Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 9

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+14], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0 + N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] + N[(4.0 * 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 99.9%

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

   3. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.9%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

   if 2e14 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

   5. Applied rewrites 96.4%

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

4. Final simplification 98.1%

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

Alternative 3: 97.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.9%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

   if 2e14 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*l* N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

      15. pow-sqr N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 96.4%

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

4. Final simplification 98.1%

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

Alternative 4: 97.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 1000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1000000000000.0)
   (fma (* b b) (* b b) -1.0)
   (* (* (fma (* b b) 2.0 (* a a)) a) a)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1000000000000.0) {
		tmp = fma((b * b), (b * b), -1.0);
	} else {
		tmp = (fma((b * b), 2.0, (a * a)) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 1000000000000.0)
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	else
		tmp = Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1000000000000.0], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1e12

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 99.2%

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

   if 1e12 < (*.f64 a a)

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

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

Alternative 5: 94.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+14) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = ((b * b) * b) * b
    end if
    code = tmp
end function

Java

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

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+14:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = ((b * b) * b) * b
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+14)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = ((b * b) * b) * b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.9%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

   if 2e14 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*l* N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

      15. pow-sqr N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left({b}^{2} \cdot b\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 90.9%

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

4. Final simplification 95.3%

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

Alternative 6: 94.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 1000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1000000000000.0)
   (fma (* b b) (* b b) -1.0)
   (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1000000000000.0) {
		tmp = fma((b * b), (b * b), -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 1000000000000.0)
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1000000000000.0], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1e12

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 99.2%

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

   if 1e12 < (*.f64 a a)

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 89.4%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 89.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 1000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 0.005) (fma (* b b) 4.0 -1.0) (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 0.005) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 0.005)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 0.005], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 0.0050000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 76.1%

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

   if 0.0050000000000000001 < (*.f64 a a)

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 88.6%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 88.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.6% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(b \cdot b, 4, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma (* b b) 4.0 -1.0))

C

double code(double a, double b) {
	return fma((b * b), 4.0, -1.0);
}

Julia

function code(a, b)
	return fma(Float64(b * b), 4.0, -1.0)
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 71.0

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

5. Applied rewrites 71.0%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 51.9%

   \[\leadsto \mathsf{fma}\left(b \cdot b, 4, -1\right) \]
9. Add Preprocessing

Alternative 9: 24.8% accurate, 131.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0;
end

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 71.0

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

5. Applied rewrites 71.0%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 25.2%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))