Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 12

Speedup: N/A×

• 60.5% 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, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	return fma(t_0, t_0, fma(4.0, Float64(b * b), -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 + N[(4.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. sub-neg N/A

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

   16. lift-*.f64 N/A

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

   17. lower-fma.f64 N/A

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

   18. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 82.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{elif}\;a \cdot a \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e-11)
   (fma (* b b) 4.0 -1.0)
   (if (<= (* a a) 2e+55) (* (* b b) (* b b)) (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-11) {
		tmp = fma((b * b), 4.0, -1.0);
	} else if ((a * a) <= 2e+55) {
		tmp = (b * b) * (b * b);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5e-11)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	elseif (Float64(a * a) <= 2e+55)
		tmp = Float64(Float64(b * b) * Float64(b * b));
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5e-11], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], If[LessEqual[N[(a * a), $MachinePrecision], 2e+55], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a a) < 5.00000000000000018e-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

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

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 82.7%

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

   if 5.00000000000000018e-11 < (*.f64 a a) < 2.00000000000000002e55

   1. Initial program 99.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.2

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

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 65.2%

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

   if 2.00000000000000002e55 < (*.f64 a a)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 92.8

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 92.8%

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

Alternative 3: 98.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a + a, 4\right)\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-20)
   (fma (* a a) (* a a) -1.0)
   (fma (fma b b (fma a (+ a a) 4.0)) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-20) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma(fma(b, b, fma(a, (a + a), 4.0)), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-20)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(fma(b, b, fma(a, Float64(a + a), 4.0)), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-20], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b + N[(a * N[(a + a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999945e-21

   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 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. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

   if 9.99999999999999945e-21 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-in N/A

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

       5. +-commutative N/A

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

       6. metadata-eval N/A

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

       7. pow-sqr N/A

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

       8. distribute-lft-in N/A

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

       9. associate-+r+ N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

   11. Applied rewrites 98.9%

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

Alternative 4: 97.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 2 \cdot \left(a \cdot a\right)\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-9)
   (fma (* a a) (* a a) -1.0)
   (* (* (fma b b (* 2.0 (* a a))) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-9) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = (fma(b, b, (2.0 * (a * a))) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-9)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(fma(b, b, Float64(2.0 * Float64(a * a))) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-9], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e-9

   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 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. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   if 5.0000000000000001e-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 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. Applied rewrites 98.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 2 \cdot \left(a \cdot a\right)\right) \cdot b\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-9)
   (fma (* a a) (* a a) -1.0)
   (fma (fma b b (* a a)) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-9) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma(fma(b, b, (a * a)), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-9)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(fma(b, b, Float64(a * a)), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-9], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e-9

   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 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. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   if 5.0000000000000001e-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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. sub-neg N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.4

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

   10. Applied rewrites 98.4%

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

Alternative 6: 94.2% accurate, 4.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+17)
   (fma (* a a) (* a a) -1.0)
   (fma (* b b) (fma b b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+17) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+17)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+17], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e17

   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 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. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   if 1e17 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 92.4

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

   5. Applied rewrites 92.4%

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

Alternative 7: 99.2% 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. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. sub-neg N/A

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

   16. lift-*.f64 N/A

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

   17. lower-fma.f64 N/A

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

   18. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 99.8%

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

Alternative 8: 94.2% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+17) (fma (* a a) (* a a) -1.0) (fma (* b b) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+17) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma((b * b), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+17)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+17], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e17

   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 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. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   if 1e17 < (*.f64 b b)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 92.4%

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

Alternative 9: 94.2% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+17) (fma (* a a) (* a a) -1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+17) {
		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) <= 1e+17)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+17], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e17

   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 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. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   if 1e17 < (*.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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

Alternative 10: 82.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 350000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 350000000000.0) (fma (* b b) 4.0 -1.0) (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 350000000000.0) {
		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) <= 350000000000.0)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 350000000000.0], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 3.5e11

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 81.0%

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

   if 3.5e11 < (*.f64 a a)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

   7. Applied rewrites 87.7%

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

Alternative 11: 51.1% 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

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

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

   3. pow-sqr N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 74.7

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 58.3%

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

Alternative 12: 24.6% accurate, 131.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 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. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 74.7

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 34.8%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))