Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 12

Speedup: N/A×

• 61.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 97.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-37}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + \left(-1 + \frac{a \cdot a + b \cdot b}{\frac{\frac{1}{b}}{b}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-37)
   (+ -1.0 (* a (* a (* a a))))
   (+ (* b (* b 4.0)) (+ -1.0 (/ (+ (* a a) (* b b)) (/ (/ 1.0 b) b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-37) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = (b * (b * 4.0)) + (-1.0 + (((a * a) + (b * b)) / ((1.0 / 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) <= 5d-37) then
        tmp = (-1.0d0) + (a * (a * (a * a)))
    else
        tmp = (b * (b * 4.0d0)) + ((-1.0d0) + (((a * a) + (b * b)) / ((1.0d0 / b) / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-37) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = (b * (b * 4.0)) + (-1.0 + (((a * a) + (b * b)) / ((1.0 / b) / b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-37:
		tmp = -1.0 + (a * (a * (a * a)))
	else:
		tmp = (b * (b * 4.0)) + (-1.0 + (((a * a) + (b * b)) / ((1.0 / b) / b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-37)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * a))));
	else
		tmp = Float64(Float64(b * Float64(b * 4.0)) + Float64(-1.0 + Float64(Float64(Float64(a * a) + Float64(b * b)) / Float64(Float64(1.0 / b) / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-37)
		tmp = -1.0 + (a * (a * (a * a)));
	else
		tmp = (b * (b * 4.0)) + (-1.0 + (((a * a) + (b * b)) / ((1.0 / b) / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-37], N[(-1.0 + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999997e-37

   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 4.9999999999999997e-37 < (*.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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      8. clear-num N/A

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

      9. flip3-+ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in a around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 97.2%

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

   9. Simplified 97.2%

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

       1. associate-/r* N/A

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

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

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

       3. /-lowering-/.f64 97.2%

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

   11. Applied egg-rr 97.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-37}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + \left(-1 + \frac{a \cdot a + b \cdot b}{\frac{\frac{1}{b}}{b}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-37}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + \left(-1 + \frac{a \cdot a + b \cdot b}{\frac{1}{b \cdot b}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-37)
   (+ -1.0 (* a (* a (* a a))))
   (+ (* b (* b 4.0)) (+ -1.0 (/ (+ (* a a) (* b b)) (/ 1.0 (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-37) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = (b * (b * 4.0)) + (-1.0 + (((a * a) + (b * b)) / (1.0 / (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) <= 5d-37) then
        tmp = (-1.0d0) + (a * (a * (a * a)))
    else
        tmp = (b * (b * 4.0d0)) + ((-1.0d0) + (((a * a) + (b * b)) / (1.0d0 / (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-37) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = (b * (b * 4.0)) + (-1.0 + (((a * a) + (b * b)) / (1.0 / (b * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-37:
		tmp = -1.0 + (a * (a * (a * a)))
	else:
		tmp = (b * (b * 4.0)) + (-1.0 + (((a * a) + (b * b)) / (1.0 / (b * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-37)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * a))));
	else
		tmp = Float64(Float64(b * Float64(b * 4.0)) + Float64(-1.0 + Float64(Float64(Float64(a * a) + Float64(b * b)) / Float64(1.0 / Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-37)
		tmp = -1.0 + (a * (a * (a * a)));
	else
		tmp = (b * (b * 4.0)) + (-1.0 + (((a * a) + (b * b)) / (1.0 / (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-37], N[(-1.0 + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999997e-37

   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 4.9999999999999997e-37 < (*.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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      8. clear-num N/A

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

      9. flip3-+ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in a around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 97.2%

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

   9. Simplified 97.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-37}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + \left(-1 + \frac{a \cdot a + b \cdot b}{\frac{1}{b \cdot b}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+46}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right) + \left(-1 + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 5e+46)
   (+ (* b (* b (* b b))) (+ -1.0 (* 4.0 (* b b))))
   (/ 1.0 (/ 1.0 (* a (* a (* a a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 5e+46) {
		tmp = (b * (b * (b * b))) + (-1.0 + (4.0 * (b * b)));
	} else {
		tmp = 1.0 / (1.0 / (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 <= 5d+46) then
        tmp = (b * (b * (b * b))) + ((-1.0d0) + (4.0d0 * (b * b)))
    else
        tmp = 1.0d0 / (1.0d0 / (a * (a * (a * a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 5e+46) {
		tmp = (b * (b * (b * b))) + (-1.0 + (4.0 * (b * b)));
	} else {
		tmp = 1.0 / (1.0 / (a * (a * (a * a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 5e+46:
		tmp = (b * (b * (b * b))) + (-1.0 + (4.0 * (b * b)))
	else:
		tmp = 1.0 / (1.0 / (a * (a * (a * a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 5e+46)
		tmp = Float64(Float64(b * Float64(b * Float64(b * b))) + Float64(-1.0 + Float64(4.0 * Float64(b * b))));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(a * Float64(a * Float64(a * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 5e+46)
		tmp = (b * (b * (b * b))) + (-1.0 + (4.0 * (b * b)));
	else
		tmp = 1.0 / (1.0 / (a * (a * (a * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 5e+46], N[(N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000002e46

   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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. cube-mult N/A

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

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

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

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

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 81.6%

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

   9. Simplified 81.6%

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

   if 5.0000000000000002e46 < 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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 96.0%

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

   9. Simplified 96.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+46}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right) + \left(-1 + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{+45}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.45e-15)
   (+ (* b (* b 4.0)) -1.0)
   (if (<= a 4.9e+45) (* b (* b (* b b))) (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.45e-15) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else if (a <= 4.9e+45) {
		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 <= 1.45d-15) then
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    else if (a <= 4.9d+45) 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 <= 1.45e-15) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else if (a <= 4.9e+45) {
		tmp = b * (b * (b * b));
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.45e-15:
		tmp = (b * (b * 4.0)) + -1.0
	elif a <= 4.9e+45:
		tmp = b * (b * (b * b))
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.45e-15)
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	elseif (a <= 4.9e+45)
		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 <= 1.45e-15)
		tmp = (b * (b * 4.0)) + -1.0;
	elseif (a <= 4.9e+45)
		tmp = b * (b * (b * b));
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.45e-15], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 4.9e+45], 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 < 1.45000000000000009e-15

   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 inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. associate-*r/ N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in b around 0

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

   10. Simplified 62.0%

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

   if 1.45000000000000009e-15 < a < 4.9000000000000002e45

   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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   8. 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. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 73.5%

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

   9. Simplified 73.5%

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

   if 4.9000000000000002e45 < 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 96.0%

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

   7. Simplified 96.0%

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

Alternative 6: 57.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-226}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.2e-226)
   (* a (* a (* a a)))
   (if (<= b 0.48) -1.0 (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.2e-226) {
		tmp = a * (a * (a * a));
	} else if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 9.2d-226) then
        tmp = a * (a * (a * a))
    else if (b <= 0.48d0) then
        tmp = -1.0d0
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.2e-226) {
		tmp = a * (a * (a * a));
	} else if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.2e-226:
		tmp = a * (a * (a * a))
	elif b <= 0.48:
		tmp = -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.2e-226)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	elseif (b <= 0.48)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.2e-226)
		tmp = a * (a * (a * a));
	elseif (b <= 0.48)
		tmp = -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.2e-226], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.48], -1.0, N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.2000000000000001e-226

   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 45.5%

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

   7. Simplified 45.5%

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

   if 9.2000000000000001e-226 < b < 0.47999999999999998

   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 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 60.3%

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

   if 0.47999999999999998 < b

   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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   8. 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. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 86.3%

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

   9. Simplified 86.3%

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

Alternative 7: 81.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+47}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 4.5e+47)
   (+ -1.0 (* (* b b) (+ 4.0 (* b b))))
   (/ 1.0 (/ 1.0 (* a (* a (* a a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 4.5e+47) {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	} else {
		tmp = 1.0 / (1.0 / (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 <= 4.5d+47) then
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + (b * b)))
    else
        tmp = 1.0d0 / (1.0d0 / (a * (a * (a * a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 4.5e+47) {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	} else {
		tmp = 1.0 / (1.0 / (a * (a * (a * a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 4.5e+47:
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)))
	else:
		tmp = 1.0 / (1.0 / (a * (a * (a * a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 4.5e+47)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(a * Float64(a * Float64(a * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 4.5e+47)
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	else
		tmp = 1.0 / (1.0 / (a * (a * (a * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 4.5e+47], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.49999999999999979e47

   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. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. metadata-eval 81.5%

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

   7. Simplified 81.5%

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

   if 4.49999999999999979e47 < 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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 96.0%

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

   9. Simplified 96.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+47}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+45}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 5e+45)
   (+ -1.0 (* (* b b) (+ 4.0 (* b b))))
   (+ -1.0 (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 5e+45) {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	} else {
		tmp = -1.0 + (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 <= 5d+45) then
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + (b * b)))
    else
        tmp = (-1.0d0) + (a * (a * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 5e+45) {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	} else {
		tmp = -1.0 + (a * (a * (a * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 5e+45:
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)))
	else:
		tmp = -1.0 + (a * (a * (a * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 5e+45)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))));
	else
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 5e+45)
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	else
		tmp = -1.0 + (a * (a * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 5e+45], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5e45

   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. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. metadata-eval 81.5%

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

   7. Simplified 81.5%

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

   if 5e45 < 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 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 96.0%

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

   7. Simplified 96.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+45}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.3% accurate, 7.2× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-5) (+ -1.0 (* a (* a (* a a)))) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-5) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d-5) then
        tmp = (-1.0d0) + (a * (a * (a * a)))
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-5) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-5:
		tmp = -1.0 + (a * (a * (a * a)))
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-5)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * a))));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-5)
		tmp = -1.0 + (a * (a * (a * a)));
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-5], N[(-1.0 + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000016e-5

   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 98.8%

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

   7. Simplified 98.8%

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

   if 2.00000000000000016e-5 < (*.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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   8. 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. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 90.2%

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

   9. Simplified 90.2%

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

4. Final simplification 94.5%

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

Alternative 10: 45.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.45e-15) -1.0 (* a (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.45e-15) {
		tmp = -1.0;
	} 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 <= 1.45d-15) then
        tmp = -1.0d0
    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 <= 1.45e-15) {
		tmp = -1.0;
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.45e-15:
		tmp = -1.0
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.45e-15)
		tmp = -1.0;
	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 <= 1.45e-15)
		tmp = -1.0;
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.45e-15], -1.0, N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.45000000000000009e-15

   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 64.3%

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

   7. Simplified 64.3%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 35.1%

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

   if 1.45000000000000009e-15 < 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 83.5%

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

   7. Simplified 83.5%

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

Alternative 11: 37.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.48) -1.0 (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		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 <= 0.48d0) 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 <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.48:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.48)
		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 <= 0.48)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.48], -1.0, N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.47999999999999998

   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 79.0%

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

   7. Simplified 79.0%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 36.5%

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

   if 0.47999999999999998 < b

   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

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. associate-*r/ N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in b around 0

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

   10. Simplified 50.2%

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

   11. Taylor expanded in b around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 50.2%

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

   13. Simplified 50.2%

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

Alternative 12: 24.3% 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 68.6%

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

7. Simplified 68.6%

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

8. Taylor expanded in a around 0

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

10. Simplified 27.3%

    \[\leadsto \color{blue}{-1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024147 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))