Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.4% → 98.2%

Time: 12.3s

Alternatives: 9

Speedup: 8.6×

• 60.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. fma-def 99.8%

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

   3. Simplified 100.0%

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

   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 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. sqr-pow 0.0%

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

      3. sqr-pow 0.0%

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

      4. fma-def 0.0%

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

      5. distribute-lft-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-lft-in 0.0%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in a around inf 96.3%

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

4. Final simplification 98.9%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[Power[a, 4.0], $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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.8%

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

   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 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. sqr-pow 0.0%

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

      3. sqr-pow 0.0%

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

      4. fma-def 0.0%

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

      5. distribute-lft-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-lft-in 0.0%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in a around inf 96.3%

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

4. Final simplification 98.8%

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

Alternative 3: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+18}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e+16)
   (pow a 4.0)
   (if (<= a 1.9e+18) (+ (+ (pow b 4.0) (* (* b b) 4.0)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e+16) {
		tmp = pow(a, 4.0);
	} else if (a <= 1.9e+18) {
		tmp = (pow(b, 4.0) + ((b * b) * 4.0)) + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-8.5d+16)) then
        tmp = a ** 4.0d0
    else if (a <= 1.9d+18) then
        tmp = ((b ** 4.0d0) + ((b * b) * 4.0d0)) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.5e+16) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 1.9e+18) {
		tmp = (Math.pow(b, 4.0) + ((b * b) * 4.0)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.5e+16:
		tmp = math.pow(a, 4.0)
	elif a <= 1.9e+18:
		tmp = (math.pow(b, 4.0) + ((b * b) * 4.0)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.5e+16)
		tmp = a ^ 4.0;
	elseif (a <= 1.9e+18)
		tmp = Float64(Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.5e+16)
		tmp = a ^ 4.0;
	elseif (a <= 1.9e+18)
		tmp = ((b ^ 4.0) + ((b * b) * 4.0)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.5e+16], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 1.9e+18], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.5e16 or 1.9e18 < a

   1. Initial program 41.3%

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

      1. associate--l+ 41.3%

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

      2. sqr-pow 41.3%

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

      3. sqr-pow 41.3%

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

      4. fma-def 41.3%

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

      5. distribute-lft-in 41.3%

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

      6. sqr-neg 41.3%

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

      7. distribute-lft-in 41.3%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in a around inf 95.6%

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

   if -8.5e16 < a < 1.9e18

   1. Initial program 98.3%

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

   2. Taylor expanded in a around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. associate-+l+ 83.5%

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

      3. +-commutative 83.5%

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

      4. associate-*r* 83.5%

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

      5. distribute-rgt-out 96.0%

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

      6. unpow2 96.0%

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

      7. +-commutative 96.0%

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

   4. Simplified 96.0%

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

   5. Taylor expanded in a around 0 97.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+18}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 4: 94.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+17}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{+19}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e+17)
   (pow a 4.0)
   (if (<= a 4.7e+19) (+ (* (* b b) (+ (* b b) 4.0)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e+17) {
		tmp = pow(a, 4.0);
	} else if (a <= 4.7e+19) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-8.5d+17)) then
        tmp = a ** 4.0d0
    else if (a <= 4.7d+19) then
        tmp = ((b * b) * ((b * b) + 4.0d0)) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.5e+17) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 4.7e+19) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.5e+17:
		tmp = math.pow(a, 4.0)
	elif a <= 4.7e+19:
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.5e+17)
		tmp = a ^ 4.0;
	elseif (a <= 4.7e+19)
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.5e+17)
		tmp = a ^ 4.0;
	elseif (a <= 4.7e+19)
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.5e+17], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 4.7e+19], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.5e17 or 4.7e19 < a

   1. Initial program 41.3%

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

      1. associate--l+ 41.3%

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

      2. sqr-pow 41.3%

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

      3. sqr-pow 41.3%

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

      4. fma-def 41.3%

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

      5. distribute-lft-in 41.3%

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

      6. sqr-neg 41.3%

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

      7. distribute-lft-in 41.3%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in a around inf 95.6%

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

   if -8.5e17 < a < 4.7e19

   1. Initial program 98.3%

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

   2. Taylor expanded in a around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. associate-+l+ 83.5%

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

      3. +-commutative 83.5%

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

      4. associate-*r* 83.5%

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

      5. distribute-rgt-out 96.0%

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

      6. unpow2 96.0%

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

      7. +-commutative 96.0%

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

   4. Simplified 96.0%

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

   5. Taylor expanded in a around 0 97.6%

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

      1. sqr-pow 97.5%

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

      2. metadata-eval 97.5%

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

      3. pow2 97.5%

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

      4. metadata-eval 97.5%

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

      5. pow2 97.5%

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

      6. distribute-lft-out 97.5%

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

   7. Applied egg-rr 97.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+17}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{+19}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5: 72.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+125}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e+125)
   (+ (* (* a a) 4.0) -1.0)
   (if (<= a 5.8e+44)
     (+ (* b (* b 4.0)) -1.0)
     (+ (* 4.0 (* (* a a) (+ a 1.0))) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.3e+125) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else if (a <= 5.8e+44) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.3d+125)) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else if (a <= 5.8d+44) then
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    else
        tmp = (4.0d0 * ((a * a) * (a + 1.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.3e+125) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else if (a <= 5.8e+44) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.3e+125:
		tmp = ((a * a) * 4.0) + -1.0
	elif a <= 5.8e+44:
		tmp = (b * (b * 4.0)) + -1.0
	else:
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.3e+125)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	elseif (a <= 5.8e+44)
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	else
		tmp = Float64(Float64(4.0 * Float64(Float64(a * a) * Float64(a + 1.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.3e+125)
		tmp = ((a * a) * 4.0) + -1.0;
	elseif (a <= 5.8e+44)
		tmp = (b * (b * 4.0)) + -1.0;
	else
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.3e+125], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 5.8e+44], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.30000000000000013e125

   1. Initial program 0.0%

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

   2. Taylor expanded in b around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. unpow2 0.0%

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

   4. Simplified 0.0%

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

   5. Taylor expanded in a around 0 89.2%

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

      1. unpow2 89.2%

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

   7. Simplified 89.2%

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

   if -2.30000000000000013e125 < a < 5.8000000000000004e44

   1. Initial program 94.1%

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

   2. Taylor expanded in a around 0 75.3%

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

      1. +-commutative 75.3%

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

      2. associate-+l+ 75.3%

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

      3. +-commutative 75.3%

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

      4. associate-*r* 75.3%

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

      5. distribute-rgt-out 85.5%

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

      6. unpow2 85.5%

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

      7. +-commutative 85.5%

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

   4. Simplified 85.5%

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

   5. Taylor expanded in a around 0 87.4%

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

   6. Taylor expanded in b around 0 66.0%

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

      1. unpow2 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*r* 66.0%

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

      4. *-commutative 66.0%

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

   8. Simplified 66.0%

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

   if 5.8000000000000004e44 < a

   1. Initial program 54.3%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in a around 0 85.4%

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

      1. distribute-lft-out 85.4%

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

      2. unpow2 85.4%

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

   7. Simplified 85.4%

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

      1. cube-mult 85.4%

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

      2. distribute-rgt1-in 85.4%

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

   9. Applied egg-rr 85.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+125}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \]

Alternative 6: 87.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+150}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{+84}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.05e+150)
   (+ (* (* a a) 4.0) -1.0)
   (if (<= a 1.06e+84)
     (+ (* (* b b) (+ (* b b) 4.0)) -1.0)
     (+ (* 4.0 (* (* a a) (+ a 1.0))) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.05e+150) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else if (a <= 1.06e+84) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.05d+150)) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else if (a <= 1.06d+84) then
        tmp = ((b * b) * ((b * b) + 4.0d0)) + (-1.0d0)
    else
        tmp = (4.0d0 * ((a * a) * (a + 1.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.05e+150) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else if (a <= 1.06e+84) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.05e+150:
		tmp = ((a * a) * 4.0) + -1.0
	elif a <= 1.06e+84:
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0
	else:
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.05e+150)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	elseif (a <= 1.06e+84)
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)) + -1.0);
	else
		tmp = Float64(Float64(4.0 * Float64(Float64(a * a) * Float64(a + 1.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.05e+150)
		tmp = ((a * a) * 4.0) + -1.0;
	elseif (a <= 1.06e+84)
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	else
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.05e+150], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 1.06e+84], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.04999999999999999e150

   1. Initial program 0.0%

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

   2. Taylor expanded in b around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. unpow2 0.0%

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

   4. Simplified 0.0%

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

   5. Taylor expanded in a around 0 97.7%

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

      1. unpow2 97.7%

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

   7. Simplified 97.7%

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

   if -1.04999999999999999e150 < a < 1.05999999999999993e84

   1. Initial program 91.4%

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

   2. Taylor expanded in a around 0 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-+l+ 73.0%

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

      3. +-commutative 73.0%

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

      4. associate-*r* 73.0%

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

      5. distribute-rgt-out 82.7%

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

      6. unpow2 82.7%

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

      7. +-commutative 82.7%

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

   4. Simplified 82.7%

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

   5. Taylor expanded in a around 0 85.1%

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

      1. sqr-pow 85.0%

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

      2. metadata-eval 85.0%

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

      3. pow2 85.0%

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

      4. metadata-eval 85.0%

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

      5. pow2 85.0%

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

      6. distribute-lft-out 85.0%

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

   7. Applied egg-rr 85.0%

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

   if 1.05999999999999993e84 < a

   1. Initial program 51.9%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in a around 0 92.9%

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

      1. distribute-lft-out 92.9%

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

      2. unpow2 92.9%

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

   7. Simplified 92.9%

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

      1. cube-mult 92.9%

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

      2. distribute-rgt1-in 92.9%

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

   9. Applied egg-rr 92.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+150}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{+84}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \]

Alternative 7: 70.1% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+272) (+ (* (* a a) 4.0) -1.0) (* b (* b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+272) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+272) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+272) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+272:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+272)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+272)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+272], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999973e272

   1. Initial program 74.7%

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

   2. Taylor expanded in b around 0 61.8%

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

      1. associate-*r* 61.8%

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

      2. unpow2 61.8%

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

   4. Simplified 61.8%

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

   5. Taylor expanded in a around 0 61.5%

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

      1. unpow2 61.5%

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

   7. Simplified 61.5%

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

   if 4.99999999999999973e272 < (*.f64 b b)

   1. Initial program 56.5%

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

   2. Taylor expanded in a around 0 46.4%

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

      1. +-commutative 46.4%

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

      2. associate-+l+ 46.4%

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

      3. +-commutative 46.4%

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

      4. associate-*r* 46.4%

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

      5. distribute-rgt-out 69.6%

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

      6. unpow2 69.6%

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

      7. +-commutative 69.6%

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

   4. Simplified 69.6%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 96.1%

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

      1. unpow2 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-*r* 96.1%

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

      4. *-commutative 96.1%

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

   8. Simplified 96.1%

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

   9. Taylor expanded in b around inf 96.1%

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

       1. unpow2 96.1%

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

       2. associate-*r* 96.1%

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

       3. *-commutative 96.1%

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

   11. Simplified 96.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 8: 39.1% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0145:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.0145) -1.0 (* b (* b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.0145) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.0145d0) then
        tmp = -1.0d0
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.0145) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.0145:
		tmp = -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.0145)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.0145)
		tmp = -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.0145], -1.0, N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0145000000000000007

   1. Initial program 73.7%

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

   2. Taylor expanded in a around 0 51.3%

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

      1. +-commutative 51.3%

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

      2. associate-+l+ 51.3%

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

      3. +-commutative 51.3%

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

      4. associate-*r* 51.3%

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

      5. distribute-rgt-out 57.2%

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

      6. unpow2 57.2%

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

      7. +-commutative 57.2%

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

   4. Simplified 57.2%

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

   5. Taylor expanded in a around 0 62.8%

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

   6. Taylor expanded in b around 0 31.6%

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

   if 0.0145000000000000007 < b

   1. Initial program 59.2%

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

   2. Taylor expanded in a around 0 55.6%

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

      1. +-commutative 55.6%

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

      2. associate-+l+ 55.6%

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

      3. +-commutative 55.6%

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

      4. associate-*r* 55.6%

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

      5. distribute-rgt-out 62.8%

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

      6. unpow2 62.8%

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

      7. +-commutative 62.8%

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

   4. Simplified 62.8%

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

   5. Taylor expanded in a around 0 87.7%

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

   6. Taylor expanded in b around 0 45.5%

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

      1. unpow2 45.5%

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

      2. *-commutative 45.5%

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

      3. associate-*r* 45.5%

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

      4. *-commutative 45.5%

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

   8. Simplified 45.5%

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

   9. Taylor expanded in b around inf 45.5%

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

       1. unpow2 45.5%

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

       2. associate-*r* 45.5%

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

       3. *-commutative 45.5%

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

   11. Simplified 45.5%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0145:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 9: 26.0% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 69.8%

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

2. Taylor expanded in a around 0 52.4%

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

   1. +-commutative 52.4%

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

   2. associate-+l+ 52.4%

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

   3. +-commutative 52.4%

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

   4. associate-*r* 52.4%

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

   5. distribute-rgt-out 58.7%

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

   6. unpow2 58.7%

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

   7. +-commutative 58.7%

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

4. Simplified 58.7%

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

5. Taylor expanded in a around 0 69.5%

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

6. Taylor expanded in b around 0 23.3%

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

7. Final simplification 23.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023274 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))