Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 8.2s

Alternatives: 12

Speedup: N/A×

• 60.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(N[(a * a), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $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. associate--l+ N/A

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. sub-neg N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

   16. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. pow2 N/A

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

   2. pow-lowering-pow.f64 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 99.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;a \cdot a \leq 0.2:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot \left(b \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b (* b b)))))
   (if (<= (* a a) 0.2)
     (+ (+ (* (* b b) 4.0) -1.0) t_0)
     (+ t_0 (* (* a a) (+ (* a a) (* b (* b 2.0))))))))

C

double code(double a, double b) {
	double t_0 = b * (b * (b * b));
	double tmp;
	if ((a * a) <= 0.2) {
		tmp = (((b * b) * 4.0) + -1.0) + t_0;
	} else {
		tmp = t_0 + ((a * a) * ((a * a) + (b * (b * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * (b * b))
    if ((a * a) <= 0.2d0) then
        tmp = (((b * b) * 4.0d0) + (-1.0d0)) + t_0
    else
        tmp = t_0 + ((a * a) * ((a * a) + (b * (b * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * (b * b));
	double tmp;
	if ((a * a) <= 0.2) {
		tmp = (((b * b) * 4.0) + -1.0) + t_0;
	} else {
		tmp = t_0 + ((a * a) * ((a * a) + (b * (b * 2.0))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * (b * b))
	tmp = 0
	if (a * a) <= 0.2:
		tmp = (((b * b) * 4.0) + -1.0) + t_0
	else:
		tmp = t_0 + ((a * a) * ((a * a) + (b * (b * 2.0))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * Float64(b * b)))
	tmp = 0.0
	if (Float64(a * a) <= 0.2)
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + -1.0) + t_0);
	else
		tmp = Float64(t_0 + Float64(Float64(a * a) * Float64(Float64(a * a) + Float64(b * Float64(b * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * (b * b));
	tmp = 0.0;
	if ((a * a) <= 0.2)
		tmp = (((b * b) * 4.0) + -1.0) + t_0;
	else
		tmp = t_0 + ((a * a) * ((a * a) + (b * (b * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * a), $MachinePrecision], 0.2], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(b * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 0.20000000000000001

   1. Initial program 100.0%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. pow-plus N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

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

   if 0.20000000000000001 < (*.f64 a a)

   1. Initial program 99.9%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. pow2 N/A

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

      2. pow-lowering-pow.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

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

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

   9. Simplified 99.9%

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

   10. Taylor expanded in b around inf

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

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. pow-plus N/A

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

       4. pow-plus N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 0.2:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right) + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot \left(b \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(b \cdot b + \left(4 + \left(a \cdot a\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+29)
   (+ (+ (* (* b b) 4.0) -1.0) (* (* a a) (+ (* a a) (* b b))))
   (+ -1.0 (* b (* b (+ (* b b) (+ 4.0 (* (* a a) 2.0))))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+29) {
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * ((a * a) + (b * b)));
	} else {
		tmp = -1.0 + (b * (b * ((b * b) + (4.0 + ((a * a) * 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+29) then
        tmp = (((b * b) * 4.0d0) + (-1.0d0)) + ((a * a) * ((a * a) + (b * b)))
    else
        tmp = (-1.0d0) + (b * (b * ((b * b) + (4.0d0 + ((a * a) * 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+29) {
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * ((a * a) + (b * b)));
	} else {
		tmp = -1.0 + (b * (b * ((b * b) + (4.0 + ((a * a) * 2.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+29:
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * ((a * a) + (b * b)))
	else:
		tmp = -1.0 + (b * (b * ((b * b) + (4.0 + ((a * a) * 2.0)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+29)
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + -1.0) + Float64(Float64(a * a) * Float64(Float64(a * a) + Float64(b * b))));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(Float64(b * b) + Float64(4.0 + Float64(Float64(a * a) * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+29)
		tmp = (((b * b) * 4.0) + -1.0) + ((a * a) * ((a * a) + (b * b)));
	else
		tmp = -1.0 + (b * (b * ((b * b) + (4.0 + ((a * a) * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+29], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(N[(b * b), $MachinePrecision] + N[(4.0 + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e29

   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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.1%

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

   7. Simplified 99.1%

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

   if 5.0000000000000001e29 < (*.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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sum3-define N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{sum3}\left(\left({b}^{2} \cdot \left(4 + 2 \cdot {a}^{2}\right)\right), \left({b}^{\left(3 + \color{blue}{1}\right)}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. pow-plus N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. sum3-define N/A

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

   7. Simplified 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(b \cdot b + \left(4 + \left(a \cdot a\right) \cdot 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-9}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(b \cdot b + \left(4 + \left(a \cdot a\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e-9)
   (+ -1.0 (* a (* a (* a a))))
   (+ -1.0 (* b (* b (+ (* b b) (+ 4.0 (* (* a a) 2.0))))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e-9) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = -1.0 + (b * (b * ((b * b) + (4.0 + ((a * a) * 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d-9) then
        tmp = (-1.0d0) + (a * (a * (a * a)))
    else
        tmp = (-1.0d0) + (b * (b * ((b * b) + (4.0d0 + ((a * a) * 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e-9) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = -1.0 + (b * (b * ((b * b) + (4.0 + ((a * a) * 2.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e-9:
		tmp = -1.0 + (a * (a * (a * a)))
	else:
		tmp = -1.0 + (b * (b * ((b * b) + (4.0 + ((a * a) * 2.0)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e-9)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * a))));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(Float64(b * b) + Float64(4.0 + Float64(Float64(a * a) * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e-9)
		tmp = -1.0 + (a * (a * (a * a)));
	else
		tmp = -1.0 + (b * (b * ((b * b) + (4.0 + ((a * a) * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e-9], N[(-1.0 + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(N[(b * b), $MachinePrecision] + N[(4.0 + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000025e-9

   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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   if 4.00000000000000025e-9 < (*.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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sum3-define N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{sum3}\left(\left({b}^{2} \cdot \left(4 + 2 \cdot {a}^{2}\right)\right), \left({b}^{\left(3 + \color{blue}{1}\right)}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. pow-plus N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. sum3-define N/A

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

   7. Simplified 97.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-9}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(b \cdot b + \left(4 + \left(a \cdot a\right) \cdot 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return (((b * b) * 4.0) + -1.0) + (t_0 * t_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) + (-1.0d0)) + (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. sub-neg N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

   16. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + -1\right) + \left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) \]
6. Add Preprocessing

Alternative 6: 57.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;b \leq 1.3 \cdot 10^{-168}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-33}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= b 1.3e-168)
     t_0
     (if (<= b 1.7e-33) -1.0 (if (<= b 6e+15) t_0 (* b (* b (* b b))))))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (b <= 1.3e-168) {
		tmp = t_0;
	} else if (b <= 1.7e-33) {
		tmp = -1.0;
	} else if (b <= 6e+15) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    if (b <= 1.3d-168) then
        tmp = t_0
    else if (b <= 1.7d-33) then
        tmp = -1.0d0
    else if (b <= 6d+15) then
        tmp = t_0
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (b <= 1.3e-168) {
		tmp = t_0;
	} else if (b <= 1.7e-33) {
		tmp = -1.0;
	} else if (b <= 6e+15) {
		tmp = t_0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if b <= 1.3e-168:
		tmp = t_0
	elif b <= 1.7e-33:
		tmp = -1.0
	elif b <= 6e+15:
		tmp = t_0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (b <= 1.3e-168)
		tmp = t_0;
	elseif (b <= 1.7e-33)
		tmp = -1.0;
	elseif (b <= 6e+15)
		tmp = t_0;
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	tmp = 0.0;
	if (b <= 1.3e-168)
		tmp = t_0;
	elseif (b <= 1.7e-33)
		tmp = -1.0;
	elseif (b <= 6e+15)
		tmp = t_0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.3e-168], t$95$0, If[LessEqual[b, 1.7e-33], -1.0, If[LessEqual[b, 6e+15], t$95$0, N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.3e-168 or 1.7e-33 < b < 6e15

   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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(\left(2 + 1\right) + 1\right)} \]

      3. pow-plus N/A

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

      4. pow-plus N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 48.8%

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

   7. Simplified 48.8%

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

   if 1.3e-168 < b < 1.7e-33

   1. Initial program 100.0%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. 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%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 65.6%

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

   if 6e15 < 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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. pow-plus N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 96.0%

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

   7. Simplified 96.0%

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

Alternative 7: 97.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+29}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+29)
   (+ -1.0 (* a (* a (* a a))))
   (* (* b b) (+ (* b b) (* (* a a) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+29) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+29) then
        tmp = (-1.0d0) + (a * (a * (a * a)))
    else
        tmp = (b * b) * ((b * b) + ((a * a) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+29) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+29:
		tmp = -1.0 + (a * (a * (a * a)))
	else:
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+29)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * a))));
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + Float64(Float64(a * a) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+29)
		tmp = -1.0 + (a * (a * (a * a)));
	else
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+29], N[(-1.0 + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e29

   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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   if 5.0000000000000001e29 < (*.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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. pow2 N/A

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

      2. pow-lowering-pow.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

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

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

   9. Simplified 92.2%

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

   10. Taylor expanded in b around inf

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

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. pow-plus N/A

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

       4. pow-plus N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 92.2%

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

   12. Simplified 92.2%

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

   13. Taylor expanded in b around inf

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l* N/A

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

       4. *-lft-identity N/A

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

       5. fma-define N/A

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

       6. associate-*l/ N/A

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

       7. associate-/l* N/A

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

       8. metadata-eval N/A

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

       9. pow-plus N/A

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

       10. unpow3 N/A

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

       11. unpow2 N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. associate-/l* N/A

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

       15. *-inverses N/A

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

       16. *-rgt-identity N/A

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

       17. fma-define N/A

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

   15. Simplified 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+29}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.55:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 42000000000:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 0.55)
   -1.0
   (if (<= a 42000000000.0) (* b (* b 4.0)) (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.55) {
		tmp = -1.0;
	} else if (a <= 42000000000.0) {
		tmp = b * (b * 4.0);
	} 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 <= 0.55d0) then
        tmp = -1.0d0
    else if (a <= 42000000000.0d0) then
        tmp = b * (b * 4.0d0)
    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 <= 0.55) {
		tmp = -1.0;
	} else if (a <= 42000000000.0) {
		tmp = b * (b * 4.0);
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.55:
		tmp = -1.0
	elif a <= 42000000000.0:
		tmp = b * (b * 4.0)
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.55)
		tmp = -1.0;
	elseif (a <= 42000000000.0)
		tmp = Float64(b * Float64(b * 4.0));
	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 <= 0.55)
		tmp = -1.0;
	elseif (a <= 42000000000.0)
		tmp = b * (b * 4.0);
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.55], -1.0, If[LessEqual[a, 42000000000.0], N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 0.55000000000000004

   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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. metadata-eval 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 31.6%

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

   if 0.55000000000000004 < a < 4.2e10

   1. Initial program 100.0%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. pow-plus N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

          \[\leadsto \left(b \cdot b\right) \cdot 4 \]

       3. associate-*l* N/A

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

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

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

       5. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

   if 4.2e10 < 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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(\left(2 + 1\right) + 1\right)} \]

      3. pow-plus N/A

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

      4. pow-plus N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 87.2%

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

   7. Simplified 87.2%

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

Alternative 9: 94.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+29}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+29)
   (+ -1.0 (* a (* a (* a a))))
   (* b (* b (+ (* b b) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+29) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = b * (b * ((b * b) + 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+29) then
        tmp = (-1.0d0) + (a * (a * (a * a)))
    else
        tmp = b * (b * ((b * b) + 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+29) {
		tmp = -1.0 + (a * (a * (a * a)));
	} else {
		tmp = b * (b * ((b * b) + 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+29:
		tmp = -1.0 + (a * (a * (a * a)))
	else:
		tmp = b * (b * ((b * b) + 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+29)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * a))));
	else
		tmp = Float64(b * Float64(b * Float64(Float64(b * b) + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+29)
		tmp = -1.0 + (a * (a * (a * a)));
	else
		tmp = b * (b * ((b * b) + 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+29], N[(-1.0 + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e29

   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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   if 5.0000000000000001e29 < (*.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. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(\left(2 + 1\right) + 1\right)}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 + 1\right)} \cdot b\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      4. pow-plus N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({b}^{2} \cdot b\right) \cdot b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, 4\right), -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, 4\right), -1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(b \cdot {b}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{4}\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

      10. *-lowering-*.f64 95.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

   7. Simplified 95.8%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(\left(b \cdot b\right) \cdot 4 + -1\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      2. cube-unmult N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{3} \cdot b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, 4\right), -1\right)\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(3 + 1\right)}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{4}\right), -1\right)\right) \]

      5. pow-lowering-pow.f64 95.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(b, 4\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

   9. Applied egg-rr 95.8%

      \[\leadsto \color{blue}{{b}^{4}} + \left(\left(b \cdot b\right) \cdot 4 + -1\right) \]

   10. Taylor expanded in b around inf

       \[\leadsto \color{blue}{{b}^{4} \cdot \left(1 + 4 \cdot \frac{1}{{b}^{2}}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(1 + 4 \cdot \frac{1}{{b}^{2}}\right) \cdot \color{blue}{{b}^{4}} \]

       2. +-commutative N/A

          \[\leadsto \left(4 \cdot \frac{1}{{b}^{2}} + 1\right) \cdot {\color{blue}{b}}^{4} \]

       3. distribute-lft1-in N/A

          \[\leadsto \left(4 \cdot \frac{1}{{b}^{2}}\right) \cdot {b}^{4} + \color{blue}{{b}^{4}} \]

       4. associate-*l* N/A

          \[\leadsto 4 \cdot \left(\frac{1}{{b}^{2}} \cdot {b}^{4}\right) + {\color{blue}{b}}^{4} \]

       5. fma-define N/A

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{1}{{b}^{2}} \cdot {b}^{4}}, {b}^{4}\right) \]

       6. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(4, \frac{1 \cdot {b}^{4}}{\color{blue}{{b}^{2}}}, {b}^{4}\right) \]

       7. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(4, \frac{{b}^{4}}{{\color{blue}{b}}^{2}}, {b}^{4}\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(4, \frac{{b}^{\left(3 + 1\right)}}{{b}^{2}}, {b}^{4}\right) \]

       9. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(4, \frac{{b}^{3} \cdot b}{{\color{blue}{b}}^{2}}, {b}^{4}\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{fma}\left(4, \frac{\left(\left(b \cdot b\right) \cdot b\right) \cdot b}{{b}^{2}}, {b}^{4}\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(4, \frac{\left({b}^{2} \cdot b\right) \cdot b}{{b}^{2}}, {b}^{4}\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(4, \frac{{b}^{2} \cdot \left(b \cdot b\right)}{{\color{blue}{b}}^{2}}, {b}^{4}\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(4, \frac{{b}^{2} \cdot {b}^{2}}{{b}^{2}}, {b}^{4}\right) \]

       14. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(4, {b}^{2} \cdot \color{blue}{\frac{{b}^{2}}{{b}^{2}}}, {b}^{4}\right) \]

       15. *-inverses N/A

           \[\leadsto \mathsf{fma}\left(4, {b}^{2} \cdot 1, {b}^{4}\right) \]

       16. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) \]

       17. fma-undefine N/A

           \[\leadsto 4 \cdot {b}^{2} + \color{blue}{{b}^{4}} \]

       18. unpow2 N/A

           \[\leadsto 4 \cdot \left(b \cdot b\right) + {b}^{4} \]

       19. associate-*r* N/A

           \[\leadsto \left(4 \cdot b\right) \cdot b + {\color{blue}{b}}^{4} \]

       20. metadata-eval N/A

           \[\leadsto \left(4 \cdot b\right) \cdot b + {b}^{\left(3 + \color{blue}{1}\right)} \]

       21. pow-plus N/A

           \[\leadsto \left(4 \cdot b\right) \cdot b + {b}^{3} \cdot \color{blue}{b} \]

   12. Simplified 95.8%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+29}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2e+21) (+ (* (* b b) 4.0) -1.0) (* a (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2e+21) {
		tmp = ((b * b) * 4.0) + -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 * a) <= 2d+21) then
        tmp = ((b * b) * 4.0d0) + (-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 * a) <= 2e+21) {
		tmp = ((b * b) * 4.0) + -1.0;
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 2e+21:
		tmp = ((b * b) * 4.0) + -1.0
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 2e+21)
		tmp = Float64(Float64(Float64(b * b) * 4.0) + -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 * a) <= 2e+21)
		tmp = ((b * b) * 4.0) + -1.0;
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 2e+21], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 2e21

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto {\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(b \cdot b\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{b} \cdot b\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot \color{blue}{b}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(\left(b \cdot b\right) \cdot 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(b \cdot b\right) \cdot 4 + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({b}^{4}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(3 + 1\right)}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{4}\right), -1\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(\left(2 + 1\right) + 1\right)}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 + 1\right)} \cdot b\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      4. pow-plus N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({b}^{2} \cdot b\right) \cdot b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, 4\right), -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, 4\right), -1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(b \cdot {b}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{4}\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

      10. *-lowering-*.f64 99.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(\left(b \cdot b\right) \cdot 4 + -1\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{4 \cdot {b}^{2} - 1} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 4 \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. metadata-eval 72.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right) \]

   10. Simplified 72.7%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot 4 + -1} \]

   if 2e21 < (*.f64 a a)

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto {\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(b \cdot b\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{b} \cdot b\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot \color{blue}{b}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(\left(b \cdot b\right) \cdot 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(b \cdot b\right) \cdot 4 + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {a}^{\left(3 + \color{blue}{1}\right)} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(\left(2 + 1\right) + 1\right)} \]

      3. pow-plus N/A

         \[\leadsto {a}^{\left(2 + 1\right)} \cdot \color{blue}{a} \]

      4. pow-plus N/A

         \[\leadsto \left({a}^{2} \cdot a\right) \cdot a \]

      5. *-commutative N/A

         \[\leadsto \left(a \cdot {a}^{2}\right) \cdot a \]

      6. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot {a}^{2}\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      10. *-lowering-*.f64 88.3%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 88.3%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 37.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.48) -1.0 (* b (* b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.48d0) then
        tmp = -1.0d0
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.48:
		tmp = -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.48)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.48)
		tmp = -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.48], -1.0, N[(b * N[(b * 4.0), $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. associate--l+ N/A

         \[\leadsto {\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(b \cdot b\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{b} \cdot b\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot \color{blue}{b}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(\left(b \cdot b\right) \cdot 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(b \cdot b\right) \cdot 4 + -1\right)} \]
   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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{\left(3 + 1\right)}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{\left(\left(2 + 1\right) + 1\right)}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{\left(2 + 1\right)} \cdot a\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      6. pow-plus N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({a}^{2} \cdot a\right) \cdot a\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot {a}^{2}\right) \cdot a\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      13. metadata-eval 77.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), -1\right) \]

   7. Simplified 77.9%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1} \]
   9. Step-by-step derivation

   10. Simplified 32.2%

       \[\leadsto \color{blue}{-1} \]

   if 0.47999999999999998 < 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. associate--l+ N/A

         \[\leadsto {\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(b \cdot b\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{b} \cdot b\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot \color{blue}{b}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(\left(b \cdot b\right) \cdot 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(b \cdot b\right) \cdot 4 + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({b}^{4}\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(3 + 1\right)}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{4}\right), -1\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(\left(2 + 1\right) + 1\right)}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 + 1\right)} \cdot b\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      4. pow-plus N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({b}^{2} \cdot b\right) \cdot b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, 4\right), -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, 4\right), -1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \left(b \cdot {b}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)}, -1\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{4}\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

      10. *-lowering-*.f64 94.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

   7. Simplified 94.6%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(\left(b \cdot b\right) \cdot 4 + -1\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{4 \cdot {b}^{2} - 1} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 4 \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. metadata-eval 52.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right) \]

   10. Simplified 52.0%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot 4 + -1} \]

   11. Taylor expanded in b around inf

       \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {b}^{2} \cdot \color{blue}{4} \]

       2. unpow2 N/A

          \[\leadsto \left(b \cdot b\right) \cdot 4 \]

       3. associate-*l* N/A

          \[\leadsto b \cdot \color{blue}{\left(b \cdot 4\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot 4\right)}\right) \]

       5. *-lowering-*.f64 52.0%

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{4}\right)\right) \]

   13. Simplified 52.0%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 24.6% 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. associate--l+ N/A

      \[\leadsto {\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(b \cdot b\right) - 1\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(b \cdot b\right)} - 1\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(b \cdot b\right) - 1\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(b \cdot b\right) - 1\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{b} \cdot b\right) - 1\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot \color{blue}{b}\right) - 1\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(b \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(b \cdot b\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(\left(b \cdot b\right) \cdot 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   16. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(b \cdot b\right) \cdot 4 + -1\right)} \]
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

      \[\leadsto \mathsf{+.f64}\left(\left({a}^{\left(3 + 1\right)}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left({a}^{\left(\left(2 + 1\right) + 1\right)}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   5. pow-plus N/A

      \[\leadsto \mathsf{+.f64}\left(\left({a}^{\left(2 + 1\right)} \cdot a\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

   6. pow-plus N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left({a}^{2} \cdot a\right) \cdot a\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot {a}^{2}\right) \cdot a\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

   13. metadata-eval 67.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), -1\right) \]

7. Simplified 67.9%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1} \]
9. Step-by-step derivation

10. Simplified 24.1%

    \[\leadsto \color{blue}{-1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))