Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 10

Speedup: 1.0×

• 61.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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: 96.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a \cdot a, 2, 4\right)\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-30)
   (/ 1.0 (/ 1.0 (- (* (* (fma (* b b) 2.0 (* 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) <= 5e-30) {
		tmp = 1.0 / (1.0 / (((fma((b * b), 2.0, (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) <= 5e-30)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * 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], 5e-30], N[(1.0 / N[(1.0 / N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $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) < 4.99999999999999972e-30

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. pow-sqr N/A

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

      15. distribute-lft-in N/A

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

      16. unpow2 N/A

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

   7. Applied rewrites 100.0%

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

   if 4.99999999999999972e-30 < (*.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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

   5. Applied rewrites 98.1%

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

Alternative 3: 97.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a \cdot a, 2, 4\right)\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-30)
   (- (* (* (* 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) <= 5e-30) {
		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) <= 5e-30)
		tmp = Float64(Float64(Float64(Float64(a * a) * 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], 5e-30], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $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) < 4.99999999999999972e-30

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. pow-sqr N/A

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

      15. distribute-lft-in N/A

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

      16. unpow2 N/A

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

   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%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]

   if 4.99999999999999972e-30 < (*.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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

   5. Applied rewrites 98.1%

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

Alternative 4: 98.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2000000000000.0)
   (fma (* b b) (fma b b 4.0) -1.0)
   (* (* (fma (* b b) 2.0 (* a a)) a) a)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2000000000000.0) {
		tmp = fma((b * b), fma(b, b, 4.0), -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) <= 2000000000000.0)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -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], 2000000000000.0], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $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) < 2e12

   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%

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

   if 2e12 < (*.f64 a a)

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. 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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. pow-sqr N/A

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

      15. distribute-lft-in N/A

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

      16. unpow2 N/A

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

   5. Applied rewrites 98.4%

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

Alternative 5: 94.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2000000000000.0)
   (fma (* b b) (fma b b 4.0) -1.0)
   (- (* (* (* a a) a) a) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2000000000000.0) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = (((a * a) * a) * a) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 2000000000000.0)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 2000000000000.0], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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%

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

   if 2e12 < (*.f64 a a)

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. 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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. pow-sqr N/A

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

      15. distribute-lft-in N/A

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

      16. unpow2 N/A

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

   5. Applied rewrites 98.4%

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

   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 93.5%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2000000000000.0)
   (fma (* b b) (fma b b 4.0) -1.0)
   (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2000000000000.0) {
		tmp = fma((b * b), 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) <= 2000000000000.0)
		tmp = fma(Float64(b * b), fma(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], 2000000000000.0], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $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) < 2e12

   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%

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

   if 2e12 < (*.f64 a a)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.6

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 93.5%

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

Alternative 7: 94.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2000000000000.0)
   (fma (* b b) (* b b) -1.0)
   (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2000000000000.0) {
		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) <= 2000000000000.0)
		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], 2000000000000.0], 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) < 2e12

   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 98.2%

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

   if 2e12 < (*.f64 a a)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.6

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 93.5%

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

Alternative 8: 83.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2000000000000.0) (fma (* b b) 4.0 -1.0) (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2000000000000.0) {
		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) <= 2000000000000.0)
		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], 2000000000000.0], 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) < 2e12

   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 0

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

   8. Applied rewrites 76.5%

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

   if 2e12 < (*.f64 a a)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.6

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 93.5%

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

Alternative 9: 51.3% 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 72.6

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

5. Applied rewrites 72.6%

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

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 54.4%

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

Alternative 10: 25.0% 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 72.6

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

5. Applied rewrites 72.6%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 26.7%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))