Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 13.3s

Alternatives: 10

Speedup: 3.3×

• 61.1% 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, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate--l+ N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. sub-neg N/A

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

4. Applied egg-rr 99.9%

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

Alternative 2: 51.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot 4\\ \mathbf{if}\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + t\_0 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) 4.0)))
   (if (<= (+ (pow (fma a a (* b b)) 2.0) t_0) 2e-9) -1.0 t_0)))

C

double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if ((pow(fma(a, a, (b * b)), 2.0) + t_0) <= 2e-9) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * 4.0)
	tmp = 0.0
	if (Float64((fma(a, a, Float64(b * b)) ^ 2.0) + t_0) <= 2e-9)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$0), $MachinePrecision], 2e-9], -1.0, t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (*.f64 b b))) < 2.00000000000000012e-9

   1. Initial program 100.0%

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-out N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 99.3%

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

   if 2.00000000000000012e-9 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (*.f64 b b)))

   1. Initial program 99.8%

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-out N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval 62.4

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

   5. Simplified 62.4%

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

   6. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-*l* N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. unpow2 N/A

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

      19. lower-fma.f64 62.5

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

   8. Simplified 62.5%

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

   9. Taylor expanded in b around 0

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

   11. Simplified 36.3%

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

4. Final simplification 52.8%

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

Alternative 3: 97.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(a, a, 0\right) \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(2, \mathsf{fma}\left(a, a, 0\right), 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, b \cdot 2, \mathsf{fma}\left(a, a, 0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fma a a 0.0) 2e+48)
   (fma (* b b) (fma b b (fma 2.0 (fma a a 0.0) 4.0)) -1.0)
   (* a (* a (fma b (* b 2.0) (fma a a 0.0))))))

C

double code(double a, double b) {
	double tmp;
	if (fma(a, a, 0.0) <= 2e+48) {
		tmp = fma((b * b), fma(b, b, fma(2.0, fma(a, a, 0.0), 4.0)), -1.0);
	} else {
		tmp = a * (a * fma(b, (b * 2.0), fma(a, a, 0.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (fma(a, a, 0.0) <= 2e+48)
		tmp = fma(Float64(b * b), fma(b, b, fma(2.0, fma(a, a, 0.0), 4.0)), -1.0);
	else
		tmp = Float64(a * Float64(a * fma(b, Float64(b * 2.0), fma(a, a, 0.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a + 0.0), $MachinePrecision], 2e+48], N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(2.0 * N[(a * a + 0.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(b * N[(b * 2.0), $MachinePrecision] + N[(a * a + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 2.00000000000000009e48

   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

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. lower-fma.f64 N/A

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

   5. Simplified 97.9%

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

   if 2.00000000000000009e48 < (*.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. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-rgt-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

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

   5. Simplified 98.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(a, a, 0\right) \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(2, \mathsf{fma}\left(a, a, 0\right), 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, b \cdot 2, \mathsf{fma}\left(a, a, 0\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(a, a, 0\right) \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, b \cdot 2, \mathsf{fma}\left(a, a, 0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fma a a 0.0) 2e+48)
   (fma b (* b (fma b b 4.0)) -1.0)
   (* a (* a (fma b (* b 2.0) (fma a a 0.0))))))

C

double code(double a, double b) {
	double tmp;
	if (fma(a, a, 0.0) <= 2e+48) {
		tmp = fma(b, (b * fma(b, b, 4.0)), -1.0);
	} else {
		tmp = a * (a * fma(b, (b * 2.0), fma(a, a, 0.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (fma(a, a, 0.0) <= 2e+48)
		tmp = fma(b, Float64(b * fma(b, b, 4.0)), -1.0);
	else
		tmp = Float64(a * Float64(a * fma(b, Float64(b * 2.0), fma(a, a, 0.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a + 0.0), $MachinePrecision], 2e+48], N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(b * N[(b * 2.0), $MachinePrecision] + N[(a * a + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 2.00000000000000009e48

   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

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.7

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

   5. Simplified 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate--l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 97.8

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

   7. Applied egg-rr 97.8%

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

   if 2.00000000000000009e48 < (*.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. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-rgt-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

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

   5. Simplified 98.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(a, a, 0\right) \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, b \cdot 2, \mathsf{fma}\left(a, a, 0\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(a, a, 0\right) \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(a, a, b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fma a a 0.0) 2e+48)
   (fma b (* b (fma b b 4.0)) -1.0)
   (* a (* a (fma a a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (fma(a, a, 0.0) <= 2e+48) {
		tmp = fma(b, (b * fma(b, b, 4.0)), -1.0);
	} else {
		tmp = a * (a * fma(a, a, (b * b)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (fma(a, a, 0.0) <= 2e+48)
		tmp = fma(b, Float64(b * fma(b, b, 4.0)), -1.0);
	else
		tmp = Float64(a * Float64(a * fma(a, a, Float64(b * b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a + 0.0), $MachinePrecision], 2e+48], N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 2.00000000000000009e48

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-out N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval 97.7

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

   5. Simplified 97.7%

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

   if 2.00000000000000009e48 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. clear-num N/A

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

      14. flip-+ N/A

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

      15. lift-+.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.2

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-rgt-in N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. *-rgt-identity N/A

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

      17. *-lft-identity N/A

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

      18. *-inverses N/A

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

      19. associate-*l/ N/A

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

      20. associate-*r/ N/A

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

   10. Simplified 98.3%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(a, a, 0\right) \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(a, a, b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a, a, 0\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+74) (fma a (* a (fma a a 0.0)) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+74) {
		tmp = fma(a, (a * fma(a, a, 0.0)), -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+74)
		tmp = fma(a, Float64(a * fma(a, a, 0.0)), -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+74], N[(a * N[(a * N[(a * a + 0.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999963e74

   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 b around 0

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 94.3

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

   5. Simplified 94.3%

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

   if 4.99999999999999963e74 < (*.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

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.1

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

   5. Simplified 97.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a, a, 0\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(a, a, 0\right) \leq 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(a, a, 0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fma a a 0.0) 1e+22)
   (fma b (* b 4.0) -1.0)
   (* a (* a (fma a a 0.0)))))

C

double code(double a, double b) {
	double tmp;
	if (fma(a, a, 0.0) <= 1e+22) {
		tmp = fma(b, (b * 4.0), -1.0);
	} else {
		tmp = a * (a * fma(a, a, 0.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (fma(a, a, 0.0) <= 1e+22)
		tmp = fma(b, Float64(b * 4.0), -1.0);
	else
		tmp = Float64(a * Float64(a * fma(a, a, 0.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a + 0.0), $MachinePrecision], 1e+22], N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(a * a + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1e22

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-out N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 77.6%

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

   if 1e22 < (*.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

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.6

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

   5. Simplified 88.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(a, a, 0\right) \leq 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(a, a, 0\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	return fma(t_0, t_0, -1.0);
}

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate--l+ N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. sub-neg N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in b around 0

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

7. Simplified 99.4%

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

Alternative 9: 51.2% accurate, 10.9× speedup

Math

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

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(b, Float64(b * 4.0), -1.0)
end

Wolfram

code[a_, b_] := N[(b * N[(b * 4.0), $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. 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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. pow-sqr N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. distribute-lft-out N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 72.1

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

5. Simplified 72.1%

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

6. Taylor expanded in b around 0

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

8. Simplified 52.5%

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

Alternative 10: 25.1% 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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. pow-sqr N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. distribute-lft-out N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 72.1

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

5. Simplified 72.1%

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

6. Taylor expanded in b around 0

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

8. Simplified 26.5%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))