Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.5% → 99.9%

Time: 11.8s

Alternatives: 13

Speedup: 5.5×

• 60.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.5% 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, 2.6× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (fma
  b
  (fma (fma a (fma a 2.0 4.0) 12.0) 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, fma(fma(a, fma(a, 2.0, 4.0), 12.0), 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, fma(fma(a, fma(a, 2.0, 4.0), 12.0), 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[(N[(a * N[(a * 2.0 + 4.0), $MachinePrecision] + 12.0), $MachinePrecision] * b + N[(b * N[(b * b), $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 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

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

5. Simplified 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 99.5

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.5

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

7. Applied egg-rr 99.5%

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

Alternative 2: 51.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 10^{-13}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* b b) (* a a)) 2.0)
       (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
      1e-13)
   -1.0
   (* 12.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 1e-13) {
		tmp = -1.0;
	} else {
		tmp = 12.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((b * b) + (a * a)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (a + 3.0d0))))) <= 1d-13) then
        tmp = -1.0d0
    else
        tmp = 12.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 1e-13) {
		tmp = -1.0;
	} else {
		tmp = 12.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 1e-13:
		tmp = -1.0
	else:
		tmp = 12.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0))))) <= 1e-13)
		tmp = -1.0;
	else
		tmp = Float64(12.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((b * b) + (a * a)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 1e-13)
		tmp = -1.0;
	else
		tmp = 12.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $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], 1e-13], -1.0, N[(12.0 * N[(b * b), $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))))) < 1e-13

   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 b around 0

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

   5. Simplified 100.0%

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

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

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Simplified 99.7%

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

   if 1e-13 < (+.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 62.3%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 81.4%

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

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

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 35.2

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

   8. Simplified 35.2%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{12 \cdot {b}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 35.7

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

   11. Simplified 35.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 10^{-13}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 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

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

5. Simplified 99.5%

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

Alternative 4: 98.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(12, b \cdot b, -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b)))) (fma t_0 t_0 (fma 12.0 (* b b) -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	return fma(t_0, t_0, fma(12.0, (b * b), -1.0));
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	return fma(t_0, t_0, fma(12.0, Float64(b * b), -1.0))
end

Wolfram

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

Derivation

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

4. Applied egg-rr 39.0%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 42.0

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

7. Simplified 42.0%

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

9. Applied egg-rr 99.4%

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

Alternative 5: 82.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{elif}\;b \cdot b \leq 10^{+67}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-15)
   (fma (* a a) 4.0 -1.0)
   (if (<= (* b b) 1e+67) (* (* a a) (* a a)) (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-15) {
		tmp = fma((a * a), 4.0, -1.0);
	} else if ((b * b) <= 1e+67) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-15)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	elseif (Float64(b * b) <= 1e+67)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-15], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 1e+67], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 4.99999999999999999e-15

   1. Initial program 81.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 99.2%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in a around 0

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

   11. Simplified 75.8%

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

   if 4.99999999999999999e-15 < (*.f64 b b) < 9.99999999999999983e66

   1. Initial program 82.1%

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

   3. Taylor expanded in 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 59.5

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

   5. Simplified 59.5%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
   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 59.5

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

   7. Applied egg-rr 59.5%

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

   if 9.99999999999999983e66 < (*.f64 b b)

   1. Initial program 57.0%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 95.5

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

   5. Simplified 95.5%

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

Alternative 6: 82.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{elif}\;b \cdot b \leq 10^{+67}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-15)
   (fma (* a a) 4.0 -1.0)
   (if (<= (* b b) 1e+67) (* a (* a (* a a))) (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-15) {
		tmp = fma((a * a), 4.0, -1.0);
	} else if ((b * b) <= 1e+67) {
		tmp = a * (a * (a * a));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-15)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	elseif (Float64(b * b) <= 1e+67)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-15], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 1e+67], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 4.99999999999999999e-15

   1. Initial program 81.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 99.2%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in a around 0

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

   11. Simplified 75.8%

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

   if 4.99999999999999999e-15 < (*.f64 b b) < 9.99999999999999983e66

   1. Initial program 82.1%

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

   3. Taylor expanded in 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 59.5

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

   5. Simplified 59.5%

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

   if 9.99999999999999983e66 < (*.f64 b b)

   1. Initial program 57.0%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 95.5

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

   5. Simplified 95.5%

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

Alternative 7: 93.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+67)
   (fma (* a a) (fma a (+ a -4.0) 4.0) -1.0)
   (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+67) {
		tmp = fma((a * a), fma(a, (a + -4.0), 4.0), -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+67)
		tmp = fma(Float64(a * a), fma(a, Float64(a + -4.0), 4.0), -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+67], 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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999983e66

   1. Initial program 82.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 99.2%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 93.5

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

   8. Simplified 93.5%

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

   if 9.99999999999999983e66 < (*.f64 b b)

   1. Initial program 57.0%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 95.5

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

   5. Simplified 95.5%

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

Alternative 8: 94.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -600000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -600000.0)
     t_0
     (if (<= a 2.1e+19) (fma (* b b) (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 <= -600000.0) {
		tmp = t_0;
	} else if (a <= 2.1e+19) {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -600000.0)
		tmp = t_0;
	elseif (a <= 2.1e+19)
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -600000.0], t$95$0, If[LessEqual[a, 2.1e+19], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6e5 or 2.1e19 < a

   1. Initial program 44.4%

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

   3. Taylor expanded in a around 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 91.7

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

   5. Simplified 91.7%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
   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 91.7

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

   7. Applied egg-rr 91.7%

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

   if -6e5 < a < 2.1e19

   1. Initial program 99.1%

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

   3. Taylor expanded in 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 98.3

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

   5. Simplified 98.3%

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

Alternative 9: 98.3% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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

4. Applied egg-rr 39.0%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 42.0

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

7. Simplified 42.0%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in b around 0

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

12. Simplified 98.2%

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

Alternative 10: 82.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -2300:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6500:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 12, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -2300.0) t_0 (if (<= a 6500.0) (fma (* b 12.0) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -2300.0) {
		tmp = t_0;
	} else if (a <= 6500.0) {
		tmp = fma((b * 12.0), b, -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 <= -2300.0)
		tmp = t_0;
	elseif (a <= 6500.0)
		tmp = fma(Float64(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), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2300.0], t$95$0, If[LessEqual[a, 6500.0], N[(N[(b * 12.0), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2300 or 6500 < a

   1. Initial program 46.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 89.3

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

   5. Simplified 89.3%

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

   if -2300 < a < 6500

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 74.4%

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

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

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.4

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

   8. Simplified 74.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 74.4

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

   10. Applied egg-rr 74.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2300:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 6500:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 12, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+292) (fma (* a a) 4.0 -1.0) (* 12.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+292) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = 12.0 * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+292)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(12.0 * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+292], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e292

   1. Initial program 80.5%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 99.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 80.0

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

   8. Simplified 80.0%

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

   9. Taylor expanded in a around 0

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

   11. Simplified 59.7%

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

   if 1e292 < (*.f64 b b)

   1. Initial program 46.2%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 100.0%

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

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

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.2

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

   8. Simplified 97.2%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{12 \cdot {b}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 97.2

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

   11. Simplified 97.2%

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

Alternative 12: 52.0% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

5. Simplified 86.1%

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

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

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 51.4

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

8. Simplified 51.4%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 51.4

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

10. Applied egg-rr 51.4%

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

11. Final simplification 51.4%

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

Alternative 13: 25.1% 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 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

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

5. Simplified 86.1%

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

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

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 51.4

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

8. Simplified 51.4%

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

9. Taylor expanded in b around 0

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

11. Simplified 25.5%

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

Reproduce

herbie shell --seed 2024208 
(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))