Bouland and Aaronson, Equation (24)

Percentage Accurate: 72.9% → 99.9%

Time: 10.3s

Alternatives: 12

Speedup: 5.5×

• 61.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 72.9% 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, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, -a, 4\right)\right), -1\right)\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (fma
  b
  (* b (fma b b (fma a (fma 2.0 a 4.0) 12.0)))
  (fma (* a a) (fma a a (fma 4.0 (- a) 4.0)) -1.0)))

C

double code(double a, double b) {
	return fma(b, (b * fma(b, b, fma(a, fma(2.0, a, 4.0), 12.0))), fma((a * a), fma(a, a, fma(4.0, -a, 4.0)), -1.0));
}

Julia

function code(a, b)
	return fma(b, Float64(b * fma(b, b, fma(a, fma(2.0, a, 4.0), 12.0))), fma(Float64(a * a), fma(a, a, fma(4.0, Float64(-a), 4.0)), -1.0))
end

Wolfram

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

Derivation

1. Initial program 68.3%

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

3. Taylor expanded in b around 0

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

5. Simplified 99.9%

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

Alternative 2: 51.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 0.0001:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
      0.0001)
   -1.0
   (* b (* b 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 0.0001) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (a + 3.0d0))))) <= 0.0001d0) then
        tmp = -1.0d0
    else
        tmp = b * (b * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 0.0001) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 0.0001:
		tmp = -1.0
	else:
		tmp = b * (b * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (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(a + 3.0))))) <= 0.0001)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 0.0001)
		tmp = -1.0;
	else
		tmp = b * (b * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], -1.0, N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 96.7%

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

   if 1.00000000000000005e-4 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 57.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 57.5

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

   5. Simplified 57.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{12 \cdot {b}^{2} - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 36.5

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

   8. Simplified 36.5%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{12 \cdot {b}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 12 \cdot \color{blue}{\left(b \cdot b\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(12 \cdot b\right) \cdot b} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \left(12 \cdot b\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(12 \cdot b\right)} \]

       5. *-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(b \cdot 12\right)} \]

       6. lower-*.f64 37.1

          \[\leadsto b \cdot \color{blue}{\left(b \cdot 12\right)} \]

   11. Simplified 37.1%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot 12\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 0.0001:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.1%

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 99.2

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

   8. Simplified 99.2%

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

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

   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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{{b}^{3}}, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \mathsf{neg}\left(a\right), 4\right)\right), -1\right)\right) \]
   7. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in a around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 99.2

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

   11. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 4: 99.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

3. Taylor expanded in b around 0

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

5. Simplified 99.9%

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

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{fma}\left(b, \color{blue}{{b}^{3}}, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \mathsf{neg}\left(a\right), 4\right)\right), -1\right)\right) \]
7. Step-by-step derivation

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 99.2

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

8. Simplified 99.2%

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

Alternative 5: 97.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \mathsf{fma}\left(a, a, 2 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{if}\;a \leq -49000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 520:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, b, 12\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (fma a a (* 2.0 (* b b)))))))
   (if (<= a -49000.0)
     t_0
     (if (<= a 520.0) (fma (* b (fma b b 12.0)) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * fma(a, a, (2.0 * (b * b))));
	double tmp;
	if (a <= -49000.0) {
		tmp = t_0;
	} else if (a <= 520.0) {
		tmp = fma((b * fma(b, b, 12.0)), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * fma(a, a, Float64(2.0 * Float64(b * b)))))
	tmp = 0.0
	if (a <= -49000.0)
		tmp = t_0;
	elseif (a <= 520.0)
		tmp = fma(Float64(b * fma(b, b, 12.0)), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a + N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -49000.0], t$95$0, If[LessEqual[a, 520.0], N[(N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -49000 or 520 < a

   1. Initial program 39.4%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

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

   8. Simplified 99.5%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 99.5

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

   11. Simplified 99.5%

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

   12. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. distribute-rgt-in N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 97.9

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

   14. Simplified 97.9%

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

   if -49000 < a < 520

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 98.9

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

   5. Simplified 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 98.9

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

   7. Applied egg-rr 98.9%

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

Alternative 6: 93.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+86)
   (fma (* a a) (fma a (+ a -4.0) 4.0) -1.0)
   (* b (* b (fma b b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+86) {
		tmp = fma((a * a), fma(a, (a + -4.0), 4.0), -1.0);
	} else {
		tmp = b * (b * fma(b, b, 12.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+86)
		tmp = fma(Float64(a * a), fma(a, Float64(a + -4.0), 4.0), -1.0);
	else
		tmp = Float64(b * Float64(b * fma(b, b, 12.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+86], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e86

   1. Initial program 75.4%

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 95.3

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

   5. Simplified 95.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 95.3

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

   8. Simplified 95.3%

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

   if 1e86 < (*.f64 b b)

   1. Initial program 58.7%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. lower-fma.f64 95.0

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

   11. Simplified 95.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+86)
   (fma (* a a) (fma a a 4.0) -1.0)
   (* b (* b (fma b b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+86) {
		tmp = fma((a * a), fma(a, a, 4.0), -1.0);
	} else {
		tmp = b * (b * fma(b, b, 12.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+86)
		tmp = fma(Float64(a * a), fma(a, a, 4.0), -1.0);
	else
		tmp = Float64(b * Float64(b * fma(b, b, 12.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+86], N[(N[(a * a), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e86

   1. Initial program 75.4%

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 95.3

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

   5. Simplified 95.3%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 94.7%

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

   if 1e86 < (*.f64 b b)

   1. Initial program 58.7%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. lower-fma.f64 95.0

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

   11. Simplified 95.0%

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

Alternative 8: 82.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+27}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 31000000000000:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 12, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.1e+27)
   (* a (* a (* a a)))
   (if (<= a 31000000000000.0) (fma b (* b 12.0) -1.0) (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.1e+27) {
		tmp = a * (a * (a * a));
	} else if (a <= 31000000000000.0) {
		tmp = fma(b, (b * 12.0), -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.1e+27)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	elseif (a <= 31000000000000.0)
		tmp = fma(b, Float64(b * 12.0), -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.1e+27], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 31000000000000.0], N[(b * N[(b * 12.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.09999999999999995e27

   1. Initial program 55.5%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

   if -2.09999999999999995e27 < a < 3.1e13

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{12 \cdot {b}^{2} - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 80.6

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

   8. Simplified 80.6%

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

   if 3.1e13 < a

   1. Initial program 25.5%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 94.0

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

   5. Simplified 94.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 94.0

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

   7. Applied egg-rr 94.0%

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

Alternative 9: 82.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -2.1 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 31000000000000:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 12, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -2.1e+27)
     t_0
     (if (<= a 31000000000000.0) (fma b (* b 12.0) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -2.1e+27) {
		tmp = t_0;
	} else if (a <= 31000000000000.0) {
		tmp = fma(b, (b * 12.0), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (a <= -2.1e+27)
		tmp = t_0;
	elseif (a <= 31000000000000.0)
		tmp = fma(b, Float64(b * 12.0), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.1e+27], t$95$0, If[LessEqual[a, 31000000000000.0], N[(b * N[(b * 12.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.09999999999999995e27 or 3.1e13 < a

   1. Initial program 38.2%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.9

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

   5. Simplified 92.9%

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

   if -2.09999999999999995e27 < a < 3.1e13

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{12 \cdot {b}^{2} - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 80.6

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

   8. Simplified 80.6%

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

Alternative 10: 68.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+291) (fma (* a a) 4.0 -1.0) (* b (* b 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+291) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+291)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(b * Float64(b * 12.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+291], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e291

   1. Initial program 71.7%

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 85.1

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

   5. Simplified 85.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 65.9%

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

   if 5.0000000000000001e291 < (*.f64 b b)

   1. Initial program 58.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{12 \cdot {b}^{2} - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 97.5

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

   8. Simplified 97.5%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{12 \cdot {b}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 12 \cdot \color{blue}{\left(b \cdot b\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(12 \cdot b\right) \cdot b} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \left(12 \cdot b\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(12 \cdot b\right)} \]

       5. *-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(b \cdot 12\right)} \]

       6. lower-*.f64 97.5

          \[\leadsto b \cdot \color{blue}{\left(b \cdot 12\right)} \]

   11. Simplified 97.5%

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

Alternative 11: 50.8% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.3%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
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 67.6

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

5. Simplified 67.6%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{12 \cdot {b}^{2} - 1} \]
7. Step-by-step derivation

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 51.8

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

8. Simplified 51.8%

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

Alternative 12: 24.1% 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 68.3%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
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 67.6

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

5. Simplified 67.6%

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

6. Taylor expanded in b around 0

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

8. Simplified 24.7%

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

Reproduce

herbie shell --seed 2024210 
(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))