Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 13

Speedup: 1.0×

• 61.2% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[(b * 4.0), $MachinePrecision] * b + N[(N[Power[N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(b \cdot 4, b, {\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2} - 1\right) \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(b * b + 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e-19)
   (- (fma (* b b) 4.0 (pow b 4.0)) 1.0)
   (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-19) {
		tmp = fma((b * b), 4.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = ((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-19)
		tmp = Float64(fma(Float64(b * b), 4.0, (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(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-19], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 5.0000000000000004e-19

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.0000000000000004e-19 < (*.f64 a a)

   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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 98.5%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, 4, {b}^{3}\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e-19)
   (fma (fma b 4.0 (pow b 3.0)) b -1.0)
   (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-19) {
		tmp = fma(fma(b, 4.0, pow(b, 3.0)), b, -1.0);
	} else {
		tmp = ((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-19)
		tmp = fma(fma(b, 4.0, (b ^ 3.0)), b, -1.0);
	else
		tmp = Float64(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-19], N[(N[(b * 4.0 + N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 5.0000000000000004e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, 4, {b}^{3}\right), b, -1\right) \]

   if 5.0000000000000004e-19 < (*.f64 a a)

   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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 98.5%

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

Alternative 5: 98.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, b, b \cdot 4\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e-19)
   (fma (fma (* b b) b (* b 4.0)) b -1.0)
   (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-19) {
		tmp = fma(fma((b * b), b, (b * 4.0)), b, -1.0);
	} else {
		tmp = ((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-19)
		tmp = fma(fma(Float64(b * b), b, Float64(b * 4.0)), b, -1.0);
	else
		tmp = Float64(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-19], N[(N[(N[(b * b), $MachinePrecision] * b + N[(b * 4.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 5.0000000000000004e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   if 5.0000000000000004e-19 < (*.f64 a a)

   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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 98.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, b, b \cdot 4\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, b, b \cdot 4\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2e-8)
   (fma (fma (* b b) b (* b 4.0)) b -1.0)
   (* (* (fma (* b b) 2.0 (* a a)) a) a)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2e-8) {
		tmp = fma(fma((b * b), b, (b * 4.0)), b, -1.0);
	} else {
		tmp = (fma((b * b), 2.0, (a * a)) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 2e-8)
		tmp = fma(fma(Float64(b * b), b, Float64(b * 4.0)), b, -1.0);
	else
		tmp = Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 2e-8], N[(N[(N[(b * b), $MachinePrecision] * b + N[(b * 4.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 2e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   if 2e-8 < (*.f64 a a)

   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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. neg-sub0 N/A

         \[\leadsto \left(\left(\color{blue}{\left(0 - -2 \cdot \frac{{b}^{2}}{{a}^{2}}\right)} + 1\right) \cdot {a}^{3}\right) \cdot a \]

      9. associate--r- N/A

         \[\leadsto \left(\color{blue}{\left(0 - \left(-2 \cdot \frac{{b}^{2}}{{a}^{2}} - 1\right)\right)} \cdot {a}^{3}\right) \cdot a \]

      10. neg-sub0 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, b, b \cdot 4\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, b, b \cdot 4\right), b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-15)
   (- (* (* a a) (* a a)) 1.0)
   (fma (fma (* b b) b (* b 4.0)) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-15) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = fma(fma((b * b), b, (b * 4.0)), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-15)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(fma(Float64(b * b), b, Float64(b * 4.0)), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-15], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b + N[(b * 4.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999999e-15

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 99.9%

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

   if 4.99999999999999999e-15 < (*.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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 91.6%

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

   9. Applied rewrites 91.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, b, b \cdot 4\right), b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-15)
   (- (* (* 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) <= 5e-15) {
		tmp = ((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) <= 5e-15)
		tmp = Float64(Float64(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], 5e-15], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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) < 4.99999999999999999e-15

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 99.9%

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

   if 4.99999999999999999e-15 < (*.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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 91.6%

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

Alternative 9: 93.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 1.95 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1.95e+117)
   (fma (* (fma b b 4.0) b) b -1.0)
   (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1.95e+117) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 1.95e+117)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1.95e+117], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1.94999999999999995e117

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 92.8

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 92.8%

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

   if 1.94999999999999995e117 < (*.f64 a a)

   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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-pow.f64 97.8

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

   10. Applied rewrites 97.8%

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

   12. Applied rewrites 97.8%

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

Alternative 10: 93.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 1.95 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1.95e+117)
   (fma (* b b) (fma b b 4.0) -1.0)
   (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1.95e+117) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 1.95e+117)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1.95e+117], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1.94999999999999995e117

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 92.8

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

   5. Applied rewrites 92.8%

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

   if 1.94999999999999995e117 < (*.f64 a a)

   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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-pow.f64 97.8

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

   10. Applied rewrites 97.8%

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

   12. Applied rewrites 97.8%

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

Alternative 11: 82.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 4, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 4e+32) (fma (* b 4.0) b -1.0) (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 4e+32) {
		tmp = fma((b * 4.0), b, -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 4e+32)
		tmp = fma(Float64(b * 4.0), b, -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 4e+32], N[(N[(b * 4.0), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.00000000000000021e32

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.0%

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

   if 4.00000000000000021e32 < (*.f64 a a)

   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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in a around inf

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

      1. lower-pow.f64 90.4

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

   10. Applied rewrites 90.4%

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

   12. Applied rewrites 90.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 4, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.6% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[(b * 4.0), $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 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 72.3

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

5. Applied rewrites 72.3%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 51.9%

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

9. Final simplification 51.9%

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

Alternative 13: 25.0% accurate, 131.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 72.3

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

5. Applied rewrites 72.3%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 24.4%

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

Reproduce

herbie shell --seed 2024325 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))