Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 11

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: 97.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e-56)
   (fma (* (fma b b 4.0) b) b -1.0)
   (- (fma (* b b) 4.0 (* (* (fma (* b b) 2.0 (* a a)) a) a)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-56) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma((b * b), 4.0, ((fma((b * b), 2.0, (a * a)) * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5e-56)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = Float64(fma(Float64(b * b), 4.0, Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5e-56], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.99999999999999997e-56

   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

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

      1. lower-pow.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   9. 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-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if 4.99999999999999997e-56 < (*.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

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

      1. lower-pow.f64 97.3

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

   5. Applied rewrites 97.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.3

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.7

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

   10. Applied rewrites 98.7%

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

Alternative 3: 97.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot \left(a - b\right)\right) \cdot \left(a - b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e-56)
   (fma (* (fma b b 4.0) b) b -1.0)
   (- (+ (* (* (* a a) (- a b)) (- a b)) (* 4.0 (* b b))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-56) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = ((((a * a) * (a - b)) * (a - b)) + (4.0 * (b * b))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5e-56)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * Float64(a - b)) * Float64(a - b)) + Float64(4.0 * Float64(b * b))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5e-56], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.99999999999999997e-56

   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

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

      1. lower-pow.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   9. 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-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if 4.99999999999999997e-56 < (*.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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 98.4

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

   9. Applied rewrites 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot \left(a - b\right)\right) \cdot \left(a - b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) - \left(\left(b - a\right) \cdot \left(a - b\right)\right) \cdot \left(a \cdot a\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e-56)
   (fma (* (fma b b 4.0) b) b -1.0)
   (- (- (* 4.0 (* b b)) (* (* (- b a) (- a b)) (* a a))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-56) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = ((4.0 * (b * b)) - (((b - a) * (a - b)) * (a * a))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5e-56)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) - Float64(Float64(Float64(b - a) * Float64(a - b)) * Float64(a * a))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5e-56], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(b - a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.99999999999999997e-56

   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

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

      1. lower-pow.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   9. 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-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if 4.99999999999999997e-56 < (*.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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) - \left(\left(b - a\right) \cdot \left(a - b\right)\right) \cdot \left(a \cdot a\right)\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 96.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e-56)
   (fma (* (fma b b 4.0) b) b -1.0)
   (- (fma (* b b) 4.0 (* (* (* a a) a) a)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-56) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma((b * b), 4.0, (((a * a) * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5e-56)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = Float64(fma(Float64(b * b), 4.0, Float64(Float64(Float64(a * a) * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5e-56], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.99999999999999997e-56

   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

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

      1. lower-pow.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   9. 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-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if 4.99999999999999997e-56 < (*.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

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

      1. lower-pow.f64 97.3

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

   5. Applied rewrites 97.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.3

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.7

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

   10. Applied rewrites 98.7%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 97.3%

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

Alternative 7: 98.1% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 10.0)
   (fma (* a a) (* a a) -1.0)
   (fma (fma (* 2.0 a) a (fma b b 4.0)) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 10.0) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma(fma((2.0 * a), a, fma(b, b, 4.0)), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 10.0)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(fma(Float64(2.0 * a), a, fma(b, b, 4.0)), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 10.0], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(2.0 * a), $MachinePrecision] * a + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 10

   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

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{{a}^{4} - 1} \]

      2. lower-pow.f64 100.0

         \[\leadsto \color{blue}{{a}^{4}} - 1 \]

   10. Applied rewrites 100.0%

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

   12. Applied rewrites 99.9%

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

   if 10 < (*.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 a around inf

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

      1. lower-pow.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right) - 1} \]
   9. 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. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. distribute-lft-in N/A

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

      13. associate-+r+ N/A

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

      14. *-commutative N/A

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

   10. Applied rewrites 95.8%

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

Alternative 8: 94.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+15)
   (fma (* a a) (* a a) -1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+15) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+15)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+15], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e15

   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

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{{a}^{4} - 1} \]

      2. lower-pow.f64 100.0

         \[\leadsto \color{blue}{{a}^{4}} - 1 \]

   10. Applied rewrites 100.0%

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

   12. Applied rewrites 99.9%

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

   if 1e15 < (*.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 a around inf

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

      1. lower-pow.f64 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   9. 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-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 90.6

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

   10. Applied rewrites 90.6%

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

Alternative 9: 84.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+306) (fma (* a a) (* a a) -1.0) (fma (* 4.0 b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+306) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma((4.0 * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+306)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(4.0 * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+306], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999993e306

   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

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

      1. lower-pow.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{{a}^{4} - 1} \]

      2. lower-pow.f64 85.3

         \[\leadsto \color{blue}{{a}^{4}} - 1 \]

   10. Applied rewrites 85.3%

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

   12. Applied rewrites 85.2%

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

   if 4.99999999999999993e306 < (*.f64 b b)

   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 inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   9. 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-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 100.0%

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

Alternative 10: 50.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-pow.f64 89.7

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

5. Applied rewrites 89.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 89.7

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

7. Applied rewrites 89.7%

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

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
9. 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-in N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 63.5

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

10. Applied rewrites 63.5%

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

11. Taylor expanded in b around 0

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

13. Applied rewrites 51.5%

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

Alternative 11: 24.6% 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 inf

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

   1. lower-pow.f64 89.7

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

5. Applied rewrites 89.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 89.7

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

7. Applied rewrites 89.7%

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

8. Taylor expanded in b around 0

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

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{{a}^{4} - 1} \]

   2. lower-pow.f64 74.2

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]

10. Applied rewrites 74.2%

    \[\leadsto \color{blue}{{a}^{4} - 1} \]

11. Taylor expanded in a around 0

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

13. Applied rewrites 24.5%

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

Reproduce

herbie shell --seed 2024312 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))