Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.5% → 99.2%

Time: 8.3s

Alternatives: 12

Speedup: 5.5×

• 60.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(\left(\left(1 - a\right) \cdot a\right) \cdot a\right) \cdot 4 + {\left(b \cdot b + a \cdot a\right)}^{2}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2e+73)
   (- (+ (* (* (* (- 1.0 a) a) a) 4.0) (pow (+ (* b b) (* a a)) 2.0)) 1.0)
   (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2e+73) {
		tmp = (((((1.0 - a) * a) * a) * 4.0) + pow(((b * b) + (a * a)), 2.0)) - 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 <= 2d+73) then
        tmp = (((((1.0d0 - a) * a) * a) * 4.0d0) + (((b * b) + (a * a)) ** 2.0d0)) - 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 <= 2e+73) {
		tmp = (((((1.0 - a) * a) * a) * 4.0) + Math.pow(((b * b) + (a * a)), 2.0)) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2e+73:
		tmp = (((((1.0 - a) * a) * a) * 4.0) + math.pow(((b * b) + (a * a)), 2.0)) - 1.0
	else:
		tmp = (a * a) * (a * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2e+73)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(1.0 - a) * a) * a) * 4.0) + (Float64(Float64(b * b) + Float64(a * a)) ^ 2.0)) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 2e+73)
		tmp = (((((1.0 - a) * a) * a) * 4.0) + (((b * b) + (a * a)) ^ 2.0)) - 1.0;
	else
		tmp = (a * a) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2e+73], N[(N[(N[(N[(N[(N[(1.0 - a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.99999999999999997e73

   1. Initial program 87.6%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.99999999999999997e73 < a

   1. Initial program 4.5%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 4.5

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 100.0

         \[\leadsto \color{blue}{{a}^{4}} \]

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(\left(\left(1 - a\right) \cdot a\right) \cdot a\right) \cdot 4 + {\left(b \cdot b + a \cdot a\right)}^{2}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \left(1 - a\right) \cdot 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right) \cdot \left(b \cdot b\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+14)
   (fma (* a a) (fma a a (* (- 1.0 a) 4.0)) -1.0)
   (-
    (fma (* 4.0 a) a (* (fma (fma 2.0 a 4.0) a (fma b b 12.0)) (* b b)))
    1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+14) {
		tmp = fma((a * a), fma(a, a, ((1.0 - a) * 4.0)), -1.0);
	} else {
		tmp = fma((4.0 * a), a, (fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)) * (b * b))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+14)
		tmp = fma(Float64(a * a), fma(a, a, Float64(Float64(1.0 - a) * 4.0)), -1.0);
	else
		tmp = Float64(fma(Float64(4.0 * a), a, Float64(fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)) * Float64(b * b))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+14], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(N[(1.0 - a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * a), $MachinePrecision] * a + N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e14

   1. Initial program 82.4%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2e14 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot 4, a, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right) \cdot \left(b \cdot b\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \left(1 - a\right) \cdot 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right) \cdot \left(b \cdot b\right)\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-44)
   (fma (* a a) 4.0 -1.0)
   (if (<= (* b b) 2e+14) (* (* a a) (* a a)) (* (* b b) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-44) {
		tmp = fma((a * a), 4.0, -1.0);
	} else if ((b * b) <= 2e+14) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-44)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	elseif (Float64(b * b) <= 2e+14)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-44], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 2e+14], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 1.99999999999999991e-44

   1. Initial program 81.0%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(a \cdot a, 4, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 79.9%

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

   if 1.99999999999999991e-44 < (*.f64 b b) < 2e14

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 76.9

         \[\leadsto \color{blue}{{a}^{4}} \]

   8. Applied rewrites 76.9%

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

   10. Applied rewrites 76.5%

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

   if 2e14 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 91.0

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

   8. Applied rewrites 91.0%

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

   10. Applied rewrites 90.9%

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

Alternative 4: 94.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \left(1 - a\right) \cdot 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+14)
   (fma (* a a) (fma a a (* (- 1.0 a) 4.0)) -1.0)
   (fma (* (* b b) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+14) {
		tmp = fma((a * a), fma(a, a, ((1.0 - a) * 4.0)), -1.0);
	} else {
		tmp = fma(((b * b) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+14)
		tmp = fma(Float64(a * a), fma(a, a, Float64(Float64(1.0 - a) * 4.0)), -1.0);
	else
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+14], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(N[(1.0 - a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e14

   1. Initial program 82.4%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2e14 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 90.9

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

   8. Applied rewrites 90.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.9%

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+14)
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (fma (* (* b b) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+14) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma(((b * b) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+14)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+14], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e14

   1. Initial program 82.4%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 50.1

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

   8. Applied rewrites 50.1%

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

   9. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

   11. Applied rewrites 99.9%

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

   if 2e14 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 90.9

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

   8. Applied rewrites 90.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.9%

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \left(a - 4\right) \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+14)
   (fma (* a a) (* (- a 4.0) a) -1.0)
   (fma (* (* b b) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+14) {
		tmp = fma((a * a), ((a - 4.0) * a), -1.0);
	} else {
		tmp = fma(((b * b) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+14)
		tmp = fma(Float64(a * a), Float64(Float64(a - 4.0) * a), -1.0);
	else
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+14], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e14

   1. Initial program 82.4%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 99.1%

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

   if 2e14 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 90.9

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

   8. Applied rewrites 90.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.9%

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -1.7e+18)
     t_0
     (if (<= a 1.6e+15) (fma (* (* b b) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -1.7e+18) {
		tmp = t_0;
	} else if (a <= 1.6e+15) {
		tmp = fma(((b * b) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -1.7e+18)
		tmp = t_0;
	elseif (a <= 1.6e+15)
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.7e+18], t$95$0, If[LessEqual[a, 1.6e+15], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7e18 or 1.6e15 < a

   1. Initial program 42.7%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 92.1

         \[\leadsto \color{blue}{{a}^{4}} \]

   8. Applied rewrites 92.1%

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

   10. Applied rewrites 92.0%

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

   if -1.7e18 < a < 1.6e15

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 96.8

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

   8. Applied rewrites 96.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 96.8%

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+14)
   (fma (* a a) (fma a a 4.0) -1.0)
   (fma (* (* b b) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+14) {
		tmp = fma((a * a), fma(a, a, 4.0), -1.0);
	} else {
		tmp = fma(((b * b) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+14)
		tmp = fma(Float64(a * a), fma(a, a, 4.0), -1.0);
	else
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+14], N[(N[(a * a), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e14

   1. Initial program 82.4%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, 4\right), -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 98.4%

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

   if 2e14 < (*.f64 b b)

   1. Initial program 64.5%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 90.9

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

   8. Applied rewrites 90.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.9%

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 82.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 900000:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -35.0) t_0 (if (<= a 900000.0) (fma 12.0 (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -35.0) {
		tmp = t_0;
	} else if (a <= 900000.0) {
		tmp = fma(12.0, (b * b), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -35.0)
		tmp = t_0;
	elseif (a <= 900000.0)
		tmp = fma(12.0, Float64(b * b), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -35.0], t$95$0, If[LessEqual[a, 900000.0], N[(12.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -35 or 9e5 < a

   1. Initial program 46.3%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 88.3

         \[\leadsto \color{blue}{{a}^{4}} \]

   8. Applied rewrites 88.3%

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

   10. Applied rewrites 88.2%

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

   if -35 < a < 9e5

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, 12\right)}, {b}^{2}, -1\right) \]

       11. unpow2 N/A

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

       12. lower-*.f64 99.6

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

   11. Applied rewrites 99.6%

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

   12. Taylor expanded in b around 0

       \[\leadsto \mathsf{fma}\left(12, \color{blue}{b} \cdot b, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 75.3%

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

Alternative 10: 68.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+307) (fma (* a a) 4.0 -1.0) (fma 12.0 (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+307) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = fma(12.0, (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+307)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(12.0, Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+307], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(12.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999986e306

   1. Initial program 79.4%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 77.9

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

   8. Applied rewrites 77.9%

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

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(a \cdot a, 4, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 58.8%

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

   if 9.99999999999999986e306 < (*.f64 b b)

   1. Initial program 55.4%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, 12\right)}, {b}^{2}, -1\right) \]

       11. unpow2 N/A

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

       12. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in b around 0

       \[\leadsto \mathsf{fma}\left(12, \color{blue}{b} \cdot b, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

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

Alternative 11: 51.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(12, b \cdot b, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma 12.0 (* b b) -1.0))

C

double code(double a, double b) {
	return fma(12.0, (b * b), -1.0);
}

Julia

function code(a, b)
	return fma(12.0, Float64(b * b), -1.0)
end

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in b around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 83.5

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

5. Applied rewrites 83.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 81.4%

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

9. Taylor expanded in a around 0

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. pow-sqr N/A

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

    4. distribute-rgt-in N/A

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

    5. *-commutative N/A

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

    6. metadata-eval N/A

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

    7. lower-fma.f64 N/A

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

    8. +-commutative N/A

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

    9. unpow2 N/A

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

    10. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, 12\right)}, {b}^{2}, -1\right) \]

    11. unpow2 N/A

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

    12. lower-*.f64 70.8

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

11. Applied rewrites 70.8%

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

12. Taylor expanded in b around 0

    \[\leadsto \mathsf{fma}\left(12, \color{blue}{b} \cdot b, -1\right) \]
13. Step-by-step derivation

14. Applied rewrites 51.7%

    \[\leadsto \mathsf{fma}\left(12, \color{blue}{b} \cdot b, -1\right) \]
15. Add Preprocessing

Alternative 12: 24.6% accurate, 155.0× 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 73.3%

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

3. Taylor expanded in b around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 83.5

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

5. Applied rewrites 83.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 81.4%

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

9. Taylor expanded in a around 0

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. pow-sqr N/A

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

    4. distribute-rgt-in N/A

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

    5. *-commutative N/A

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

    6. metadata-eval N/A

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

    7. lower-fma.f64 N/A

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

    8. +-commutative N/A

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

    9. unpow2 N/A

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

    10. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, 12\right)}, {b}^{2}, -1\right) \]

    11. unpow2 N/A

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

    12. lower-*.f64 70.8

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

11. Applied rewrites 70.8%

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

12. Taylor expanded in b around 0

    \[\leadsto -1 \]
13. Step-by-step derivation

14. Applied rewrites 25.0%

    \[\leadsto -1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))