Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.4% → 98.3%

Time: 9.7s

Alternatives: 8

Speedup: 9.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = (math.pow(a, 3.0) * (a + -4.0)) + -1.0
	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(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(Float64((a ^ 3.0) * Float64(a + -4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = ((a ^ 3.0) * (a + -4.0)) + -1.0;
	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[(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[(t$95$0 + -1.0), $MachinePrecision], N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $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 3 a))))) < +inf.0

   1. Initial program 99.8%

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

   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. Taylor expanded in b around 0 32.5%

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

      1. unpow2 32.5%

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

      2. associate-*l* 32.5%

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

   4. Simplified 32.5%

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

   5. Taylor expanded in a around inf 32.5%

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

      1. unpow2 32.5%

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

      2. mul-1-neg 32.5%

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

      3. distribute-rgt-neg-in 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in a around 0 32.5%

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

      1. *-commutative 32.5%

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

      2. metadata-eval 32.5%

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

      3. pow-plus 32.5%

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

      4. distribute-lft-out 97.1%

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

   10. Simplified 97.1%

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

4. Final simplification 99.1%

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

Alternative 2: 93.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+89}:\\ \;\;\;\;\left({a}^{4} + \left(a \cdot a\right) \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+89) (+ (+ (pow a 4.0) (* (* a a) 4.0)) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+89) {
		tmp = (pow(a, 4.0) + ((a * a) * 4.0)) + -1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+89) then
        tmp = ((a ** 4.0d0) + ((a * a) * 4.0d0)) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+89) {
		tmp = (Math.pow(a, 4.0) + ((a * a) * 4.0)) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+89:
		tmp = (math.pow(a, 4.0) + ((a * a) * 4.0)) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+89)
		tmp = Float64(Float64((a ^ 4.0) + Float64(Float64(a * a) * 4.0)) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+89)
		tmp = ((a ^ 4.0) + ((a * a) * 4.0)) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+89], N[(N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999995e88

   1. Initial program 83.3%

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

   2. Taylor expanded in b around 0 78.7%

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

      1. unpow2 78.7%

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

      2. associate-*l* 78.7%

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

   4. Simplified 78.7%

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

   5. Taylor expanded in a around 0 93.4%

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

      1. unpow2 93.4%

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

   7. Simplified 93.4%

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

   if 9.99999999999999995e88 < (*.f64 b b)

   1. Initial program 61.8%

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

      1. associate--l+ 61.8%

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

      2. fma-def 61.8%

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

      3. +-commutative 61.8%

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

      4. sqr-neg 61.8%

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

      5. +-commutative 61.8%

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

      6. fma-def 64.7%

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

      7. sqr-neg 64.7%

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

      8. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in b around inf 97.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+89}:\\ \;\;\;\;\left({a}^{4} + \left(a \cdot a\right) \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 3: 93.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+89}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+89) (+ (* (* a a) (+ (* a a) 4.0)) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+89) {
		tmp = ((a * a) * ((a * a) + 4.0)) + -1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+89) then
        tmp = ((a * a) * ((a * a) + 4.0d0)) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+89) {
		tmp = ((a * a) * ((a * a) + 4.0)) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+89:
		tmp = ((a * a) * ((a * a) + 4.0)) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+89)
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(a * a) + 4.0)) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+89)
		tmp = ((a * a) * ((a * a) + 4.0)) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+89], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999995e88

   1. Initial program 83.3%

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

   2. Taylor expanded in b around 0 78.7%

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

      1. unpow2 78.7%

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

      2. associate-*l* 78.7%

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

   4. Simplified 78.7%

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

   5. Taylor expanded in a around 0 93.4%

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

      1. unpow2 93.4%

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

   7. Simplified 93.4%

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

      1. +-commutative 93.4%

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

      2. metadata-eval 93.4%

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

      3. pow-sqr 93.3%

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

      4. pow-prod-down 93.3%

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

      5. pow2 93.3%

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

      6. distribute-rgt-out 93.3%

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

   9. Applied egg-rr 93.3%

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

   if 9.99999999999999995e88 < (*.f64 b b)

   1. Initial program 61.8%

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

      1. associate--l+ 61.8%

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

      2. fma-def 61.8%

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

      3. +-commutative 61.8%

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

      4. sqr-neg 61.8%

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

      5. +-commutative 61.8%

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

      6. fma-def 64.7%

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

      7. sqr-neg 64.7%

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

      8. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in b around inf 97.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+89}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 4: 93.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+89}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+89)
   (+ (* (* a a) (+ (* a a) 4.0)) -1.0)
   (+ (* (* b b) (* b b)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+89) {
		tmp = ((a * a) * ((a * a) + 4.0)) + -1.0;
	} else {
		tmp = ((b * b) * (b * b)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+89) then
        tmp = ((a * a) * ((a * a) + 4.0d0)) + (-1.0d0)
    else
        tmp = ((b * b) * (b * b)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+89) {
		tmp = ((a * a) * ((a * a) + 4.0)) + -1.0;
	} else {
		tmp = ((b * b) * (b * b)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+89:
		tmp = ((a * a) * ((a * a) + 4.0)) + -1.0
	else:
		tmp = ((b * b) * (b * b)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+89)
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(a * a) + 4.0)) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(b * b)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+89)
		tmp = ((a * a) * ((a * a) + 4.0)) + -1.0;
	else
		tmp = ((b * b) * (b * b)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+89], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999995e88

   1. Initial program 83.3%

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

   2. Taylor expanded in b around 0 78.7%

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

      1. unpow2 78.7%

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

      2. associate-*l* 78.7%

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

   4. Simplified 78.7%

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

   5. Taylor expanded in a around 0 93.4%

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

      1. unpow2 93.4%

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

   7. Simplified 93.4%

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

      1. +-commutative 93.4%

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

      2. metadata-eval 93.4%

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

      3. pow-sqr 93.3%

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

      4. pow-prod-down 93.3%

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

      5. pow2 93.3%

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

      6. distribute-rgt-out 93.3%

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

   9. Applied egg-rr 93.3%

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

   if 9.99999999999999995e88 < (*.f64 b b)

   1. Initial program 61.8%

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

   2. Taylor expanded in a around 0 63.8%

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

      1. +-commutative 63.8%

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

      2. +-commutative 63.8%

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

      3. associate-+l+ 63.8%

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

      4. +-commutative 63.8%

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

      5. unpow2 63.8%

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

      6. unpow2 63.8%

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

      7. associate-*r* 63.8%

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

      8. distribute-rgt-out 78.1%

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

   4. Simplified 78.1%

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

   5. Taylor expanded in a around 0 97.3%

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

      1. unpow2 97.3%

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

   7. Simplified 97.3%

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

      1. +-commutative 97.3%

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

      2. metadata-eval 97.3%

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

      3. pow-sqr 97.2%

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

      4. pow-prod-down 97.2%

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

      5. pow2 97.2%

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

      6. distribute-rgt-out 97.2%

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

   9. Applied egg-rr 97.2%

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

   10. Taylor expanded in b around inf 97.2%

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

       1. unpow2 97.2%

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

   12. Simplified 97.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+89}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \end{array} \]

Alternative 5: 73.6% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1.0:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = 2.0 * (a * (a * (b * b)))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1

   1. Initial program 84.1%

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

   2. Taylor expanded in b around 0 84.0%

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

      1. unpow2 84.0%

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

      2. associate-*l* 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in a around 0 74.0%

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

      1. unpow2 74.0%

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

   7. Simplified 74.0%

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

   if 1 < (*.f64 b b)

   1. Initial program 64.1%

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

      1. associate--l+ 64.1%

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

      2. fma-def 64.1%

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

      3. +-commutative 64.1%

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

      4. sqr-neg 64.1%

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

      5. +-commutative 64.1%

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

      6. fma-def 66.5%

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

      7. sqr-neg 66.5%

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

      8. +-commutative 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in a around inf 44.1%

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

   5. Taylor expanded in b around inf 56.6%

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

      1. unpow2 56.6%

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

      2. unpow2 56.6%

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

      3. associate-*l* 70.4%

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

      4. *-commutative 70.4%

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

      5. associate-*l* 59.4%

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

   7. Simplified 59.4%

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

   8. Taylor expanded in b around 0 70.4%

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

      1. unpow2 70.4%

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

   10. Simplified 70.4%

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

4. Final simplification 72.3%

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

Alternative 6: 83.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 500:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 500.0) (+ (* (* a a) 4.0) -1.0) (+ (* (* b b) (* b b)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 500.0) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = ((b * b) * (b * b)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 500.0d0) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = ((b * b) * (b * b)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 500.0) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = ((b * b) * (b * b)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 500.0:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = ((b * b) * (b * b)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 500.0)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(b * b)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 500.0)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = ((b * b) * (b * b)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 500.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 500

   1. Initial program 83.5%

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

   2. Taylor expanded in b around 0 83.4%

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

      1. unpow2 83.4%

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

      2. associate-*l* 83.4%

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

   4. Simplified 83.4%

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

   5. Taylor expanded in a around 0 73.5%

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

      1. unpow2 73.5%

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

   7. Simplified 73.5%

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

   if 500 < (*.f64 b b)

   1. Initial program 64.6%

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

   2. Taylor expanded in a around 0 61.8%

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

      1. +-commutative 61.8%

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

      2. +-commutative 61.8%

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

      3. associate-+l+ 61.8%

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

      4. +-commutative 61.8%

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

      5. unpow2 61.8%

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

      6. unpow2 61.8%

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

      7. associate-*r* 61.8%

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

      8. distribute-rgt-out 74.1%

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

   4. Simplified 74.1%

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

   5. Taylor expanded in a around 0 90.6%

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

      1. unpow2 90.6%

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

   7. Simplified 90.6%

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

      1. +-commutative 90.6%

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

      2. metadata-eval 90.6%

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

      3. pow-sqr 90.5%

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

      4. pow-prod-down 90.5%

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

      5. pow2 90.5%

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

      6. distribute-rgt-out 90.5%

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

   9. Applied egg-rr 90.5%

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

   10. Taylor expanded in b around inf 90.2%

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

       1. unpow2 90.2%

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

   12. Simplified 90.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 500:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \end{array} \]

Alternative 7: 70.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+301) (+ (* (* a a) 4.0) -1.0) (+ (* b (* b 12.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+301) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = (b * (b * 12.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+301) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = (b * (b * 12.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+301) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = (b * (b * 12.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+301:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = (b * (b * 12.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+301)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	else
		tmp = Float64(Float64(b * Float64(b * 12.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+301)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = (b * (b * 12.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+301], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000011e301

   1. Initial program 79.6%

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

   2. Taylor expanded in b around 0 67.0%

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

      1. unpow2 67.0%

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

      2. associate-*l* 67.0%

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

   4. Simplified 67.0%

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

   5. Taylor expanded in a around 0 60.4%

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

      1. unpow2 60.4%

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

   7. Simplified 60.4%

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

   if 2.00000000000000011e301 < (*.f64 b b)

   1. Initial program 58.7%

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

   2. Taylor expanded in a around 0 55.6%

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

      1. +-commutative 55.6%

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

      2. +-commutative 55.6%

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

      3. associate-+l+ 55.6%

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

      4. +-commutative 55.6%

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

      5. unpow2 55.6%

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

      6. unpow2 55.6%

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

      7. associate-*r* 55.6%

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

      8. distribute-rgt-out 79.4%

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

   4. Simplified 79.4%

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

   5. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow-sqr 100.0%

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

      4. pow-prod-down 100.0%

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

      5. pow2 100.0%

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

      6. distribute-rgt-out 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*r* 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) + -1\\ \end{array} \]

Alternative 8: 52.8% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

2. Taylor expanded in b around 0 55.1%

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

   1. unpow2 55.1%

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

   2. associate-*l* 55.1%

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

4. Simplified 55.1%

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

5. Taylor expanded in a around 0 52.9%

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

   1. unpow2 52.9%

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

7. Simplified 52.9%

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

8. Final simplification 52.9%

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

Reproduce

herbie shell --seed 2023280 
(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))