Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 13.8s

Alternatives: 16

Speedup: 3.3×

• 60.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 1.0;
end

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 1.0;
end

Wolfram

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

Alternative 1: 99.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. distribute-lft-in N/A

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

   3. pow-sqr N/A

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

   4. metadata-eval N/A

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

   5. associate-+r+ N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. distribute-lft-in N/A

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

   3. pow-sqr N/A

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

   4. metadata-eval N/A

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

   5. associate-+r+ N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

5. Simplified 99.9%

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. associate-*r* N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 99.9

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

7. Applied egg-rr 99.9%

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

Alternative 3: 98.4% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-9)
   (fma a (* a (* a a)) -1.0)
   (fma (* b b) (fma b b (fma 2.0 (* a a) 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-9) {
		tmp = fma(a, (a * (a * a)), -1.0);
	} else {
		tmp = fma((b * b), fma(b, b, fma(2.0, (a * a), 4.0)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-9)
		tmp = fma(a, Float64(a * Float64(a * a)), -1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, fma(2.0, Float64(a * a), 4.0)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-9], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(2.0 * N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.00000000000000006e-9

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 99.7

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

   5. Simplified 99.7%

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

   if 1.00000000000000006e-9 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 95.7%

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

Alternative 4: 99.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sub-neg N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 N/A

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

   14. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 99.9

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

6. Applied egg-rr 99.9%

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

Alternative 5: 99.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sub-neg N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 N/A

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

   14. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 6: 97.7% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+19)
   (fma a (* a (* a a)) -1.0)
   (* b (* b (fma b b (* 2.0 (* a a)))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+19) {
		tmp = fma(a, (a * (a * a)), -1.0);
	} else {
		tmp = b * (b * fma(b, b, (2.0 * (a * a))));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+19)
		tmp = fma(a, Float64(a * Float64(a * a)), -1.0);
	else
		tmp = Float64(b * Float64(b * fma(b, b, Float64(2.0 * Float64(a * a)))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+19], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b + N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e19

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 98.9

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

   5. Simplified 98.9%

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

   if 1e19 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

   5. Simplified 96.4%

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

   6. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 96.4

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

   8. Simplified 96.4%

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

Alternative 7: 97.7% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+19)
   (fma a (* a (* a a)) -1.0)
   (fma (fma a a (* b b)) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+19) {
		tmp = fma(a, (a * (a * a)), -1.0);
	} else {
		tmp = fma(fma(a, a, (b * b)), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+19)
		tmp = fma(a, Float64(a * Float64(a * a)), -1.0);
	else
		tmp = fma(fma(a, a, Float64(b * b)), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+19], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e19

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 98.9

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

   5. Simplified 98.9%

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

   if 1e19 < (*.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. associate--l+ N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 99.9%

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

   8. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 96.0

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

   10. Simplified 96.0%

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

Alternative 8: 93.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(a \cdot a\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1e+127)
   (fma b (* b (fma b b 4.0)) -1.0)
   (fma a (* a (* a a)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1e+127) {
		tmp = fma(b, (b * fma(b, b, 4.0)), -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+127)
		tmp = fma(b, Float64(b * fma(b, b, 4.0)), -1.0);
	else
		tmp = fma(a, Float64(a * Float64(a * a)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1e+127], N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 9.99999999999999955e126

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-out N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. metadata-eval 94.9

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

   5. Simplified 94.9%

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

   if 9.99999999999999955e126 < (*.f64 a a)

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 98.0

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

   5. Simplified 98.0%

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

Alternative 9: 99.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sub-neg N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 N/A

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

   14. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in b around 0

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

7. Simplified 99.3%

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

   1. accelerator-lowering-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 99.3

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

9. Applied egg-rr 99.3%

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

Alternative 10: 99.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sub-neg N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 N/A

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

   14. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in b around 0

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

7. Simplified 99.3%

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

Alternative 11: 92.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(a \cdot a\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1e+127)
   (fma (* b b) (* b b) -1.0)
   (fma a (* a (* a a)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1e+127) {
		tmp = fma((b * b), (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+127)
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	else
		tmp = fma(a, Float64(a * Float64(a * a)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1e+127], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 9.99999999999999955e126

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. pow-sqr N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 94.9

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

   8. Simplified 94.9%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 94.0

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

   11. Simplified 94.0%

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

   if 9.99999999999999955e126 < (*.f64 a a)

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 98.0

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

   5. Simplified 98.0%

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

Alternative 12: 94.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(a \cdot a\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+62) (fma a (* a (* a a)) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+62) {
		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) <= 2e+62)
		tmp = fma(a, Float64(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], 2e+62], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000007e62

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 97.6

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

   5. Simplified 97.6%

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

   if 2.00000000000000007e62 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 92.3

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

   5. Simplified 92.3%

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

Alternative 13: 82.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2e-11) (fma (* b b) 4.0 -1.0) (* a (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2e-11) {
		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) <= 2e-11)
		tmp = fma(Float64(b * 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], 2e-11], N[(N[(b * b), $MachinePrecision] * 4.0 + -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) < 1.99999999999999988e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. pow-sqr N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Simplified 83.0%

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

   if 1.99999999999999988e-11 < (*.f64 a a)

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 90.6

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

   5. Simplified 90.6%

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

Alternative 14: 51.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.24:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 0.24) -1.0 (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.24) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.24d0) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.24) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.24:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.24)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.24)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.24], -1.0, N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.23999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 58.6%

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

   if 0.23999999999999999 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. pow-sqr N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 87.9

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

   8. Simplified 87.9%

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

   9. Taylor expanded in b around 0

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

   11. Simplified 53.0%

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

   12. Taylor expanded in b around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 53.0

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

   14. Simplified 53.0%

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

Alternative 15: 51.4% accurate, 10.9× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. distribute-lft-in N/A

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

   3. pow-sqr N/A

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

   4. metadata-eval N/A

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

   5. associate-+r+ N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

5. Simplified 99.9%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. accelerator-lowering-fma.f64 72.7

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

8. Simplified 72.7%

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

9. Taylor expanded in b around 0

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

11. Simplified 56.4%

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

Alternative 16: 25.7% accurate, 131.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. metadata-eval 70.6

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

5. Simplified 70.6%

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

6. Taylor expanded in a around 0

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

8. Simplified 31.9%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))