right)} - 1 \]

      3. associate-*r* 61.9%

         \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right)\right) - 1 \]

      4. distribute-rgt-out 82.5%

         \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)}\right) - 1 \]

      5. unpow2 82.5%

         \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(-12 \cdot a + 4\right)\right) - 1 \]

      6. +-commutative 82.5%

         \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 + -12 \cdot a\right)}\right) - 1 \]

   4. Simplified 82.5%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)\right)} - 1 \]

   5. Taylor expanded in a around 0 100.0%

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

      1. metadata-eval 100.0%

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

      2. pow-sqr 100.0%

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

      3. pow2 100.0%

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

      4. pow2 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(b \cdot \left(b \cdot \left(a \cdot -3\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \end{array} \]

Alternative 2: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+31}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+38}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+31)
   (pow a 4.0)
   (if (<= a 4.5e+38) (+ (+ (pow b 4.0) (* (* b b) 4.0)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+31) {
		tmp = pow(a, 4.0);
	} else if (a <= 4.5e+38) {
		tmp = (pow(b, 4.0) + ((b * b) * 4.0)) + -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 <= (-2.4d+31)) then
        tmp = a ** 4.0d0
    else if (a <= 4.5d+38) then
        tmp = ((b ** 4.0d0) + ((b * b) * 4.0d0)) + (-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 <= -2.4e+31) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 4.5e+38) {
		tmp = (Math.pow(b, 4.0) + ((b * b) * 4.0)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.4e+31:
		tmp = math.pow(a, 4.0)
	elif a <= 4.5e+38:
		tmp = (math.pow(b, 4.0) + ((b * b) * 4.0)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+31)
		tmp = a ^ 4.0;
	elseif (a <= 4.5e+38)
		tmp = Float64(Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.4e+31)
		tmp = a ^ 4.0;
	elseif (a <= 4.5e+38)
		tmp = ((b ^ 4.0) + ((b * b) * 4.0)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+31], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 4.5e+38], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.39999999999999982e31 or 4.4999999999999998e38 < a

   1. Initial program 43.7%

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

      1. associate--l+ 43.7%

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

      2. sqr-pow 43.7%

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

      3. sqr-pow 43.7%

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

      4. fma-def 43.7%

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

      5. fma-neg 43.7%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in a around inf 96.6%

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

   if -2.39999999999999982e31 < a < 4.4999999999999998e38

   1. Initial program 97.2%

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

   2. Taylor expanded in a around 0 82.8%

      \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   3. Step-by-step derivation

      1. associate-+r+ 82.8%

         \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

      2. +-commutative 82.8%

         \[\leadsto \color{blue}{\left({b}^{4} + \left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right)\right)} - 1 \]

      3. associate-*r* 82.8%

         \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right)\right) - 1 \]

      4. distribute-rgt-out 96.0%

         \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)}\right) - 1 \]

      5. unpow2 96.0%

         \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(-12 \cdot a + 4\right)\right) - 1 \]

      6. +-commutative 96.0%

         \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 + -12 \cdot a\right)}\right) - 1 \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)\right)} - 1 \]

   5. Taylor expanded in a around 0 98.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+31}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+38}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 3: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+25}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8e+25)
   (pow a 4.0)
   (if (<= a 1.5e+39) (+ (* b (* b (fma b b 4.0))) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8e+25) {
		tmp = pow(a, 4.0);
	} else if (a <= 1.5e+39) {
		tmp = (b * (b * fma(b, b, 4.0))) + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8e+25)
		tmp = a ^ 4.0;
	elseif (a <= 1.5e+39)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8e+25], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 1.5e+39], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.00000000000000072e25 or 1.5e39 < a

   1. Initial program 43.7%

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

      1. associate--l+ 43.7%

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

      2. sqr-pow 43.7%

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

      3. sqr-pow 43.7%

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

      4. fma-def 43.7%

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

      5. fma-neg 43.7%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in a around inf 96.6%

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

   if -8.00000000000000072e25 < a < 1.5e39

   1. Initial program 97.2%

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

   2. Taylor expanded in a around 0 82.8%

      \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   3. Step-by-step derivation

      1. associate-+r+ 82.8%

         \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

      2. +-commutative 82.8%

         \[\leadsto \color{blue}{\left({b}^{4} + \left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right)\right)} - 1 \]

      3. associate-*r* 82.8%

         \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right)\right) - 1 \]

      4. distribute-rgt-out 96.0%

         \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)}\right) - 1 \]

      5. unpow2 96.0%

         \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(-12 \cdot a + 4\right)\right) - 1 \]

      6. +-commutative 96.0%

         \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 + -12 \cdot a\right)}\right) - 1 \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)\right)} - 1 \]

   5. Taylor expanded in a around 0 98.7%

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

   6. Taylor expanded in b around 0 98.7%

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

      1. unpow2 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow-sqr 98.6%

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

      4. unpow2 98.6%

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

      5. unpow2 98.6%

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

      6. distribute-rgt-in 98.6%

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

      7. associate-*l* 98.6%

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

      8. +-commutative 98.6%

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

      9. fma-def 98.6%

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

   8. Simplified 98.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+25}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 4: 93.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -56000000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{+40}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -56000000000.0)
   (pow a 4.0)
   (if (<= a 2.15e+40) (+ (* (* b b) (+ (* b b) 4.0)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -56000000000.0) {
		tmp = pow(a, 4.0);
	} else if (a <= 2.15e+40) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -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 <= (-56000000000.0d0)) then
        tmp = a ** 4.0d0
    else if (a <= 2.15d+40) then
        tmp = ((b * b) * ((b * b) + 4.0d0)) + (-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 <= -56000000000.0) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 2.15e+40) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -56000000000.0:
		tmp = math.pow(a, 4.0)
	elif a <= 2.15e+40:
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -56000000000.0)
		tmp = a ^ 4.0;
	elseif (a <= 2.15e+40)
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -56000000000.0)
		tmp = a ^ 4.0;
	elseif (a <= 2.15e+40)
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -56000000000.0], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 2.15e+40], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.6e10 or 2.1500000000000001e40 < a

   1. Initial program 43.7%

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

      1. associate--l+ 43.7%

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

      2. sqr-pow 43.7%

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

      3. sqr-pow 43.7%

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

      4. fma-def 43.7%

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

      5. fma-neg 43.7%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in a around inf 96.6%

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

   if -5.6e10 < a < 2.1500000000000001e40

   1. Initial program 97.2%

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

   2. Taylor expanded in a around 0 82.8%

      \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   3. Step-by-step derivation

      1. associate-+r+ 82.8%

         \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

      2. +-commutative 82.8%

         \[\leadsto \color{blue}{\left({b}^{4} + \left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right)\right)} - 1 \]

      3. associate-*r* 82.8%

         \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right)\right) - 1 \]

      4. distribute-rgt-out 96.0%

         \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)}\right) - 1 \]

      5. unpow2 96.0%

         \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(-12 \cdot a + 4\right)\right) - 1 \]

      6. +-commutative 96.0%

         \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 + -12 \cdot a\right)}\right) - 1 \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)\right)} - 1 \]

   5. Taylor expanded in a around 0 98.7%

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

      1. metadata-eval 98.7%

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

      2. pow-sqr 98.6%

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

      3. pow2 98.6%

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

      4. pow2 98.6%

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

      5. distribute-lft-out 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -56000000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{+40}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5: 53.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5000000000:\\ \;\;\;\;-12 \cdot \left(\left(b \cdot b\right) \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5000000000.0)
   (+ (* -12.0 (* (* b b) a)) -1.0)
   (+ (* b (* b 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5000000000.0) {
		tmp = (-12.0 * ((b * b) * a)) + -1.0;
	} else {
		tmp = (b * (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 (a <= (-5000000000.0d0)) then
        tmp = ((-12.0d0) * ((b * b) * a)) + (-1.0d0)
    else
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5000000000.0) {
		tmp = (-12.0 * ((b * b) * a)) + -1.0;
	} else {
		tmp = (b * (b * 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5000000000.0:
		tmp = (-12.0 * ((b * b) * a)) + -1.0
	else:
		tmp = (b * (b * 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5000000000.0)
		tmp = Float64(Float64(-12.0 * Float64(Float64(b * b) * a)) + -1.0);
	else
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5000000000.0)
		tmp = (-12.0 * ((b * b) * a)) + -1.0;
	else
		tmp = (b * (b * 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5000000000.0], N[(N[(-12.0 * N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5e9

   1. Initial program 23.6%

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

   2. Taylor expanded in a around 0 47.2%

      \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   3. Step-by-step derivation

      1. associate-+r+ 47.2%

         \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

      2. +-commutative 47.2%

         \[\leadsto \color{blue}{\left({b}^{4} + \left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right)\right)} - 1 \]

      3. associate-*r* 47.2%

         \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right)\right) - 1 \]

      4. distribute-rgt-out 47.2%

         \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)}\right) - 1 \]

      5. unpow2 47.2%

         \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(-12 \cdot a + 4\right)\right) - 1 \]

      6. +-commutative 47.2%

         \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 + -12 \cdot a\right)}\right) - 1 \]

   4. Simplified 47.2%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)\right)} - 1 \]

   5. Taylor expanded in a around inf 42.5%

      \[\leadsto \color{blue}{-12 \cdot \left(a \cdot {b}^{2}\right)} - 1 \]
   6. Step-by-step derivation

      1. unpow2 42.5%

         \[\leadsto -12 \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right) - 1 \]

   7. Simplified 42.5%

      \[\leadsto \color{blue}{-12 \cdot \left(a \cdot \left(b \cdot b\right)\right)} - 1 \]

   if -5e9 < a

   1. Initial program 90.8%

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

   2. Taylor expanded in a around 0 63.5%

      \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   3. Step-by-step derivation

      1. associate-+r+ 63.5%

         \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

      2. +-commutative 63.5%

         \[\leadsto \color{blue}{\left({b}^{4} + \left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right)\right)} - 1 \]

      3. associate-*r* 63.5%

         \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right)\right) - 1 \]

      4. distribute-rgt-out 73.7%

         \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)}\right) - 1 \]

      5. unpow2 73.7%

         \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(-12 \cdot a + 4\right)\right) - 1 \]

      6. +-commutative 73.7%

         \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 + -12 \cdot a\right)}\right) - 1 \]

   4. Simplified 73.7%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)\right)} - 1 \]

   5. Taylor expanded in a around 0 82.5%

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

   6. Taylor expanded in b around 0 82.5%

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

      1. unpow2 82.5%

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

      2. metadata-eval 82.5%

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

      3. pow-sqr 82.4%

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

      4. unpow2 82.4%

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

      5. unpow2 82.4%

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

      6. distribute-rgt-in 82.4%

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

      7. associate-*l* 82.4%

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

      8. +-commutative 82.4%

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

      9. fma-def 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in b around 0 59.8%

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

       1. unpow2 59.8%

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

       2. *-commutative 59.8%

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

       3. associate-*r* 59.8%

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

   11. Simplified 59.8%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5000000000:\\ \;\;\;\;-12 \cdot \left(\left(b \cdot b\right) \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \]

Alternative 6: 69.4% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

2. Taylor expanded in a around 0 59.8%

   \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
3. Step-by-step derivation

   1. associate-+r+ 59.8%

      \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

   2. +-commutative 59.8%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right)\right)} - 1 \]

   3. associate-*r* 59.8%

      \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right)\right) - 1 \]

   4. distribute-rgt-out 67.6%

      \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)}\right) - 1 \]

   5. unpow2 67.6%

      \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(-12 \cdot a + 4\right)\right) - 1 \]

   6. +-commutative 67.6%

      \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 + -12 \cdot a\right)}\right) - 1 \]

4. Simplified 67.6%

   \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)\right)} - 1 \]

5. Taylor expanded in a around 0 73.5%

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

   1. metadata-eval 73.5%

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

   2. pow-sqr 73.4%

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

   3. pow2 73.4%

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

   4. pow2 73.4%

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

   5. distribute-lft-out 73.4%

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

7. Applied egg-rr 73.4%

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

8. Final simplification 73.4%

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

Alternative 7: 50.7% accurate, 18.6× speedup

Math

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

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(b * Float64(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[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 75.3%

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

2. Taylor expanded in a around 0 59.8%

   \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
3. Step-by-step derivation

   1. associate-+r+ 59.8%

      \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

   2. +-commutative 59.8%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right)\right)} - 1 \]

   3. associate-*r* 59.8%

      \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right)\right) - 1 \]

   4. distribute-rgt-out 67.6%

      \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)}\right) - 1 \]

   5. unpow2 67.6%

      \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(-12 \cdot a + 4\right)\right) - 1 \]

   6. +-commutative 67.6%

      \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 + -12 \cdot a\right)}\right) - 1 \]

4. Simplified 67.6%

   \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)\right)} - 1 \]

5. Taylor expanded in a around 0 73.5%

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

6. Taylor expanded in b around 0 73.5%

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

   1. unpow2 73.5%

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

   2. metadata-eval 73.5%

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

   3. pow-sqr 73.4%

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

   4. unpow2 73.4%

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

   5. unpow2 73.4%

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

   6. distribute-rgt-in 73.4%

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

   7. associate-*l* 73.5%

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

   8. +-commutative 73.5%

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

   9. fma-def 73.5%

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

8. Simplified 73.5%

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

9. Taylor expanded in b around 0 51.6%

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

    1. unpow2 51.6%

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

    2. *-commutative 51.6%

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

    3. associate-*r* 51.6%

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

11. Simplified 51.6%

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

12. Final simplification 51.6%

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

Alternative 8: 24.9% accurate, 130.0× 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 75.3%

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

2. Taylor expanded in a around 0 59.8%

   \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
3. Step-by-step derivation

   1. associate-+r+ 59.8%

      \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

   2. +-commutative 59.8%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right)\right)} - 1 \]

   3. associate-*r* 59.8%

      \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right)\right) - 1 \]

   4. distribute-rgt-out 67.6%

      \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)}\right) - 1 \]

   5. unpow2 67.6%

      \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(-12 \cdot a + 4\right)\right) - 1 \]

   6. +-commutative 67.6%

      \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 + -12 \cdot a\right)}\right) - 1 \]

4. Simplified 67.6%

   \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)\right)} - 1 \]

5. Taylor expanded in b around 0 27.5%

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

6. Final simplification 27.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023285 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))