Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 6

Speedup: 1.0×

• 61.0% 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

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

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]

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. Add Preprocessing

Alternative 2: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a \cdot a, 2, 4\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.01)
   (fma (* b b) 4.0 (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0))
   (- (* (* (fma b b (fma (* a a) 2.0 4.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.01) {
		tmp = fma((b * b), 4.0, (((fma((b * b), 2.0, (a * a)) * a) * a) - 1.0));
	} else {
		tmp = ((fma(b, b, fma((a * a), 2.0, 4.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.01)
		tmp = fma(Float64(b * b), 4.0, Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a) - 1.0));
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(Float64(a * a), 2.0, 4.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.01], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b + N[(N[(a * a), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0100000000000000002

   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 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. count-2-rev N/A

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

      4. distribute-lft-in N/A

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

      5. count-2-rev N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 0.0100000000000000002 < (*.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 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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+r+ N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 97.8%

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

Alternative 3: 98.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.01:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a \cdot a, 2, 4\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.01)
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* (fma b b (fma (* a a) 2.0 4.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.01) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((fma(b, b, fma((a * a), 2.0, 4.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.01)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(Float64(a * a), 2.0, 4.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.01], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + N[(N[(a * a), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0100000000000000002

   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

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

      1. lower-pow.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.5%

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

   if 0.0100000000000000002 < (*.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 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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+r+ N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 97.8%

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

Alternative 4: 94.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.01:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.01)
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.01) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.01)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.01], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0100000000000000002

   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

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

      1. lower-pow.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.5%

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

   if 0.0100000000000000002 < (*.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 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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+r+ N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.7%

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

Alternative 5: 84.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2.2 \cdot 10^{+306}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2.2e+306)
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2.2e+306) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2.2d+306) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2.2e+306) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2.2e+306:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2.2e+306)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2.2e+306)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2.2e+306], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.2e306

   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

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

      1. lower-pow.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 80.7%

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

   if 2.2e306 < (*.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 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto 4 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 97.4%

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

Alternative 6: 50.2% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 69.8

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

5. Applied rewrites 69.8%

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

6. Taylor expanded in b around 0

   \[\leadsto 4 \cdot \color{blue}{{b}^{2}} - 1 \]
7. Step-by-step derivation

8. Applied rewrites 50.5%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))