Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.9% → 99.9%

Time: 7.3s

Alternatives: 10

Speedup: 5.2×

• 59.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a - 4, a, 4\right)\\ \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), b \cdot b, \mathsf{fma}\left(a \cdot a, t\_0, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (- a 4.0) a 4.0)))
   (if (<= (* b b) 2e-49)
     (fma (* t_0 a) a -1.0)
     (fma
      (fma b b (fma a (fma 2.0 a 4.0) 12.0))
      (* b b)
      (fma (* a a) t_0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma((a - 4.0), a, 4.0);
	double tmp;
	if ((b * b) <= 2e-49) {
		tmp = fma((t_0 * a), a, -1.0);
	} else {
		tmp = fma(fma(b, b, fma(a, fma(2.0, a, 4.0), 12.0)), (b * b), fma((a * a), t_0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a - 4.0), a, 4.0)
	tmp = 0.0
	if (Float64(b * b) <= 2e-49)
		tmp = fma(Float64(t_0 * a), a, -1.0);
	else
		tmp = fma(fma(b, b, fma(a, fma(2.0, a, 4.0), 12.0)), Float64(b * b), fma(Float64(a * a), t_0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 2e-49], N[(N[(t$95$0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b + N[(a * N[(2.0 * a + 4.0), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.99999999999999987e-49

   1. Initial program 89.2%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

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

   if 1.99999999999999987e-49 < (*.f64 b b)

   1. Initial program 69.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) + {a}^{4}\right) + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)} - 1 \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right)} - 1 \]

      3. +-commutative N/A

         \[\leadsto \left({b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) + \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}\right) - 1 \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) \cdot {b}^{2}} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right), {b}^{2}, \left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right)} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), b \cdot b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(1 - a, 4, a \cdot a\right), -1\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), b \cdot b, \mathsf{fma}\left(a \cdot a, 4 + a \cdot \left(a - 4\right), -1\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), b \cdot b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right)\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e-18)
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (fma (* (fma b b (fma (fma 2.0 a 4.0) a 12.0)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e-18) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, fma(fma(2.0, a, 4.0), a, 12.0)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e-18)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, fma(fma(2.0, a, 4.0), a, 12.0)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e-18], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e-18

   1. Initial program 89.5%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

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

   if 4.0000000000000003e-18 < (*.f64 b b)

   1. Initial program 67.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) + {a}^{4}\right) + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)} - 1 \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right)} - 1 \]

      3. +-commutative N/A

         \[\leadsto \left({b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) + \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}\right) - 1 \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right)\right) \cdot {b}^{2}} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot {a}^{2} + \left(4 \cdot \left(3 + a\right) + {b}^{2}\right), {b}^{2}, \left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right)} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), b \cdot b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(1 - a, 4, a \cdot a\right), -1\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), b \cdot b, -1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), b \cdot b, -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 97.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right)\right) \cdot b, b, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1000000.0)
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000.0) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1000000.0)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1000000.0], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e6

   1. Initial program 88.4%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.4%

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

   10. Applied rewrites 99.5%

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

   if 1e6 < (*.f64 b b)

   1. Initial program 67.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(12 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(12 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left({b}^{2} + 12\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {b}^{2} \cdot \left({b}^{2} + 12\right) + \color{blue}{-1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} + 12, -1\right)} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      10. unpow2 N/A

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

      11. lower-fma.f64 91.2

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

   5. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 91.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, \color{blue}{b}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1000000.0)
   (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000.0) {
		tmp = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1000000.0)
		tmp = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1000000.0], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e6

   1. Initial program 88.4%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.4%

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

   if 1e6 < (*.f64 b b)

   1. Initial program 67.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(12 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(12 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left({b}^{2} + 12\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {b}^{2} \cdot \left({b}^{2} + 12\right) + \color{blue}{-1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} + 12, -1\right)} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      10. unpow2 N/A

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

      11. lower-fma.f64 91.2

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

   5. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 91.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, \color{blue}{b}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.95 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.95e+90)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 2.4e+153)
     (fma (* (fma b b 12.0) b) b -1.0)
     (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.95e+90) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 2.4e+153) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.95e+90)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 2.4e+153)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.95e+90], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.4e+153], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.9499999999999998e90

   1. Initial program 57.1%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.4%

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

   if -3.9499999999999998e90 < a < 2.39999999999999992e153

   1. Initial program 92.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(12 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(12 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left({b}^{2} + 12\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {b}^{2} \cdot \left({b}^{2} + 12\right) + \color{blue}{-1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} + 12, -1\right)} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      10. unpow2 N/A

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

      11. lower-fma.f64 87.4

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

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.4%

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

   if 2.39999999999999992e153 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.2%

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

Alternative 6: 87.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.95 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.95e+90)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 2.4e+153)
     (fma (* b b) (fma b b 12.0) -1.0)
     (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.95e+90) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 2.4e+153) {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.95e+90)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 2.4e+153)
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.95e+90], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.4e+153], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.9499999999999998e90

   1. Initial program 57.1%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.4%

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

   if -3.9499999999999998e90 < a < 2.39999999999999992e153

   1. Initial program 92.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(12 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(12 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left({b}^{2} + 12\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {b}^{2} \cdot \left({b}^{2} + 12\right) + \color{blue}{-1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} + 12, -1\right)} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      10. unpow2 N/A

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

      11. lower-fma.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if 2.39999999999999992e153 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.2%

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

Alternative 7: 72.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.95 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(12 \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.95e+90)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 2.4e+153) (fma (* 12.0 b) b -1.0) (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.95e+90) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 2.4e+153) {
		tmp = fma((12.0 * b), b, -1.0);
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.95e+90)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 2.4e+153)
		tmp = fma(Float64(12.0 * b), b, -1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.95e+90], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.4e+153], N[(N[(12.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.9499999999999998e90

   1. Initial program 57.1%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.4%

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

   if -3.9499999999999998e90 < a < 2.39999999999999992e153

   1. Initial program 92.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(12 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(12 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left({b}^{2} + 12\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {b}^{2} \cdot \left({b}^{2} + 12\right) + \color{blue}{-1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} + 12, -1\right)} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      10. unpow2 N/A

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

      11. lower-fma.f64 87.4

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

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.2%

       \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]

   if 2.39999999999999992e153 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.2%

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

Alternative 8: 69.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(12 \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+280) (fma (* a a) 4.0 -1.0) (fma (* 12.0 b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+280) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = fma((12.0 * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+280)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(Float64(12.0 * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+280], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(12.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000002e280

   1. Initial program 83.3%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.9%

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

   if 5.0000000000000002e280 < (*.f64 b b)

   1. Initial program 63.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(12 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(12 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left({b}^{2} + 12\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {b}^{2} \cdot \left({b}^{2} + 12\right) + \color{blue}{-1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} + 12, -1\right)} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

      10. unpow2 N/A

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

      11. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 97.4%

       \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(12 \cdot b, b, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma (* 12.0 b) b -1.0))

C

double code(double a, double b) {
	return fma((12.0 * b), b, -1.0);
}

Julia

function code(a, b)
	return fma(Float64(12.0 * b), b, -1.0)
end

Wolfram

code[a_, b_] := N[(N[(12.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \left(12 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

   3. pow-sqr N/A

      \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(12 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

   5. +-commutative N/A

      \[\leadsto {b}^{2} \cdot \color{blue}{\left({b}^{2} + 12\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto {b}^{2} \cdot \left({b}^{2} + 12\right) + \color{blue}{-1} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} + 12, -1\right)} \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} + 12, -1\right) \]

   10. unpow2 N/A

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

   11. lower-fma.f64 73.6

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

5. Applied rewrites 73.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 73.6%

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

8. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]
9. Step-by-step derivation

10. Applied rewrites 56.4%

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

Alternative 10: 25.4% accurate, 155.0× 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 78.0%

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

3. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 28.8%

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

Reproduce

herbie shell --seed 2024323 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))