Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 8.9s

Alternatives: 11

Speedup: N/A×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot a + b \cdot b\\ b \cdot \left(b \cdot 4\right) + \left(t\_0 \cdot t\_0 + -1\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* a a) (* b b)))) (+ (* b (* b 4.0)) (+ (* t_0 t_0) -1.0))))

C

double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return (b * (b * 4.0)) + ((t_0 * t_0) + -1.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    t_0 = (a * a) + (b * b)
    code = (b * (b * 4.0d0)) + ((t_0 * t_0) + (-1.0d0))
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return (b * (b * 4.0)) + ((t_0 * t_0) + -1.0);
}

Python

def code(a, b):
	t_0 = (a * a) + (b * b)
	return (b * (b * 4.0)) + ((t_0 * t_0) + -1.0)

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b))
	return Float64(Float64(b * Float64(b * 4.0)) + Float64(Float64(t_0 * t_0) + -1.0))
end

MATLAB

function tmp = code(a, b)
	t_0 = (a * a) + (b * b);
	tmp = (b * (b * 4.0)) + ((t_0 * t_0) + -1.0);
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

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

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   19. metadata-eval 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 97.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-68}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-68)
   (+ -1.0 (* a (* a (* a a))))
   (+ -1.0 (* (* b b) (+ 4.0 (+ (* b b) (* (* a a) 2.0)))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-68) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d-68) then
        tmp = (-1.0d0) + (a * (a * (a * a)))
    else
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + ((b * b) + ((a * a) * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-68) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e-68:
		tmp = -1.0 + (a * (a * (a * a)))
	else:
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-68)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * a))));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(Float64(b * b) + Float64(Float64(a * a) * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e-68)
		tmp = -1.0 + (a * (a * (a * a)));
	else
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-68], N[(-1.0 + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.00000000000000007e-68

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{4}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if 1.00000000000000007e-68 < (*.f64 b b)

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sum3-define N/A

         \[\leadsto \mathsf{sum3}\left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right), \color{blue}{\left({b}^{4}\right)}, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sum3}\left(\left(\left(2 \cdot {a}^{2}\right) \cdot {b}^{2} + 4 \cdot {b}^{2}\right), \left({b}^{4}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{sum3}\left(\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4\right)\right), \left({\color{blue}{b}}^{4}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sum3}\left(\left({b}^{2} \cdot \left(4 + 2 \cdot {a}^{2}\right)\right), \left({b}^{4}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. sum3-define N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. distribute-lft-in N/A

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

      11. associate-+r+ N/A

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

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

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

   7. Simplified 97.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-68}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.8:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+59}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 0.8)
   (+ (* b (* b 4.0)) -1.0)
   (if (<= a 2.4e+59) (* (* b b) (+ 4.0 (* b b))) (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.8) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else if (a <= 2.4e+59) {
		tmp = (b * b) * (4.0 + (b * b));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 0.8d0) then
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    else if (a <= 2.4d+59) then
        tmp = (b * b) * (4.0d0 + (b * b))
    else
        tmp = a * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 0.8) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else if (a <= 2.4e+59) {
		tmp = (b * b) * (4.0 + (b * b));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.8:
		tmp = (b * (b * 4.0)) + -1.0
	elif a <= 2.4e+59:
		tmp = (b * b) * (4.0 + (b * b))
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.8)
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	elseif (a <= 2.4e+59)
		tmp = Float64(Float64(b * b) * Float64(4.0 + Float64(b * b)));
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 0.8)
		tmp = (b * (b * 4.0)) + -1.0;
	elseif (a <= 2.4e+59)
		tmp = (b * b) * (4.0 + (b * b));
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.8], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.4e+59], N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 0.80000000000000004

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. *-commutative N/A

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

      14. distribute-lft-out N/A

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

      15. +-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. metadata-eval 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in b around 0

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

   10. Simplified 57.8%

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

   if 0.80000000000000004 < a < 2.4000000000000002e59

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. *-commutative N/A

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

      14. distribute-lft-out N/A

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

      15. +-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. metadata-eval 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. fma-define N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. lft-mult-inverse N/A

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

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{\left(2 \cdot 2\right)}\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{2} \cdot {b}^{2}\right) \]

      13. fma-define N/A

          \[\leadsto 4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      14. distribute-rgt-in N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(4 + {b}^{2}\right)}\right) \]

      16. unpow2 N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{4} + {b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      19. unpow2 N/A

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

      20. *-lowering-*.f64 70.6%

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

   10. Simplified 70.6%

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

   if 2.4000000000000002e59 < a

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 97.1%

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

   7. Simplified 97.1%

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

Alternative 4: 93.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{+119}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1e+119)
   (+ -1.0 (* b (* b (+ 4.0 (* b b)))))
   (* a (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1e+119) {
		tmp = -1.0 + (b * (b * (4.0 + (b * b))));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 1d+119) then
        tmp = (-1.0d0) + (b * (b * (4.0d0 + (b * b))))
    else
        tmp = a * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1e+119) {
		tmp = -1.0 + (b * (b * (4.0 + (b * b))));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 1e+119:
		tmp = -1.0 + (b * (b * (4.0 + (b * b))))
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 1e+119)
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(4.0 + Float64(b * b)))));
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 1e+119)
		tmp = -1.0 + (b * (b * (4.0 + (b * b))));
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1e+119], N[(-1.0 + N[(b * N[(b * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 9.99999999999999944e118

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. *-commutative N/A

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

      14. distribute-lft-out N/A

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

      15. +-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. metadata-eval 95.8%

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

   7. Simplified 95.8%

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

   if 9.99999999999999944e118 < (*.f64 a a)

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{+119}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.92:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+59}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 0.92)
   (+ (* b (* b 4.0)) -1.0)
   (if (<= a 2.4e+59) (* b (* b (* b b))) (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.92) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else if (a <= 2.4e+59) {
		tmp = b * (b * (b * b));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 0.92d0) then
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    else if (a <= 2.4d+59) then
        tmp = b * (b * (b * b))
    else
        tmp = a * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 0.92) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else if (a <= 2.4e+59) {
		tmp = b * (b * (b * b));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.92:
		tmp = (b * (b * 4.0)) + -1.0
	elif a <= 2.4e+59:
		tmp = b * (b * (b * b))
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.92)
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	elseif (a <= 2.4e+59)
		tmp = Float64(b * Float64(b * Float64(b * b)));
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 0.92)
		tmp = (b * (b * 4.0)) + -1.0;
	elseif (a <= 2.4e+59)
		tmp = b * (b * (b * b));
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.92], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.4e+59], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 0.92000000000000004

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. *-commutative N/A

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

      14. distribute-lft-out N/A

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

      15. +-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. metadata-eval 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in b around 0

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

   10. Simplified 57.8%

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

   if 0.92000000000000004 < a < 2.4000000000000002e59

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
   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 70.5%

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

   7. Simplified 70.5%

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

   if 2.4000000000000002e59 < a

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 97.1%

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

   7. Simplified 97.1%

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

Alternative 6: 47.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{-52}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+59}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.8e-52)
   -1.0
   (if (<= a 4.2e+59) (* b (* b (* b b))) (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.8e-52) {
		tmp = -1.0;
	} else if (a <= 4.2e+59) {
		tmp = b * (b * (b * b));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 1.8d-52) then
        tmp = -1.0d0
    else if (a <= 4.2d+59) then
        tmp = b * (b * (b * b))
    else
        tmp = a * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.8e-52) {
		tmp = -1.0;
	} else if (a <= 4.2e+59) {
		tmp = b * (b * (b * b));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.8e-52:
		tmp = -1.0
	elif a <= 4.2e+59:
		tmp = b * (b * (b * b))
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.8e-52)
		tmp = -1.0;
	elseif (a <= 4.2e+59)
		tmp = Float64(b * Float64(b * Float64(b * b)));
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.8e-52)
		tmp = -1.0;
	elseif (a <= 4.2e+59)
		tmp = b * (b * (b * b));
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.8e-52], -1.0, If[LessEqual[a, 4.2e+59], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.79999999999999994e-52

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{4}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. metadata-eval 68.7%

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

   7. Simplified 68.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 36.8%

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

   if 1.79999999999999994e-52 < a < 4.19999999999999968e59

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
   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 64.1%

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

   7. Simplified 64.1%

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

   if 4.19999999999999968e59 < a

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 97.1%

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

   7. Simplified 97.1%

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

Alternative 7: 46.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{-10}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+59}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 3.8e-10)
   -1.0
   (if (<= a 2.4e+59) (* 4.0 (* b b)) (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 3.8e-10) {
		tmp = -1.0;
	} else if (a <= 2.4e+59) {
		tmp = 4.0 * (b * b);
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 3.8d-10) then
        tmp = -1.0d0
    else if (a <= 2.4d+59) then
        tmp = 4.0d0 * (b * b)
    else
        tmp = a * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 3.8e-10) {
		tmp = -1.0;
	} else if (a <= 2.4e+59) {
		tmp = 4.0 * (b * b);
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 3.8e-10:
		tmp = -1.0
	elif a <= 2.4e+59:
		tmp = 4.0 * (b * b)
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 3.8e-10)
		tmp = -1.0;
	elseif (a <= 2.4e+59)
		tmp = Float64(4.0 * Float64(b * b));
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 3.8e-10)
		tmp = -1.0;
	elseif (a <= 2.4e+59)
		tmp = 4.0 * (b * b);
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 3.8e-10], -1.0, If[LessEqual[a, 2.4e+59], N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.7999999999999998e-10

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{4}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. metadata-eval 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 37.1%

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

   if 3.7999999999999998e-10 < a < 2.4000000000000002e59

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. *-commutative N/A

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

      14. distribute-lft-out N/A

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

      15. +-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. metadata-eval 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. fma-define N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma}\left(4, 1 \cdot {\color{blue}{b}}^{2}, {b}^{4}\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{\left(2 \cdot 2\right)}\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{2} \cdot {b}^{2}\right) \]

      13. fma-define N/A

          \[\leadsto 4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      14. distribute-rgt-in N/A

          \[\leadsto {b}^{2} \cdot \color{blue}{\left(4 + {b}^{2}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(4 + {b}^{2}\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{4} + {b}^{2}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{4} + {b}^{2}\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      20. *-lowering-*.f64 67.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   10. Simplified 67.5%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{4}\right) \]
   12. Step-by-step derivation

   13. Simplified 52.8%

       \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{4} \]

   if 2.4000000000000002e59 < a

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot b\right) \cdot b\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      19. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
   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

         \[\leadsto {a}^{\left(2 \cdot \color{blue}{2}\right)} \]

      2. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      4. associate-*l* N/A

         \[\leadsto a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot {a}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      8. *-lowering-*.f64 97.1%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 97.1%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{-10}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+59}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10000:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 10000.0)
   (+ -1.0 (* a (* a (* a a))))
   (* (* b b) (+ 4.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 10000.0) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 10000.0d0) then
        tmp = (-1.0d0) + (a * (a * (a * a)))
    else
        tmp = (b * b) * (4.0d0 + (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 10000.0) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 10000.0:
		tmp = -1.0 + (a * (a * (a * a)))
	else:
		tmp = (b * b) * (4.0 + (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 10000.0)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * a))));
	else
		tmp = Float64(Float64(b * b) * Float64(4.0 + Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 10000.0)
		tmp = -1.0 + (a * (a * (a * a)));
	else
		tmp = (b * b) * (4.0 + (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 10000.0], N[(-1.0 + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e4

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot b\right) \cdot b\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      19. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), -1\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {a}^{4} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{4}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{\left(2 \cdot 2\right)}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2} \cdot {a}^{2}\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot {a}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. metadata-eval 98.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), -1\right) \]

   7. Simplified 98.5%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1} \]

   if 1e4 < (*.f64 b b)

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot b\right) \cdot b\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      19. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), -1\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(4 \cdot {b}^{2} + {b}^{4}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot b\right) \cdot {b}^{2} + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(b \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 4 \cdot \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(4 \cdot b\right) \cdot b\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 4 \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 4 \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 4\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 4\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left(4 + {b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(4 + {b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(4, \left({b}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(4, \left(b \cdot b\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      20. metadata-eval 90.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), -1\right) \]

   7. Simplified 90.2%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + -1} \]

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4} \cdot \left(1 + 4 \cdot \frac{1}{{b}^{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + 4 \cdot \frac{1}{{b}^{2}}\right) \cdot \color{blue}{{b}^{4}} \]

      2. +-commutative N/A

         \[\leadsto \left(4 \cdot \frac{1}{{b}^{2}} + 1\right) \cdot {\color{blue}{b}}^{4} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(4 \cdot \frac{1}{{b}^{2}}\right) \cdot {b}^{4} + \color{blue}{{b}^{4}} \]

      4. associate-*l* N/A

         \[\leadsto 4 \cdot \left(\frac{1}{{b}^{2}} \cdot {b}^{4}\right) + {\color{blue}{b}}^{4} \]

      5. fma-define N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{1}{{b}^{2}} \cdot {b}^{4}}, {b}^{4}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(4, \frac{1}{{b}^{2}} \cdot {b}^{\left(2 \cdot \color{blue}{2}\right)}, {b}^{4}\right) \]

      7. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(4, \frac{1}{{b}^{2}} \cdot \left({b}^{2} \cdot \color{blue}{{b}^{2}}\right), {b}^{4}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(4, \left(\frac{1}{{b}^{2}} \cdot {b}^{2}\right) \cdot \color{blue}{{b}^{2}}, {b}^{4}\right) \]

      9. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma}\left(4, 1 \cdot {\color{blue}{b}}^{2}, {b}^{4}\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{\left(2 \cdot 2\right)}\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{2} \cdot {b}^{2}\right) \]

      13. fma-define N/A

          \[\leadsto 4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      14. distribute-rgt-in N/A

          \[\leadsto {b}^{2} \cdot \color{blue}{\left(4 + {b}^{2}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(4 + {b}^{2}\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{4} + {b}^{2}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{4} + {b}^{2}\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      20. *-lowering-*.f64 89.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   10. Simplified 89.9%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10000:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{+59}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 7.8e+59) (+ -1.0 (* (* b b) (* b b))) (* a (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 7.8e+59) {
		tmp = -1.0 + ((b * b) * (b * b));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 7.8d+59) then
        tmp = (-1.0d0) + ((b * b) * (b * b))
    else
        tmp = a * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 7.8e+59) {
		tmp = -1.0 + ((b * b) * (b * b));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 7.8e+59:
		tmp = -1.0 + ((b * b) * (b * b))
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 7.8e+59)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(b * b)));
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 7.8e+59)
		tmp = -1.0 + ((b * b) * (b * b));
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 7.8e+59], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.80000000000000043e59

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot b\right) \cdot b\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      19. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), -1\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right) - 1} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. sum3-define N/A

         \[\leadsto \mathsf{sum3}\left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right), \color{blue}{\left({b}^{4}\right)}, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sum3}\left(\left(\left(2 \cdot {a}^{2}\right) \cdot {b}^{2} + 4 \cdot {b}^{2}\right), \left({b}^{4}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{sum3}\left(\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4\right)\right), \left({\color{blue}{b}}^{4}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sum3}\left(\left({b}^{2} \cdot \left(4 + 2 \cdot {a}^{2}\right)\right), \left({b}^{4}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. sum3-define N/A

         \[\leadsto \left({b}^{2} \cdot \left(4 + 2 \cdot {a}^{2}\right) + {b}^{4}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left({b}^{2} \cdot \left(4 + 2 \cdot {a}^{2}\right) + {b}^{\left(2 \cdot 2\right)}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      9. pow-sqr N/A

         \[\leadsto \left({b}^{2} \cdot \left(4 + 2 \cdot {a}^{2}\right) + {b}^{2} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      10. distribute-lft-in N/A

          \[\leadsto {b}^{2} \cdot \left(\left(4 + 2 \cdot {a}^{2}\right) + {b}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      11. associate-+r+ N/A

          \[\leadsto {b}^{2} \cdot \left(4 + \left(2 \cdot {a}^{2} + {b}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left(4 + \left(2 \cdot {a}^{2} + {b}^{2}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

   7. Simplified 83.5%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + 2 \cdot \left(a \cdot a\right)\right)\right) + -1} \]

   8. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{\left({b}^{2}\right)}\right), -1\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(b \cdot b\right)\right), -1\right) \]

      2. *-lowering-*.f64 75.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right), -1\right) \]

   10. Simplified 75.9%

       \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)} + -1 \]

   if 7.80000000000000043e59 < a

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot b\right) \cdot b\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      19. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
   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

         \[\leadsto {a}^{\left(2 \cdot \color{blue}{2}\right)} \]

      2. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      4. associate-*l* N/A

         \[\leadsto a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot {a}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      8. *-lowering-*.f64 97.1%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 97.1%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{+59}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.235:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 0.235) -1.0 (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.235) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.235d0) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.235) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.235:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.235)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.235)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.235], -1.0, N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.23499999999999999

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot b\right) \cdot b\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      19. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), -1\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {a}^{4} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{4}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{\left(2 \cdot 2\right)}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2} \cdot {a}^{2}\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot {a}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. metadata-eval 99.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), -1\right) \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1} \]
   9. Step-by-step derivation

   10. Simplified 48.2%

       \[\leadsto \color{blue}{-1} \]

   if 0.23499999999999999 < (*.f64 b b)

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot b\right) \cdot b\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      19. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), -1\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(4 \cdot {b}^{2} + {b}^{4}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot b\right) \cdot {b}^{2} + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(b \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 4 \cdot {b}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 4 \cdot \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(4 \cdot b\right) \cdot b\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 4 \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 4 \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 4\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 4\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left(4 + {b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(4 + {b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(4, \left({b}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(4, \left(b \cdot b\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      20. metadata-eval 89.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), -1\right) \]

   7. Simplified 89.6%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + -1} \]

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4} \cdot \left(1 + 4 \cdot \frac{1}{{b}^{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + 4 \cdot \frac{1}{{b}^{2}}\right) \cdot \color{blue}{{b}^{4}} \]

      2. +-commutative N/A

         \[\leadsto \left(4 \cdot \frac{1}{{b}^{2}} + 1\right) \cdot {\color{blue}{b}}^{4} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(4 \cdot \frac{1}{{b}^{2}}\right) \cdot {b}^{4} + \color{blue}{{b}^{4}} \]

      4. associate-*l* N/A

         \[\leadsto 4 \cdot \left(\frac{1}{{b}^{2}} \cdot {b}^{4}\right) + {\color{blue}{b}}^{4} \]

      5. fma-define N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{1}{{b}^{2}} \cdot {b}^{4}}, {b}^{4}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(4, \frac{1}{{b}^{2}} \cdot {b}^{\left(2 \cdot \color{blue}{2}\right)}, {b}^{4}\right) \]

      7. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(4, \frac{1}{{b}^{2}} \cdot \left({b}^{2} \cdot \color{blue}{{b}^{2}}\right), {b}^{4}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(4, \left(\frac{1}{{b}^{2}} \cdot {b}^{2}\right) \cdot \color{blue}{{b}^{2}}, {b}^{4}\right) \]

      9. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma}\left(4, 1 \cdot {\color{blue}{b}}^{2}, {b}^{4}\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{\left(2 \cdot 2\right)}\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{2} \cdot {b}^{2}\right) \]

      13. fma-define N/A

          \[\leadsto 4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      14. distribute-rgt-in N/A

          \[\leadsto {b}^{2} \cdot \color{blue}{\left(4 + {b}^{2}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(4 + {b}^{2}\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{4} + {b}^{2}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{4} + {b}^{2}\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      20. *-lowering-*.f64 88.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   10. Simplified 88.8%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{4}\right) \]
   12. Step-by-step derivation

   13. Simplified 52.7%

       \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.235:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 25.8% accurate, 116.0× 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 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot b\right) \cdot b\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(4 \cdot b\right)\right), \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   16. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   19. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, 4\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), -1\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in b around 0

   \[\leadsto \color{blue}{{a}^{4} - 1} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto {a}^{4} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left({a}^{4}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left({a}^{\left(2 \cdot 2\right)}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{+.f64}\left(\left({a}^{2} \cdot {a}^{2}\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot {a}^{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   11. metadata-eval 73.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), -1\right) \]

7. Simplified 73.6%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1} \]
9. Step-by-step derivation

10. Simplified 26.3%

    \[\leadsto \color{blue}{-1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))