Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 9.7s

Alternatives: 11

Speedup: 3.1×

• 60.8% 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: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. sqr-pow N/A

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

   7. pow2 N/A

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

   8. pow-pow N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. unpow1/2 N/A

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

   14. lower-sqrt.f64 100.0

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

   15. lift-+.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-fma.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 99.9

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.9

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

4. Applied egg-rr 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. associate-+l+ N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lower-fma.f64 N/A

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 98.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{-17}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a \cdot 2, 4\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, b \cdot 2, a \cdot a\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 4e-17)
   (+ (* b (* b (fma b b (fma a (* a 2.0) 4.0)))) -1.0)
   (+ (* a (* a (fma b (* b 2.0) (* a a)))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 4e-17) {
		tmp = (b * (b * fma(b, b, fma(a, (a * 2.0), 4.0)))) + -1.0;
	} else {
		tmp = (a * (a * fma(b, (b * 2.0), (a * a)))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 4e-17)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, fma(a, Float64(a * 2.0), 4.0)))) + -1.0);
	else
		tmp = Float64(Float64(a * Float64(a * fma(b, Float64(b * 2.0), Float64(a * a)))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 4e-17], N[(N[(b * N[(b * N[(b * b + N[(a * N[(a * 2.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * N[(a * N[(b * N[(b * 2.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.00000000000000029e-17

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   6. 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. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. associate-+r+ N/A

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

      16. +-commutative N/A

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

   7. Simplified 100.0%

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

   if 4.00000000000000029e-17 < (*.f64 a a)

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.8

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in b around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   7. Simplified 98.5%

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

   8. Taylor expanded in a around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-*r* N/A

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

   10. Simplified 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{-17}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a \cdot 2, 4\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, b \cdot 2, a \cdot a\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 99.9

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.9

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

4. Applied egg-rr 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 99.9

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.9

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 5: 98.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, b \cdot 2, a \cdot a\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 4e-17)
   (fma (fma b b 4.0) (* b b) -1.0)
   (+ (* a (* a (fma b (* b 2.0) (* a a)))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 4e-17) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = (a * (a * fma(b, (b * 2.0), (a * a)))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 4e-17)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = Float64(Float64(a * Float64(a * fma(b, Float64(b * 2.0), Float64(a * a)))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 4e-17], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * N[(a * N[(b * N[(b * 2.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.00000000000000029e-17

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 99.9

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

   7. Simplified 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   9. Applied egg-rr 99.9%

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

   if 4.00000000000000029e-17 < (*.f64 a a)

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.8

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in b around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   7. Simplified 98.5%

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

   8. Taylor expanded in a around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-*r* N/A

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

   10. Simplified 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(b, b \cdot 2, a \cdot a\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1e-8)
   (fma (fma b b 4.0) (* b b) -1.0)
   (+ (* a (* a (* a a))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1e-8) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = (a * (a * (a * a))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 1e-8)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = Float64(Float64(a * Float64(a * Float64(a * a))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1e-8], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1e-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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 99.9

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

   7. Simplified 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   9. Applied egg-rr 99.9%

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

   if 1e-8 < (*.f64 a a)

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.8

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

         \[\leadsto {a}^{\color{blue}{\left(3 + 1\right)}} - 1 \]

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.6

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

   7. Simplified 91.6%

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

4. Final simplification 95.8%

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

Alternative 7: 94.2% accurate, 4.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1e-8)
   (fma (fma b b 4.0) (* b b) -1.0)
   (fma (* a a) (* a a) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1e-8) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = fma((a * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 1e-8)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1e-8], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1e-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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 99.9

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

   7. Simplified 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   9. Applied egg-rr 99.9%

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

   if 1e-8 < (*.f64 a a)

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.8

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

         \[\leadsto {a}^{\color{blue}{\left(3 + 1\right)}} - 1 \]

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.6

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

   7. Simplified 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 91.5

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

   9. Applied egg-rr 91.5%

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

Alternative 8: 93.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+87) (fma (* a a) (* a a) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+87) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+87)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+87], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999998e87

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

         \[\leadsto {a}^{\color{blue}{\left(3 + 1\right)}} - 1 \]

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 95.2

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

   7. Simplified 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 95.2

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

   9. Applied egg-rr 95.2%

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

   if 4.9999999999999998e87 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 96.5

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

   7. Simplified 96.5%

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

   8. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{b}^{3} \cdot b} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot {b}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot {b}^{3}} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.6

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

   10. Simplified 96.6%

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

Alternative 9: 82.7% accurate, 4.8× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1e-5) (fma b (* b 4.0) -1.0) (* a (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1e-5) {
		tmp = fma(b, (b * 4.0), -1.0);
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 1e-5)
		tmp = fma(b, Float64(b * 4.0), -1.0);
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1e-5], N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1.00000000000000008e-5

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 99.7

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

   7. Simplified 99.7%

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

   8. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 82.6

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

   10. Simplified 82.6%

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

   if 1.00000000000000008e-5 < (*.f64 a a)

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.8

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

         \[\leadsto {a}^{\color{blue}{\left(3 + 1\right)}} - 1 \]

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.4

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

   7. Simplified 91.4%

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

   8. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 90.9

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

   10. Simplified 90.9%

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

Alternative 10: 52.4% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 99.9

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 69.2

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

7. Simplified 69.2%

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

8. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 54.9

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

10. Simplified 54.9%

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

Alternative 11: 25.3% 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 99.9

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around inf

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

   1. metadata-eval N/A

      \[\leadsto {a}^{\color{blue}{\left(3 + 1\right)}} - 1 \]

   2. pow-plus N/A

      \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 74.4

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

7. Simplified 74.4%

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

8. Taylor expanded in a around 0

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

10. Simplified 29.6%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))