Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.4% → 98.4%

Time: 16.5s

Alternatives: 12

Speedup: 8.0×

• 61.6% 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: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + -4\right)\right) + -1\\ \end{array} \end{array} \]

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < +inf.0

   1. Initial program 99.9%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) 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. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left({a}^{2}\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \left(1 - a\right)\right)\right)\right), 1\right) \]

      14. --lowering--.f64 95.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(1, a\right)\right)\right)\right), 1\right) \]

   5. Simplified 95.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \color{blue}{\left(4 + a \cdot \left(a - 4\right)\right)}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \left(a \cdot \left(a - 4\right)\right)\right)\right), 1\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a - 4\right)\right)\right)\right), 1\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a + \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a + -4\right)\right)\right)\right), 1\right) \]

      5. +-lowering-+.f64 95.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, -4\right)\right)\right)\right), 1\right) \]

   8. Simplified 95.9%

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

4. Final simplification 98.8%

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

Alternative 2: 98.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ t_1 := a \cdot a + b \cdot b\\ \mathbf{if}\;a \leq -9.5 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;t\_1 \cdot t\_1 + \left(4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right) + b \cdot \left(b \cdot \left(a + 3\right)\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))) (t_1 (+ (* a a) (* b b))))
   (if (<= a -9.5e+88)
     t_0
     (if (<= a 1.5e+77)
       (+
        (* t_1 t_1)
        (+ (* 4.0 (+ (* a (* a (- 1.0 a))) (* b (* b (+ a 3.0))))) -1.0))
       t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double t_1 = (a * a) + (b * b);
	double tmp;
	if (a <= -9.5e+88) {
		tmp = t_0;
	} else if (a <= 1.5e+77) {
		tmp = (t_1 * t_1) + ((4.0 * ((a * (a * (1.0 - a))) + (b * (b * (a + 3.0))))) + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    t_1 = (a * a) + (b * b)
    if (a <= (-9.5d+88)) then
        tmp = t_0
    else if (a <= 1.5d+77) then
        tmp = (t_1 * t_1) + ((4.0d0 * ((a * (a * (1.0d0 - a))) + (b * (b * (a + 3.0d0))))) + (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double t_1 = (a * a) + (b * b);
	double tmp;
	if (a <= -9.5e+88) {
		tmp = t_0;
	} else if (a <= 1.5e+77) {
		tmp = (t_1 * t_1) + ((4.0 * ((a * (a * (1.0 - a))) + (b * (b * (a + 3.0))))) + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	t_1 = (a * a) + (b * b)
	tmp = 0
	if a <= -9.5e+88:
		tmp = t_0
	elif a <= 1.5e+77:
		tmp = (t_1 * t_1) + ((4.0 * ((a * (a * (1.0 - a))) + (b * (b * (a + 3.0))))) + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	t_1 = Float64(Float64(a * a) + Float64(b * b))
	tmp = 0.0
	if (a <= -9.5e+88)
		tmp = t_0;
	elseif (a <= 1.5e+77)
		tmp = Float64(Float64(t_1 * t_1) + Float64(Float64(4.0 * Float64(Float64(a * Float64(a * Float64(1.0 - a))) + Float64(b * Float64(b * Float64(a + 3.0))))) + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	t_1 = (a * a) + (b * b);
	tmp = 0.0;
	if (a <= -9.5e+88)
		tmp = t_0;
	elseif (a <= 1.5e+77)
		tmp = (t_1 * t_1) + ((4.0 * ((a * (a * (1.0 - a))) + (b * (b * (a + 3.0))))) + -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9.5e+88], t$95$0, If[LessEqual[a, 1.5e+77], N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[(4.0 * N[(N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.50000000000000059e88 or 1.4999999999999999e77 < a

   1. Initial program 29.2%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 - a\right)} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(3 + a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 29.2%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto {a}^{3} \cdot \color{blue}{a} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{{a}^{3}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      9. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 100.0%

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

   if -9.50000000000000059e88 < a < 1.4999999999999999e77

   1. Initial program 98.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 - a\right)} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(3 + a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 98.0%

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

Alternative 3: 94.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left({a}^{2}\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \left(1 - a\right)\right)\right)\right), 1\right) \]

      14. --lowering--.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(1, a\right)\right)\right)\right), 1\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \color{blue}{\left(4 + a \cdot \left(a - 4\right)\right)}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \left(a \cdot \left(a - 4\right)\right)\right)\right), 1\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a - 4\right)\right)\right)\right), 1\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a + \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a + -4\right)\right)\right)\right), 1\right) \]

      5. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, -4\right)\right)\right)\right), 1\right) \]

   8. Simplified 99.9%

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

   if 1 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 94.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

   5. Simplified 94.0%

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

4. Final simplification 96.7%

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

Alternative 4: 93.7% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left({a}^{2}\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \left(1 - a\right)\right)\right)\right), 1\right) \]

      14. --lowering--.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(1, a\right)\right)\right)\right), 1\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \color{blue}{\left(4 + a \cdot \left(a - 4\right)\right)}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \left(a \cdot \left(a - 4\right)\right)\right)\right), 1\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a - 4\right)\right)\right)\right), 1\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a + \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a + -4\right)\right)\right)\right), 1\right) \]

      5. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, -4\right)\right)\right)\right), 1\right) \]

   8. Simplified 99.9%

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

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \color{blue}{\left({a}^{2} \cdot \left(1 - 4 \cdot \frac{1}{a}\right)\right)}\right), 1\right) \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} \cdot \left(1 + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right)\right)\right)\right), 1\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} \cdot 1 + {a}^{2} \cdot \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right)\right)\right), 1\right) \]

       3. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} + {a}^{2} \cdot \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right)\right)\right), 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + {a}^{2} \cdot \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right)\right)\right), 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right) \cdot {a}^{2}\right)\right), 1\right) \]

       6. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{a}\right) \cdot {a}^{2}\right)\right), 1\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + \left(-4 \cdot \frac{1}{a}\right) \cdot {a}^{2}\right)\right), 1\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + -4 \cdot \left(\frac{1}{a} \cdot {a}^{2}\right)\right)\right), 1\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + -4 \cdot \frac{1 \cdot {a}^{2}}{a}\right)\right), 1\right) \]

       10. *-lft-identity N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + -4 \cdot \frac{{a}^{2}}{a}\right)\right), 1\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + -4 \cdot \frac{a \cdot a}{a}\right)\right), 1\right) \]

       12. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + -4 \cdot \left(a \cdot \frac{a}{a}\right)\right)\right), 1\right) \]

       13. *-inverses N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + -4 \cdot \left(a \cdot 1\right)\right)\right), 1\right) \]

       14. *-rgt-identity N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a + -4 \cdot a\right)\right), 1\right) \]

       15. distribute-rgt-in N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a + -4\right)\right)\right), 1\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a + \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right), 1\right) \]

       17. sub-neg N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a - 4\right)\right)\right), 1\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a - 4\right)\right)\right), 1\right) \]

       19. sub-neg N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a + \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right), 1\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a + -4\right)\right)\right), 1\right) \]

       21. +-lowering-+.f64 98.5%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, -4\right)\right)\right), 1\right) \]

   11. Simplified 98.5%

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

   if 1 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 94.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

   5. Simplified 94.0%

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

4. Final simplification 96.0%

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

Alternative 5: 93.4% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

   3. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(3 + 1\right)}\right), 1\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{3} \cdot a\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 98.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), 1\right) \]

   5. Simplified 98.4%

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

   if 1 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 94.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

   5. Simplified 94.0%

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

4. Final simplification 96.0%

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

Alternative 6: 58.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-196}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 0.29:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-196)
   (* a (* a (* a a)))
   (if (<= b 0.29) -1.0 (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.1e-196) {
		tmp = a * (a * (a * a));
	} else if (b <= 0.29) {
		tmp = -1.0;
	} else {
		tmp = b * (b * (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 <= 2.1d-196) then
        tmp = a * (a * (a * a))
    else if (b <= 0.29d0) then
        tmp = -1.0d0
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.1e-196) {
		tmp = a * (a * (a * a));
	} else if (b <= 0.29) {
		tmp = -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.1e-196:
		tmp = a * (a * (a * a))
	elif b <= 0.29:
		tmp = -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.1e-196)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	elseif (b <= 0.29)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.1e-196)
		tmp = a * (a * (a * a));
	elseif (b <= 0.29)
		tmp = -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.1e-196], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.29], -1.0, N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.09999999999999988e-196

   1. Initial program 72.1%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 - a\right)} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(3 + a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 72.1%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto {a}^{3} \cdot \color{blue}{a} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{{a}^{3}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      9. *-lowering-*.f64 47.7%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 47.7%

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

   if 2.09999999999999988e-196 < b < 0.28999999999999998

   1. Initial program 86.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

   5. Simplified 58.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 58.2%

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

   if 0.28999999999999998 < b

   1. Initial program 64.2%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 - a\right)} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(3 + a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 64.2%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 94.0%

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

Alternative 7: 69.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -0.42:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.0185:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -0.42) t_0 (if (<= a 0.0185) -1.0 t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -0.42) {
		tmp = t_0;
	} else if (a <= 0.0185) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    if (a <= (-0.42d0)) then
        tmp = t_0
    else if (a <= 0.0185d0) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -0.42) {
		tmp = t_0;
	} else if (a <= 0.0185) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -0.42:
		tmp = t_0
	elif a <= 0.0185:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (a <= -0.42)
		tmp = t_0;
	elseif (a <= 0.0185)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	tmp = 0.0;
	if (a <= -0.42)
		tmp = t_0;
	elseif (a <= 0.0185)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.42], t$95$0, If[LessEqual[a, 0.0185], -1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.419999999999999984 or 0.0184999999999999991 < a

   1. Initial program 46.2%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 - a\right)} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(3 + a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 46.2%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto {a}^{3} \cdot \color{blue}{a} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{{a}^{3}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      9. *-lowering-*.f64 84.7%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 84.7%

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

   if -0.419999999999999984 < a < 0.0184999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 98.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

   5. Simplified 98.6%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 47.5%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1.0d0) then
        tmp = (a * (a * (a * a))) + (-1.0d0)
    else
        tmp = b * (b * ((b * b) + 12.0d0))
    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 * (a * a))) + -1.0;
	} else {
		tmp = b * (b * ((b * b) + 12.0));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

   3. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(3 + 1\right)}\right), 1\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{3} \cdot a\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 98.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), 1\right) \]

   5. Simplified 98.4%

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

   if 1 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 94.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

   5. Simplified 94.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot b\right) \cdot \left(12 + b \cdot b\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(12 + b \cdot b\right) \cdot \left(b \cdot b\right)\right), 1\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(12 + b \cdot b\right), \left(b \cdot b\right)\right), 1\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(12, \left(b \cdot b\right)\right), \left(b \cdot b\right)\right), 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right), \left(b \cdot b\right)\right), 1\right) \]

      6. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, b\right)\right), 1\right) \]

   7. Applied egg-rr 94.0%

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

   8. Taylor expanded in b around inf

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\left(3 + 1\right)} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right) \]

      2. pow-plus N/A

         \[\leadsto \left({b}^{3} \cdot b\right) \cdot \left(\color{blue}{1} + 12 \cdot \frac{1}{{b}^{2}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot {b}^{3}\right) \cdot \left(\color{blue}{1} + 12 \cdot \frac{1}{{b}^{2}}\right) \]

      4. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left({b}^{3} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{3} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\color{blue}{1} + 12 \cdot \frac{1}{{b}^{2}}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(\left(b \cdot {b}^{2}\right) \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} \cdot \left(12 \cdot \frac{1}{{b}^{2}} + \color{blue}{1}\right)\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(\left(12 \cdot \frac{1}{{b}^{2}}\right) \cdot {b}^{2} + \color{blue}{1 \cdot {b}^{2}}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 \cdot \left(\frac{1}{{b}^{2}} \cdot {b}^{2}\right) + \color{blue}{1} \cdot {b}^{2}\right)\right)\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 \cdot 1 + 1 \cdot {b}^{2}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(1 \cdot 12 + \color{blue}{1} \cdot {b}^{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + \color{blue}{1} \cdot {b}^{2}\right)\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{\color{blue}{2}}\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      19. *-lowering-*.f64 93.6%

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   10. Simplified 93.6%

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

4. Final simplification 95.7%

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

Alternative 9: 82.6% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1.0d0) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = b * (b * ((b * b) + 12.0d0))
    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 = b * (b * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1.0:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = b * (b * ((b * b) + 12.0))
	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(b * Float64(b * Float64(Float64(b * b) + 12.0)));
	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 = b * (b * ((b * b) + 12.0));
	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[(b * N[(b * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left({a}^{2}\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \left(1 - a\right)\right)\right)\right), 1\right) \]

      14. --lowering--.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(1, a\right)\right)\right)\right), 1\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \color{blue}{4}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 77.9%

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

   if 1 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 94.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

   5. Simplified 94.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot b\right) \cdot \left(12 + b \cdot b\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(12 + b \cdot b\right) \cdot \left(b \cdot b\right)\right), 1\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(12 + b \cdot b\right), \left(b \cdot b\right)\right), 1\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(12, \left(b \cdot b\right)\right), \left(b \cdot b\right)\right), 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right), \left(b \cdot b\right)\right), 1\right) \]

      6. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, b\right)\right), 1\right) \]

   7. Applied egg-rr 94.0%

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

   8. Taylor expanded in b around inf

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\left(3 + 1\right)} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right) \]

      2. pow-plus N/A

         \[\leadsto \left({b}^{3} \cdot b\right) \cdot \left(\color{blue}{1} + 12 \cdot \frac{1}{{b}^{2}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot {b}^{3}\right) \cdot \left(\color{blue}{1} + 12 \cdot \frac{1}{{b}^{2}}\right) \]

      4. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left({b}^{3} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{3} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\color{blue}{1} + 12 \cdot \frac{1}{{b}^{2}}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(\left(b \cdot {b}^{2}\right) \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} \cdot \left(12 \cdot \frac{1}{{b}^{2}} + \color{blue}{1}\right)\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(\left(12 \cdot \frac{1}{{b}^{2}}\right) \cdot {b}^{2} + \color{blue}{1 \cdot {b}^{2}}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 \cdot \left(\frac{1}{{b}^{2}} \cdot {b}^{2}\right) + \color{blue}{1} \cdot {b}^{2}\right)\right)\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 \cdot 1 + 1 \cdot {b}^{2}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(1 \cdot 12 + \color{blue}{1} \cdot {b}^{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + \color{blue}{1} \cdot {b}^{2}\right)\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{\color{blue}{2}}\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      19. *-lowering-*.f64 93.6%

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   10. Simplified 93.6%

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

4. Final simplification 86.6%

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

Alternative 10: 67.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14.8:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 14.8) (+ (* (* a a) 4.0) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 14.8) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = b * (b * (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 <= 14.8d0) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 14.8) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 14.8:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 14.8)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 14.8)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 14.8], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14.800000000000001

   1. Initial program 75.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left({a}^{2}\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \left(1 - a\right)\right)\right)\right), 1\right) \]

      14. --lowering--.f64 77.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(1, a\right)\right)\right)\right), 1\right) \]

   5. Simplified 77.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \color{blue}{4}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 57.7%

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

   if 14.800000000000001 < b

   1. Initial program 64.2%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 - a\right)} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(3 + a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 64.2%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 94.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14.8:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.29:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.29) -1.0 (* (* b b) 12.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.29) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.29d0) then
        tmp = -1.0d0
    else
        tmp = (b * b) * 12.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.29) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.29:
		tmp = -1.0
	else:
		tmp = (b * b) * 12.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.29)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * 12.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.29)
		tmp = -1.0;
	else
		tmp = (b * b) * 12.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.29], -1.0, N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.28999999999999998

   1. Initial program 75.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 67.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

   5. Simplified 67.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 31.4%

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

   if 0.28999999999999998 < b

   1. Initial program 64.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 95.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

   5. Simplified 95.8%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot b\right) \cdot \left(12 + b \cdot b\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(12 + b \cdot b\right) \cdot \left(b \cdot b\right)\right), 1\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(12 + b \cdot b\right), \left(b \cdot b\right)\right), 1\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(12, \left(b \cdot b\right)\right), \left(b \cdot b\right)\right), 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right), \left(b \cdot b\right)\right), 1\right) \]

      6. *-lowering-*.f64 95.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, b\right)\right), 1\right) \]

   7. Applied egg-rr 95.8%

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

   8. Taylor expanded in b around inf

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\left(3 + 1\right)} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right) \]

      2. pow-plus N/A

         \[\leadsto \left({b}^{3} \cdot b\right) \cdot \left(\color{blue}{1} + 12 \cdot \frac{1}{{b}^{2}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot {b}^{3}\right) \cdot \left(\color{blue}{1} + 12 \cdot \frac{1}{{b}^{2}}\right) \]

      4. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left({b}^{3} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{3} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\color{blue}{1} + 12 \cdot \frac{1}{{b}^{2}}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(\left(b \cdot {b}^{2}\right) \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2} \cdot \left(1 + 12 \cdot \frac{1}{{b}^{2}}\right)\right)}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} \cdot \left(12 \cdot \frac{1}{{b}^{2}} + \color{blue}{1}\right)\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(\left(12 \cdot \frac{1}{{b}^{2}}\right) \cdot {b}^{2} + \color{blue}{1 \cdot {b}^{2}}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 \cdot \left(\frac{1}{{b}^{2}} \cdot {b}^{2}\right) + \color{blue}{1} \cdot {b}^{2}\right)\right)\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 \cdot 1 + 1 \cdot {b}^{2}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(1 \cdot 12 + \color{blue}{1} \cdot {b}^{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + \color{blue}{1} \cdot {b}^{2}\right)\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{\color{blue}{2}}\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      19. *-lowering-*.f64 95.0%

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   10. Simplified 95.0%

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

   11. Taylor expanded in b around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(12, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(12, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 55.6%

          \[\leadsto \mathsf{*.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   13. Simplified 55.6%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.29:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 24.8% accurate, 128.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 72.2%

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

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

   3. pow-plus N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

   4. cube-unmult N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

   8. distribute-rgt-out N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(12 + {b}^{2}\right)\right)\right), 1\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left({b}^{2}\right)\right)\right)\right), 1\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot b\right)\right)\right)\right), 1\right) \]

   16. *-lowering-*.f64 74.7%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), 1\right) \]

5. Simplified 74.7%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 23.4%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024184 
(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))