Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.7% → 99.6%

Time: 7.7s

Alternatives: 11

Speedup: 9.8×

• 60.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\\ \mathbf{if}\;a \leq -4.8 \cdot 10^{+15}:\\ \;\;\;\;t_0 + -4 \cdot {a}^{3}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+76}:\\ \;\;\;\;t_0 + \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, a + 3, a \cdot a\right) - {a}^{3}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (pow (hypot a b) 4.0)))
   (if (<= a -4.8e+15)
     (+ t_0 (* -4.0 (pow a 3.0)))
     (if (<= a 2.8e+76)
       (+ t_0 (fma 4.0 (- (fma (* b b) (+ a 3.0) (* a a)) (pow a 3.0)) -1.0))
       (pow a 4.0)))))

C

double code(double a, double b) {
	double t_0 = pow(hypot(a, b), 4.0);
	double tmp;
	if (a <= -4.8e+15) {
		tmp = t_0 + (-4.0 * pow(a, 3.0));
	} else if (a <= 2.8e+76) {
		tmp = t_0 + fma(4.0, (fma((b * b), (a + 3.0), (a * a)) - pow(a, 3.0)), -1.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = hypot(a, b) ^ 4.0
	tmp = 0.0
	if (a <= -4.8e+15)
		tmp = Float64(t_0 + Float64(-4.0 * (a ^ 3.0)));
	elseif (a <= 2.8e+76)
		tmp = Float64(t_0 + fma(4.0, Float64(fma(Float64(b * b), Float64(a + 3.0), Float64(a * a)) - (a ^ 3.0)), -1.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[a, -4.8e+15], N[(t$95$0 + N[(-4.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.8e+76], N[(t$95$0 + N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.8e15

   1. Initial program 56.2%

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

      1. associate--l+ 56.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -4.8e15 < a < 2.7999999999999999e76

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

   3. Simplified 100.0%

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

   if 2.7999999999999999e76 < a

   1. Initial program 13.8%

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

      1. sub-neg 13.8%

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

      2. fma-def 13.8%

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

      3. fma-def 15.5%

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

      4. +-commutative 15.5%

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

      5. metadata-eval 15.5%

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

   3. Simplified 15.5%

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

   4. Taylor expanded in b around 0 13.8%

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

      1. unpow2 13.8%

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

   6. Simplified 13.8%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in a around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+15}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -4 \cdot {a}^{3}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+76}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, a + 3, a \cdot a\right) - {a}^{3}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+15}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -4 \cdot {a}^{3}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+76}:\\ \;\;\;\;-1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.8e+15)
   (+ (pow (hypot a b) 4.0) (* -4.0 (pow a 3.0)))
   (if (<= a 2.8e+76)
     (+
      -1.0
      (+
       (pow (+ (* b b) (* a a)) 2.0)
       (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0))))))
     (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.8e+15) {
		tmp = pow(hypot(a, b), 4.0) + (-4.0 * pow(a, 3.0));
	} else if (a <= 2.8e+76) {
		tmp = -1.0 + (pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.8e+15) {
		tmp = Math.pow(Math.hypot(a, b), 4.0) + (-4.0 * Math.pow(a, 3.0));
	} else if (a <= 2.8e+76) {
		tmp = -1.0 + (Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.8e+15:
		tmp = math.pow(math.hypot(a, b), 4.0) + (-4.0 * math.pow(a, 3.0))
	elif a <= 2.8e+76:
		tmp = -1.0 + (math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.8e+15)
		tmp = Float64((hypot(a, b) ^ 4.0) + Float64(-4.0 * (a ^ 3.0)));
	elseif (a <= 2.8e+76)
		tmp = Float64(-1.0 + Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0))))));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.8e+15)
		tmp = (hypot(a, b) ^ 4.0) + (-4.0 * (a ^ 3.0));
	elseif (a <= 2.8e+76)
		tmp = -1.0 + ((((b * b) + (a * a)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.8e+15], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(-4.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.8e+76], N[(-1.0 + N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.8e15

   1. Initial program 56.2%

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

      1. associate--l+ 56.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -4.8e15 < a < 2.7999999999999999e76

   1. Initial program 99.8%

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

   if 2.7999999999999999e76 < a

   1. Initial program 13.8%

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

      1. sub-neg 13.8%

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

      2. fma-def 13.8%

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

      3. fma-def 15.5%

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

      4. +-commutative 15.5%

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

      5. metadata-eval 15.5%

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

   3. Simplified 15.5%

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

   4. Taylor expanded in b around 0 13.8%

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

      1. unpow2 13.8%

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

   6. Simplified 13.8%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in a around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+15}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -4 \cdot {a}^{3}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+76}:\\ \;\;\;\;-1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;-1 + t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* b b) (* a a)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))))
   (if (<= t_0 INFINITY) (+ -1.0 t_0) (+ -1.0 (* (* a a) (* a a))))))

C

double code(double a, double b) {
	double t_0 = pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = -1.0 + t_0;
	} else {
		tmp = -1.0 + ((a * a) * (a * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = -1.0 + t_0;
	} else {
		tmp = -1.0 + ((a * a) * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = -1.0 + t_0
	else:
		tmp = -1.0 + ((a * a) * (a * a))
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(-1.0 + t_0);
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((b * b) + (a * a)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = -1.0 + t_0;
	else
		tmp = -1.0 + ((a * a) * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(-1.0 + t$95$0), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a))))) < +inf.0

   1. Initial program 99.9%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a)))))

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 1.3%

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

      4. +-commutative 1.3%

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

      5. metadata-eval 1.3%

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

   3. Simplified 1.3%

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

   4. Taylor expanded in b around 0 29.8%

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

      1. unpow2 29.8%

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

   6. Simplified 29.8%

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

   7. Taylor expanded in a around 0 93.9%

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

      1. unpow2 93.9%

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

   9. Simplified 93.9%

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

       1. sqr-pow 93.9%

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

       2. metadata-eval 93.9%

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

       3. pow2 93.9%

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

       4. metadata-eval 93.9%

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

       5. pow2 93.9%

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

       6. distribute-rgt-out 93.9%

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

   11. Applied egg-rr 93.9%

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

   12. Taylor expanded in a around inf 93.9%

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

       1. unpow2 93.9%

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

   14. Simplified 93.9%

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

4. Final simplification 98.0%

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

Alternative 4: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+48} \lor \neg \left(a \leq 80000000000\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.3e+48) (not (<= a 80000000000.0)))
   (pow a 4.0)
   (+ -1.0 (+ (pow b 4.0) (* b (* b 12.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.3e+48) || !(a <= 80000000000.0)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + (pow(b, 4.0) + (b * (b * 12.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.3d+48)) .or. (.not. (a <= 80000000000.0d0))) then
        tmp = a ** 4.0d0
    else
        tmp = (-1.0d0) + ((b ** 4.0d0) + (b * (b * 12.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.3e+48) || !(a <= 80000000000.0)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (Math.pow(b, 4.0) + (b * (b * 12.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.3e+48) or not (a <= 80000000000.0):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0 + (math.pow(b, 4.0) + (b * (b * 12.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.3e+48) || !(a <= 80000000000.0))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(b * Float64(b * 12.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.3e+48) || ~((a <= 80000000000.0)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0 + ((b ^ 4.0) + (b * (b * 12.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.3e+48], N[Not[LessEqual[a, 80000000000.0]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.29999999999999998e48 or 8e10 < a

   1. Initial program 42.0%

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

      1. sub-neg 42.0%

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

      2. fma-def 42.0%

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

      3. fma-def 42.8%

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

      4. +-commutative 42.8%

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

      5. metadata-eval 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in b around 0 58.0%

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

      1. unpow2 58.0%

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

   6. Simplified 58.0%

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

   7. Taylor expanded in a around 0 97.5%

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

      1. unpow2 97.5%

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

   9. Simplified 97.5%

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

   10. Taylor expanded in a around inf 97.5%

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

   if -1.29999999999999998e48 < a < 8e10

   1. Initial program 96.1%

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

      1. sub-neg 96.1%

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

      2. fma-def 96.1%

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

      3. fma-def 96.1%

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

      4. +-commutative 96.1%

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

      5. metadata-eval 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in a around 0 75.9%

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

      1. +-commutative 75.9%

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

      2. +-commutative 75.9%

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

      3. associate-+l+ 75.9%

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

      4. +-commutative 75.9%

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

      5. associate-*r* 75.9%

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

      6. distribute-rgt-out 91.3%

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

      7. metadata-eval 91.3%

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

      8. distribute-lft-in 91.3%

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

      9. unpow2 91.3%

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

      10. distribute-rgt-in 91.3%

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

      11. metadata-eval 91.3%

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

   6. Simplified 91.3%

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

   7. Taylor expanded in a around 0 95.2%

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

      1. unpow2 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*l* 95.2%

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

   9. Simplified 95.2%

      \[\leadsto \left({b}^{4} + \color{blue}{b \cdot \left(b \cdot 12\right)}\right) + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+48} \lor \neg \left(a \leq 80000000000\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \end{array} \]

Alternative 5: 93.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+51} \lor \neg \left(a \leq 650000000000\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.9e+51) (not (<= a 650000000000.0)))
   (pow a 4.0)
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.9e+51) || !(a <= 650000000000.0)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.9d+51)) .or. (.not. (a <= 650000000000.0d0))) then
        tmp = a ** 4.0d0
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.9e+51) || !(a <= 650000000000.0)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.9e+51) or not (a <= 650000000000.0):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.9e+51) || !(a <= 650000000000.0))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.9e+51) || ~((a <= 650000000000.0)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.9e+51], N[Not[LessEqual[a, 650000000000.0]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8999999999999999e51 or 6.5e11 < a

   1. Initial program 42.0%

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

      1. sub-neg 42.0%

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

      2. fma-def 42.0%

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

      3. fma-def 42.8%

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

      4. +-commutative 42.8%

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

      5. metadata-eval 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in b around 0 58.0%

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

      1. unpow2 58.0%

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

   6. Simplified 58.0%

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

   7. Taylor expanded in a around 0 97.5%

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

      1. unpow2 97.5%

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

   9. Simplified 97.5%

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

   10. Taylor expanded in a around inf 97.5%

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

   if -1.8999999999999999e51 < a < 6.5e11

   1. Initial program 96.1%

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

      1. sub-neg 96.1%

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

      2. fma-def 96.1%

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

      3. fma-def 96.1%

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

      4. +-commutative 96.1%

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

      5. metadata-eval 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in a around 0 75.9%

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

      1. +-commutative 75.9%

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

      2. +-commutative 75.9%

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

      3. associate-+l+ 75.9%

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

      4. +-commutative 75.9%

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

      5. associate-*r* 75.9%

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

      6. distribute-rgt-out 91.3%

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

      7. metadata-eval 91.3%

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

      8. distribute-lft-in 91.3%

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

      9. unpow2 91.3%

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

      10. distribute-rgt-in 91.3%

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

      11. metadata-eval 91.3%

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

   6. Simplified 91.3%

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

   7. Taylor expanded in a around 0 95.2%

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

      1. unpow2 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*l* 95.2%

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

   9. Simplified 95.2%

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

       1. sqr-pow 95.1%

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

       2. metadata-eval 95.1%

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

       3. pow2 95.1%

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

       4. metadata-eval 95.1%

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

       5. pow2 95.1%

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

       6. associate-*r* 95.1%

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

       7. distribute-lft-out 95.1%

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

   11. Applied egg-rr 95.1%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+51} \lor \neg \left(a \leq 650000000000\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 6: 93.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+55} \lor \neg \left(a \leq 2800000000\right):\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -6e+55) (not (<= a 2800000000.0)))
   (+ -1.0 (* (* a a) (* a a)))
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -6e+55) || !(a <= 2800000000.0)) {
		tmp = -1.0 + ((a * a) * (a * a));
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-6d+55)) .or. (.not. (a <= 2800000000.0d0))) then
        tmp = (-1.0d0) + ((a * a) * (a * a))
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -6e+55) || !(a <= 2800000000.0)) {
		tmp = -1.0 + ((a * a) * (a * a));
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -6e+55) or not (a <= 2800000000.0):
		tmp = -1.0 + ((a * a) * (a * a))
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -6e+55) || !(a <= 2800000000.0))
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -6e+55) || ~((a <= 2800000000.0)))
		tmp = -1.0 + ((a * a) * (a * a));
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -6e+55], N[Not[LessEqual[a, 2800000000.0]], $MachinePrecision]], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.00000000000000033e55 or 2.8e9 < a

   1. Initial program 42.0%

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

      1. sub-neg 42.0%

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

      2. fma-def 42.0%

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

      3. fma-def 42.8%

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

      4. +-commutative 42.8%

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

      5. metadata-eval 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in b around 0 58.0%

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

      1. unpow2 58.0%

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

   6. Simplified 58.0%

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

   7. Taylor expanded in a around 0 97.5%

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

      1. unpow2 97.5%

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

   9. Simplified 97.5%

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

       1. sqr-pow 97.4%

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

       2. metadata-eval 97.4%

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

       3. pow2 97.4%

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

       4. metadata-eval 97.4%

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

       5. pow2 97.4%

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

       6. distribute-rgt-out 97.4%

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

   11. Applied egg-rr 97.4%

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

   12. Taylor expanded in a around inf 97.4%

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

       1. unpow2 97.4%

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

   14. Simplified 97.4%

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

   if -6.00000000000000033e55 < a < 2.8e9

   1. Initial program 96.1%

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

      1. sub-neg 96.1%

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

      2. fma-def 96.1%

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

      3. fma-def 96.1%

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

      4. +-commutative 96.1%

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

      5. metadata-eval 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in a around 0 75.9%

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

      1. +-commutative 75.9%

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

      2. +-commutative 75.9%

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

      3. associate-+l+ 75.9%

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

      4. +-commutative 75.9%

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

      5. associate-*r* 75.9%

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

      6. distribute-rgt-out 91.3%

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

      7. metadata-eval 91.3%

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

      8. distribute-lft-in 91.3%

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

      9. unpow2 91.3%

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

      10. distribute-rgt-in 91.3%

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

      11. metadata-eval 91.3%

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

   6. Simplified 91.3%

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

   7. Taylor expanded in a around 0 95.2%

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

      1. unpow2 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*l* 95.2%

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

   9. Simplified 95.2%

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

       1. sqr-pow 95.1%

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

       2. metadata-eval 95.1%

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

       3. pow2 95.1%

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

       4. metadata-eval 95.1%

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

       5. pow2 95.1%

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

       6. associate-*r* 95.1%

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

       7. distribute-lft-out 95.1%

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

   11. Applied egg-rr 95.1%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+55} \lor \neg \left(a \leq 2800000000\right):\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 7: 84.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+304}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+304)
   (+ -1.0 (* (* a a) (+ 4.0 (* a a))))
   (+ -1.0 (* (* b b) 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+304) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * a)));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	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+304) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (a * a)))
    else
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+304) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * a)));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+304:
		tmp = -1.0 + ((a * a) * (4.0 + (a * a)))
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+304)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(a * a))));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+304)
		tmp = -1.0 + ((a * a) * (4.0 + (a * a)));
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+304], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999999e304

   1. Initial program 71.4%

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

      1. sub-neg 71.4%

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

      2. fma-def 71.4%

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

      3. fma-def 71.4%

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

      4. +-commutative 71.4%

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

      5. metadata-eval 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in b around 0 60.1%

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

      1. unpow2 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in a around 0 81.5%

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

      1. unpow2 81.5%

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

   9. Simplified 81.5%

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

       1. sqr-pow 81.4%

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

       2. metadata-eval 81.4%

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

       3. pow2 81.4%

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

       4. metadata-eval 81.4%

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

       5. pow2 81.4%

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

       6. distribute-rgt-out 81.4%

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

   11. Applied egg-rr 81.4%

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

   if 1.9999999999999999e304 < (*.f64 b b)

   1. Initial program 63.5%

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

      1. sub-neg 63.5%

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

      2. fma-def 63.5%

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

      3. fma-def 65.1%

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

      4. +-commutative 65.1%

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

      5. metadata-eval 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in a around 0 39.7%

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

      1. +-commutative 39.7%

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

      2. +-commutative 39.7%

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

      3. associate-+l+ 39.7%

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

      4. +-commutative 39.7%

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

      5. associate-*r* 39.7%

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

      6. distribute-rgt-out 71.4%

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

      7. metadata-eval 71.4%

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

      8. distribute-lft-in 71.4%

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

      9. unpow2 71.4%

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

      10. distribute-rgt-in 71.4%

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

      11. metadata-eval 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in b around 0 71.4%

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

      1. +-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. *-commutative 71.4%

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

      4. *-commutative 71.4%

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

      5. fma-udef 71.4%

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

      6. associate-*r* 71.4%

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

      7. fma-udef 71.4%

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

      8. *-commutative 71.4%

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

      9. fma-def 71.4%

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

   9. Simplified 71.4%

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

   10. Taylor expanded in a around 0 100.0%

       \[\leadsto \color{blue}{12 \cdot {b}^{2}} + -1 \]
   11. Step-by-step derivation

       1. unpow2 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{12 \cdot \left(b \cdot b\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+304}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 8: 84.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+304}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+304)
   (+ -1.0 (* (* a a) (* a a)))
   (+ -1.0 (* (* b b) 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+304) {
		tmp = -1.0 + ((a * a) * (a * a));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	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+304) then
        tmp = (-1.0d0) + ((a * a) * (a * a))
    else
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+304) {
		tmp = -1.0 + ((a * a) * (a * a));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+304:
		tmp = -1.0 + ((a * a) * (a * a))
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+304)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+304)
		tmp = -1.0 + ((a * a) * (a * a));
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+304], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999999e304

   1. Initial program 71.4%

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

      1. sub-neg 71.4%

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

      2. fma-def 71.4%

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

      3. fma-def 71.4%

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

      4. +-commutative 71.4%

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

      5. metadata-eval 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in b around 0 60.1%

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

      1. unpow2 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in a around 0 81.5%

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

      1. unpow2 81.5%

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

   9. Simplified 81.5%

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

       1. sqr-pow 81.4%

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

       2. metadata-eval 81.4%

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

       3. pow2 81.4%

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

       4. metadata-eval 81.4%

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

       5. pow2 81.4%

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

       6. distribute-rgt-out 81.4%

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

   11. Applied egg-rr 81.4%

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

   12. Taylor expanded in a around inf 81.1%

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

       1. unpow2 81.1%

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

   14. Simplified 81.1%

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

   if 1.9999999999999999e304 < (*.f64 b b)

   1. Initial program 63.5%

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

      1. sub-neg 63.5%

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

      2. fma-def 63.5%

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

      3. fma-def 65.1%

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

      4. +-commutative 65.1%

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

      5. metadata-eval 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in a around 0 39.7%

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

      1. +-commutative 39.7%

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

      2. +-commutative 39.7%

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

      3. associate-+l+ 39.7%

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

      4. +-commutative 39.7%

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

      5. associate-*r* 39.7%

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

      6. distribute-rgt-out 71.4%

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

      7. metadata-eval 71.4%

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

      8. distribute-lft-in 71.4%

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

      9. unpow2 71.4%

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

      10. distribute-rgt-in 71.4%

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

      11. metadata-eval 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in b around 0 71.4%

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

      1. +-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. *-commutative 71.4%

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

      4. *-commutative 71.4%

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

      5. fma-udef 71.4%

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

      6. associate-*r* 71.4%

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

      7. fma-udef 71.4%

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

      8. *-commutative 71.4%

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

      9. fma-def 71.4%

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

   9. Simplified 71.4%

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

   10. Taylor expanded in a around 0 100.0%

       \[\leadsto \color{blue}{12 \cdot {b}^{2}} + -1 \]
   11. Step-by-step derivation

       1. unpow2 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{12 \cdot \left(b \cdot b\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+304}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 9: 69.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+287}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+287) (+ -1.0 (* 4.0 (* a a))) (+ -1.0 (* (* b b) 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+287) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+287) then
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    else
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+287) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+287:
		tmp = -1.0 + (4.0 * (a * a))
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+287)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+287)
		tmp = -1.0 + (4.0 * (a * a));
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+287], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e287

   1. Initial program 71.8%

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

      1. sub-neg 71.8%

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

      2. fma-def 71.8%

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

      3. fma-def 71.8%

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

      4. +-commutative 71.8%

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

      5. metadata-eval 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in b around 0 60.3%

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

      1. unpow2 60.3%

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

   6. Simplified 60.3%

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

   7. Taylor expanded in a around 0 62.0%

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

      1. unpow2 62.0%

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

   9. Simplified 62.0%

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

   if 5e287 < (*.f64 b b)

   1. Initial program 62.7%

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

      1. sub-neg 62.7%

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

      2. fma-def 62.7%

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

      3. fma-def 64.2%

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

      4. +-commutative 64.2%

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

      5. metadata-eval 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in a around 0 40.3%

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

      1. +-commutative 40.3%

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

      2. +-commutative 40.3%

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

      3. associate-+l+ 40.3%

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

      4. +-commutative 40.3%

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

      5. associate-*r* 40.3%

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

      6. distribute-rgt-out 70.1%

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

      7. metadata-eval 70.1%

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

      8. distribute-lft-in 70.1%

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

      9. unpow2 70.1%

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

      10. distribute-rgt-in 70.1%

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

      11. metadata-eval 70.1%

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

   6. Simplified 70.1%

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

   7. Taylor expanded in b around 0 67.5%

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

      1. +-commutative 67.5%

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

      2. unpow2 67.5%

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

      3. *-commutative 67.5%

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

      4. *-commutative 67.5%

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

      5. fma-udef 67.5%

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

      6. associate-*r* 67.5%

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

      7. fma-udef 67.5%

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

      8. *-commutative 67.5%

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

      9. fma-def 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in a around 0 94.7%

       \[\leadsto \color{blue}{12 \cdot {b}^{2}} + -1 \]
   11. Step-by-step derivation

       1. unpow2 94.7%

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

   12. Simplified 94.7%

       \[\leadsto \color{blue}{12 \cdot \left(b \cdot b\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+287}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 10: 51.2% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

   1. sub-neg 69.4%

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

   2. fma-def 69.4%

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

   3. fma-def 69.8%

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

   4. +-commutative 69.8%

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

   5. metadata-eval 69.8%

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

3. Simplified 69.8%

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

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. unpow2 51.4%

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

6. Simplified 51.4%

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

7. Taylor expanded in a around 0 51.3%

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

   1. unpow2 51.3%

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

9. Simplified 51.3%

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

10. Final simplification 51.3%

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

Alternative 11: 24.8% accurate, 128.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 69.4%

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

   1. sub-neg 69.4%

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

   2. fma-def 69.4%

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

   3. fma-def 69.8%

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

   4. +-commutative 69.8%

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

   5. metadata-eval 69.8%

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

3. Simplified 69.8%

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

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. unpow2 51.4%

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

6. Simplified 51.4%

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

7. Taylor expanded in a around 0 20.6%

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

8. Final simplification 20.6%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))