Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.8% → 99.0%

Time: 8.2s

Alternatives: 11

Speedup: 9.9×

• 59.8% 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: 74.8% 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: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate--l+ 75.3%

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

   2. fma-def 75.3%

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

3. Simplified 76.4%

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

4. Taylor expanded in a around 0 99.2%

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

   1. unpow2 99.2%

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

6. Simplified 99.2%

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

   1. fma-udef 99.2%

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

   2. +-commutative 99.2%

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

8. Applied egg-rr 99.2%

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

9. Final simplification 99.2%

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

Alternative 2: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-9}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-9)
   (+ (+ (* (* b b) 4.0) -1.0) (* (* a a) (* a a)))
   (+ (pow b 4.0) (* (* b b) (+ 4.0 (* (* a a) 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-9) {
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * (a * a));
	} else {
		tmp = pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d-9) then
        tmp = (((b * b) * 4.0d0) + (-1.0d0)) + ((a * a) * (a * a))
    else
        tmp = (b ** 4.0d0) + ((b * b) * (4.0d0 + ((a * a) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-9) {
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * (a * a));
	} else {
		tmp = Math.pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e-9:
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * (a * a))
	else:
		tmp = math.pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-9)
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + -1.0) + Float64(Float64(a * a) * Float64(a * a)));
	else
		tmp = Float64((b ^ 4.0) + Float64(Float64(b * b) * Float64(4.0 + Float64(Float64(a * a) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e-9)
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * (a * a));
	else
		tmp = (b ^ 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-9], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.00000000000000006e-9

   1. Initial program 83.4%

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

      1. associate--l+ 83.4%

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

      2. fma-def 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in a around 0 98.5%

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

      1. unpow2 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in a around inf 98.5%

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

      1. unpow2 98.5%

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

   9. Simplified 98.5%

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

       1. unpow2 98.5%

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

   11. Applied egg-rr 98.5%

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

   if 1.00000000000000006e-9 < (*.f64 b b)

   1. Initial program 68.0%

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

      1. associate--l+ 68.0%

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

      2. fma-def 68.0%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in a around 0 99.9%

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

      1. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in b around inf 96.6%

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

      1. unpow2 96.6%

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

      2. unpow2 96.6%

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

   9. Simplified 96.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-9}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 3: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+18} \lor \neg \left(a \leq 195000000000\right):\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.5e+18) (not (<= a 195000000000.0)))
   (+ (+ (* (* b b) 4.0) -1.0) (* (* a a) (* a a)))
   (+ (* b (* b (fma b b 4.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.5e+18) || !(a <= 195000000000.0)) {
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * (a * a));
	} else {
		tmp = (b * (b * fma(b, b, 4.0))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.5e+18) || !(a <= 195000000000.0))
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + -1.0) + Float64(Float64(a * a) * Float64(a * a)));
	else
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.5e+18], N[Not[LessEqual[a, 195000000000.0]], $MachinePrecision]], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5e18 or 1.95e11 < a

   1. Initial program 48.3%

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

      1. associate--l+ 48.3%

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

      2. fma-def 48.3%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in a around 0 99.8%

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

      1. unpow2 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in a around inf 97.0%

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

      1. unpow2 97.0%

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

   9. Simplified 97.0%

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

       1. unpow2 97.0%

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

   11. Applied egg-rr 97.0%

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

   if -2.5e18 < a < 1.95e11

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 98.7%

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

      1. unpow2 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in a around 0 97.8%

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

      1. unpow2 97.8%

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

   9. Simplified 97.8%

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

       1. associate-+r- 97.8%

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

       2. unpow2 97.8%

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

       3. distribute-rgt-out 97.8%

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

   11. Applied egg-rr 97.8%

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

   12. Taylor expanded in b around 0 97.9%

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

       1. metadata-eval 97.9%

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

       2. pow-plus 97.9%

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

       3. unpow2 97.9%

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

       4. associate-*r* 97.9%

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

       5. distribute-rgt-out 97.9%

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

       6. unpow3 97.8%

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

       7. distribute-rgt-out 97.8%

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

       8. fma-udef 97.8%

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

   14. Simplified 97.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+18} \lor \neg \left(a \leq 195000000000\right):\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \end{array} \]

Alternative 4: 96.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+18} \lor \neg \left(a \leq 10500000000\right):\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.5e+18) (not (<= a 10500000000.0)))
   (+ (+ (* (* b b) 4.0) -1.0) (* (* a a) (* a a)))
   (+ (* (* b b) (+ (* b b) 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.5e+18) || !(a <= 10500000000.0)) {
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * (a * a));
	} 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 ((a <= (-2.5d+18)) .or. (.not. (a <= 10500000000.0d0))) then
        tmp = (((b * b) * 4.0d0) + (-1.0d0)) + ((a * a) * (a * a))
    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 ((a <= -2.5e+18) || !(a <= 10500000000.0)) {
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * (a * a));
	} else {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.5e+18) or not (a <= 10500000000.0):
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * (a * a))
	else:
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.5e+18) || !(a <= 10500000000.0))
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + -1.0) + Float64(Float64(a * a) * Float64(a * a)));
	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 ((a <= -2.5e+18) || ~((a <= 10500000000.0)))
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * (a * a));
	else
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.5e+18], N[Not[LessEqual[a, 10500000000.0]], $MachinePrecision]], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $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 a < -2.5e18 or 1.05e10 < a

   1. Initial program 48.3%

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

      1. associate--l+ 48.3%

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

      2. fma-def 48.3%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in a around 0 99.8%

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

      1. unpow2 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in a around inf 97.0%

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

      1. unpow2 97.0%

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

   9. Simplified 97.0%

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

       1. unpow2 97.0%

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

   11. Applied egg-rr 97.0%

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

   if -2.5e18 < a < 1.05e10

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 98.7%

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

      1. unpow2 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in a around 0 97.8%

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

      1. unpow2 97.8%

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

   9. Simplified 97.8%

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

       1. associate-+r- 97.8%

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

       2. unpow2 97.8%

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

       3. distribute-rgt-out 97.8%

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

   11. Applied egg-rr 97.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+18} \lor \neg \left(a \leq 10500000000\right):\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \end{array} \]

Alternative 5: 87.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.8e+21)
   (* 2.0 (* b (* b (* a a))))
   (if (<= a 3.6e+102)
     (+ (* (* b b) (* b b)) -1.0)
     (+ (* 4.0 (* (* a a) (+ a 1.0))) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.8e+21) {
		tmp = 2.0 * (b * (b * (a * a)));
	} else if (a <= 3.6e+102) {
		tmp = ((b * b) * (b * b)) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -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 <= (-4.8d+21)) then
        tmp = 2.0d0 * (b * (b * (a * a)))
    else if (a <= 3.6d+102) then
        tmp = ((b * b) * (b * b)) + (-1.0d0)
    else
        tmp = (4.0d0 * ((a * a) * (a + 1.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.8e+21) {
		tmp = 2.0 * (b * (b * (a * a)));
	} else if (a <= 3.6e+102) {
		tmp = ((b * b) * (b * b)) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.8e+21:
		tmp = 2.0 * (b * (b * (a * a)))
	elif a <= 3.6e+102:
		tmp = ((b * b) * (b * b)) + -1.0
	else:
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.8e+21)
		tmp = Float64(2.0 * Float64(b * Float64(b * Float64(a * a))));
	elseif (a <= 3.6e+102)
		tmp = Float64(Float64(Float64(b * b) * Float64(b * b)) + -1.0);
	else
		tmp = Float64(Float64(4.0 * Float64(Float64(a * a) * Float64(a + 1.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.8e+21)
		tmp = 2.0 * (b * (b * (a * a)));
	elseif (a <= 3.6e+102)
		tmp = ((b * b) * (b * b)) + -1.0;
	else
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.8e+21], N[(2.0 * N[(b * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.6e+102], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.8e21

   1. Initial program 32.2%

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

      1. associate--l+ 32.2%

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

      2. fma-def 32.2%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in b around inf 59.2%

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

      1. *-commutative 59.2%

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

      2. fma-def 59.2%

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

      3. unpow2 59.2%

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

      4. +-commutative 59.2%

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

      5. +-commutative 59.2%

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

      6. fma-def 59.2%

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

      7. unpow2 59.2%

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

      8. distribute-lft-in 59.2%

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

      9. metadata-eval 59.2%

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

      10. associate-*r* 59.2%

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

      11. metadata-eval 59.2%

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

   6. Simplified 59.2%

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

   7. Taylor expanded in a around inf 59.2%

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

      1. unpow2 59.2%

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

      2. unpow2 59.2%

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

      3. associate-*r* 59.2%

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

      4. *-commutative 59.2%

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

      5. associate-*r* 59.2%

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

   9. Simplified 59.2%

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

   10. Taylor expanded in a around 0 59.2%

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

       1. unpow2 59.2%

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

       2. unpow2 59.2%

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

       3. *-commutative 59.2%

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

       4. associate-*l* 73.7%

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

   12. Simplified 73.7%

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

   if -4.8e21 < a < 3.6000000000000002e102

   1. Initial program 97.1%

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

      1. associate--l+ 97.1%

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

      2. fma-def 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in a around 0 98.7%

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

      1. unpow2 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in a around 0 94.7%

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

      1. unpow2 94.7%

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

   9. Simplified 94.7%

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

       1. associate-+r- 94.7%

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

       2. unpow2 94.7%

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

       3. distribute-rgt-out 94.7%

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

   11. Applied egg-rr 94.7%

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

   12. Taylor expanded in b around inf 94.2%

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

       1. unpow2 94.2%

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

   14. Simplified 94.2%

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

   if 3.6000000000000002e102 < a

   1. Initial program 63.0%

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

      1. associate--l+ 63.0%

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

      2. fma-def 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in b around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-*r* 100.0%

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

      3. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. unpow2 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

       1. cube-mult 100.0%

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

       2. distribute-rgt1-in 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \]

Alternative 6: 88.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.8e+21)
   (* 2.0 (* b (* b (* a a))))
   (if (<= a 3.5e+102)
     (+ (* (* b b) (+ (* b b) 4.0)) -1.0)
     (+ (* 4.0 (* (* a a) (+ a 1.0))) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.8e+21) {
		tmp = 2.0 * (b * (b * (a * a)));
	} else if (a <= 3.5e+102) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -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 <= (-4.8d+21)) then
        tmp = 2.0d0 * (b * (b * (a * a)))
    else if (a <= 3.5d+102) then
        tmp = ((b * b) * ((b * b) + 4.0d0)) + (-1.0d0)
    else
        tmp = (4.0d0 * ((a * a) * (a + 1.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.8e+21) {
		tmp = 2.0 * (b * (b * (a * a)));
	} else if (a <= 3.5e+102) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.8e+21:
		tmp = 2.0 * (b * (b * (a * a)))
	elif a <= 3.5e+102:
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0
	else:
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.8e+21)
		tmp = Float64(2.0 * Float64(b * Float64(b * Float64(a * a))));
	elseif (a <= 3.5e+102)
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)) + -1.0);
	else
		tmp = Float64(Float64(4.0 * Float64(Float64(a * a) * Float64(a + 1.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.8e+21)
		tmp = 2.0 * (b * (b * (a * a)));
	elseif (a <= 3.5e+102)
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	else
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.8e+21], N[(2.0 * N[(b * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.5e+102], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.8e21

   1. Initial program 32.2%

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

      1. associate--l+ 32.2%

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

      2. fma-def 32.2%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in b around inf 59.2%

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

      1. *-commutative 59.2%

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

      2. fma-def 59.2%

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

      3. unpow2 59.2%

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

      4. +-commutative 59.2%

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

      5. +-commutative 59.2%

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

      6. fma-def 59.2%

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

      7. unpow2 59.2%

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

      8. distribute-lft-in 59.2%

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

      9. metadata-eval 59.2%

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

      10. associate-*r* 59.2%

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

      11. metadata-eval 59.2%

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

   6. Simplified 59.2%

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

   7. Taylor expanded in a around inf 59.2%

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

      1. unpow2 59.2%

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

      2. unpow2 59.2%

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

      3. associate-*r* 59.2%

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

      4. *-commutative 59.2%

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

      5. associate-*r* 59.2%

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

   9. Simplified 59.2%

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

   10. Taylor expanded in a around 0 59.2%

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

       1. unpow2 59.2%

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

       2. unpow2 59.2%

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

       3. *-commutative 59.2%

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

       4. associate-*l* 73.7%

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

   12. Simplified 73.7%

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

   if -4.8e21 < a < 3.50000000000000011e102

   1. Initial program 97.1%

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

      1. associate--l+ 97.1%

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

      2. fma-def 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in a around 0 98.7%

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

      1. unpow2 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in a around 0 94.7%

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

      1. unpow2 94.7%

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

   9. Simplified 94.7%

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

       1. associate-+r- 94.7%

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

       2. unpow2 94.7%

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

       3. distribute-rgt-out 94.7%

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

   11. Applied egg-rr 94.7%

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

   if 3.50000000000000011e102 < a

   1. Initial program 63.0%

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

      1. associate--l+ 63.0%

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

      2. fma-def 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in b around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-*r* 100.0%

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

      3. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. unpow2 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

       1. cube-mult 100.0%

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

       2. distribute-rgt1-in 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \]

Alternative 7: 73.7% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 0.44\right):\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.4) (not (<= a 0.44)))
   (* 2.0 (* b (* b (* a a))))
   (+ (* (* b b) 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.4) || !(a <= 0.44)) {
		tmp = 2.0 * (b * (b * (a * a)));
	} 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 <= (-2.4d0)) .or. (.not. (a <= 0.44d0))) then
        tmp = 2.0d0 * (b * (b * (a * a)))
    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 <= -2.4) || !(a <= 0.44)) {
		tmp = 2.0 * (b * (b * (a * a)));
	} else {
		tmp = ((b * b) * 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.4) or not (a <= 0.44):
		tmp = 2.0 * (b * (b * (a * a)))
	else:
		tmp = ((b * b) * 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.4) || !(a <= 0.44))
		tmp = Float64(2.0 * Float64(b * Float64(b * Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.4) || ~((a <= 0.44)))
		tmp = 2.0 * (b * (b * (a * a)));
	else
		tmp = ((b * b) * 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.4], N[Not[LessEqual[a, 0.44]], $MachinePrecision]], N[(2.0 * N[(b * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.39999999999999991 or 0.440000000000000002 < a

   1. Initial program 52.2%

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

      1. associate--l+ 52.2%

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

      2. fma-def 52.2%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in b around inf 68.4%

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

      1. *-commutative 68.4%

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

      2. fma-def 68.4%

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

      3. unpow2 68.4%

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

      4. +-commutative 68.4%

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

      5. +-commutative 68.4%

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

      6. fma-def 68.4%

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

      7. unpow2 68.4%

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

      8. distribute-lft-in 68.4%

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

      9. metadata-eval 68.4%

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

      10. associate-*r* 68.4%

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

      11. metadata-eval 68.4%

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

   6. Simplified 68.4%

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

   7. Taylor expanded in a around inf 63.5%

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

      1. unpow2 63.5%

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

      2. unpow2 63.5%

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

      3. associate-*r* 63.5%

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

      4. *-commutative 63.5%

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

      5. associate-*r* 63.5%

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

   9. Simplified 63.5%

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

   10. Taylor expanded in a around 0 63.5%

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

       1. unpow2 63.5%

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

       2. unpow2 63.5%

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

       3. *-commutative 63.5%

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

       4. associate-*l* 74.1%

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

   12. Simplified 74.1%

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

   if -2.39999999999999991 < a < 0.440000000000000002

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 99.3%

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

      1. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in a around 0 99.3%

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

      1. unpow2 99.3%

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

   9. Simplified 99.3%

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

       1. associate-+r- 99.3%

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

       2. unpow2 99.3%

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

       3. distribute-rgt-out 99.3%

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

   11. Applied egg-rr 99.3%

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

   12. Taylor expanded in b around 0 75.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 0.44\right):\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \end{array} \]

Alternative 8: 85.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+21} \lor \neg \left(a \leq 7.8 \cdot 10^{+39}\right):\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.8e+21) (not (<= a 7.8e+39)))
   (* 2.0 (* b (* b (* a a))))
   (+ (* (* b b) (* b b)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -4.8e+21) || !(a <= 7.8e+39)) {
		tmp = 2.0 * (b * (b * (a * a)));
	} else {
		tmp = ((b * b) * (b * b)) + -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 <= (-4.8d+21)) .or. (.not. (a <= 7.8d+39))) then
        tmp = 2.0d0 * (b * (b * (a * a)))
    else
        tmp = ((b * b) * (b * b)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -4.8e+21) || !(a <= 7.8e+39)) {
		tmp = 2.0 * (b * (b * (a * a)));
	} else {
		tmp = ((b * b) * (b * b)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -4.8e+21) or not (a <= 7.8e+39):
		tmp = 2.0 * (b * (b * (a * a)))
	else:
		tmp = ((b * b) * (b * b)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -4.8e+21) || !(a <= 7.8e+39))
		tmp = Float64(2.0 * Float64(b * Float64(b * Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(b * b)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -4.8e+21) || ~((a <= 7.8e+39)))
		tmp = 2.0 * (b * (b * (a * a)));
	else
		tmp = ((b * b) * (b * b)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -4.8e+21], N[Not[LessEqual[a, 7.8e+39]], $MachinePrecision]], N[(2.0 * N[(b * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.8e21 or 7.8000000000000002e39 < a

   1. Initial program 47.0%

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

      1. associate--l+ 47.0%

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

      2. fma-def 47.0%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in b around inf 68.1%

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

      1. *-commutative 68.1%

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

      2. fma-def 68.1%

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

      3. unpow2 68.1%

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

      4. +-commutative 68.1%

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

      5. +-commutative 68.1%

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

      6. fma-def 68.1%

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

      7. unpow2 68.1%

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

      8. distribute-lft-in 68.1%

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

      9. metadata-eval 68.1%

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

      10. associate-*r* 68.1%

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

      11. metadata-eval 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in a around inf 68.1%

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

      1. unpow2 68.1%

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

      2. unpow2 68.1%

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

      3. associate-*r* 68.1%

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

      4. *-commutative 68.1%

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

      5. associate-*r* 68.1%

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

   9. Simplified 68.1%

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

   10. Taylor expanded in a around 0 68.1%

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

       1. unpow2 68.1%

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

       2. unpow2 68.1%

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

       3. *-commutative 68.1%

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

       4. associate-*l* 79.9%

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

   12. Simplified 79.9%

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

   if -4.8e21 < a < 7.8000000000000002e39

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 98.6%

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

      1. unpow2 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in a around 0 96.5%

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

      1. unpow2 96.5%

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

   9. Simplified 96.5%

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

       1. associate-+r- 96.5%

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

       2. unpow2 96.5%

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

       3. distribute-rgt-out 96.5%

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

   11. Applied egg-rr 96.5%

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

   12. Taylor expanded in b around inf 95.9%

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

       1. unpow2 95.9%

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

   14. Simplified 95.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+21} \lor \neg \left(a \leq 7.8 \cdot 10^{+39}\right):\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \end{array} \]

Alternative 9: 70.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+292}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+292) (+ (* a (* a 4.0)) -1.0) (* (* b b) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+292) {
		tmp = (a * (a * 4.0)) + -1.0;
	} else {
		tmp = (b * b) * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+292) then
        tmp = (a * (a * 4.0d0)) + (-1.0d0)
    else
        tmp = (b * b) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+292) {
		tmp = (a * (a * 4.0)) + -1.0;
	} else {
		tmp = (b * b) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+292:
		tmp = (a * (a * 4.0)) + -1.0
	else:
		tmp = (b * b) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+292)
		tmp = Float64(Float64(a * Float64(a * 4.0)) + -1.0);
	else
		tmp = Float64(Float64(b * b) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+292)
		tmp = (a * (a * 4.0)) + -1.0;
	else
		tmp = (b * b) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+292], N[(N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e292

   1. Initial program 80.8%

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

      1. associate--l+ 80.8%

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

      2. fma-def 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in b around 0 63.4%

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

      1. associate--l+ 63.4%

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

      2. associate-*r* 63.4%

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

      3. unpow2 63.4%

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

   6. Simplified 63.4%

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

   7. Taylor expanded in a around 0 50.5%

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

      1. sub-neg 50.5%

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

      2. unpow2 50.5%

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

      3. distribute-lft-out 50.5%

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

      4. metadata-eval 50.5%

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

   9. Simplified 50.5%

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

   10. Taylor expanded in a around 0 60.4%

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

       1. unpow2 60.4%

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

       2. *-commutative 60.4%

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

       3. associate-*r* 60.4%

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

   12. Simplified 60.4%

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

   if 2e292 < (*.f64 b b)

   1. Initial program 59.7%

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

      1. associate--l+ 59.7%

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

      2. fma-def 59.7%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around inf 97.4%

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

      1. unpow2 97.4%

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

   9. Simplified 97.4%

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

   10. Taylor expanded in b around inf 97.4%

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

       1. unpow2 97.4%

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

   12. Simplified 97.4%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+292}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \]

Alternative 10: 51.6% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.2 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 3.2e-7) -1.0 (* (* b b) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.2e-7) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 3.2d-7) then
        tmp = -1.0d0
    else
        tmp = (b * b) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.2e-7) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 3.2e-7:
		tmp = -1.0
	else:
		tmp = (b * b) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 3.2e-7)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 3.2e-7)
		tmp = -1.0;
	else
		tmp = (b * b) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 3.2e-7], -1.0, N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.2000000000000001e-7

   1. Initial program 83.4%

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

      1. associate--l+ 83.4%

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

      2. fma-def 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in b around 0 83.1%

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

      1. associate--l+ 83.1%

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

      2. associate-*r* 83.1%

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

      3. unpow2 83.1%

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

   6. Simplified 83.1%

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

   7. Taylor expanded in a around 0 49.0%

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

   if 3.2000000000000001e-7 < (*.f64 b b)

   1. Initial program 68.0%

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

      1. associate--l+ 68.0%

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

      2. fma-def 68.0%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in a around 0 99.9%

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

      1. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in a around inf 71.4%

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

      1. unpow2 71.4%

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

   9. Simplified 71.4%

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

   10. Taylor expanded in b around inf 51.4%

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

       1. unpow2 51.4%

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

   12. Simplified 51.4%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.2 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \]

Alternative 11: 26.1% 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. Step-by-step derivation

   1. associate--l+ 75.3%

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

   2. fma-def 75.3%

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

3. Simplified 76.4%

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

4. Taylor expanded in b around 0 51.4%

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

   1. associate--l+ 51.4%

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

   2. associate-*r* 51.4%

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

   3. unpow2 51.4%

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

6. Simplified 51.4%

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

7. Taylor expanded in a around 0 23.5%

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

8. Final simplification 23.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023249 
(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))