Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.4% → 98.9%

Time: 9.8s

Alternatives: 11

Speedup: 5.5×

• 60.1% 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: 74.4% 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: 98.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \left(\frac{t\_0}{\frac{1}{t\_0}} + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right) + -1 \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))))
   (+ (+ (/ t_0 (/ 1.0 t_0)) (* 4.0 (* (* b b) 3.0))) -1.0)))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	return ((t_0 / (1.0 / t_0)) + (4.0 * ((b * b) * 3.0))) + -1.0;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	return Float64(Float64(Float64(t_0 / Float64(1.0 / t_0)) + Float64(4.0 * Float64(Float64(b * b) * 3.0))) + -1.0)
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(b * b), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Initial program 75.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. clear-num N/A

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

4. Applied rewrites 75.3%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \left(\frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{\frac{1}{\mathsf{fma}\left(a, a, b \cdot b\right)}} + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right) + -1 \]
9. Add Preprocessing

Alternative 2: 52.2% accurate, 0.9× speedup

Math

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

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)))))
      5e-12)
   -1.0
   (* b (* b 12.0))))

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))))) <= 5e-12) {
		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 (((((b * b) + (a * a)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (a + 3.0d0))))) <= 5d-12) 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(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 5e-12) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	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))))) <= 5e-12:
		tmp = -1.0
	else:
		tmp = b * (b * 12.0)
	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))))) <= 5e-12)
		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 (((((b * b) + (a * a)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 5e-12)
		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[(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], 5e-12], -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))))) < 4.9999999999999997e-12

   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.8%

      \[\leadsto -1 \]

   if 4.9999999999999997e-12 < (+.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 65.9%

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

   3. Taylor expanded in 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 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 1.3%

      \[\leadsto -1 \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 36.5%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 36.2%

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

4. Final simplification 52.2%

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

Reproduce

herbie shell --seed 2024229 
(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))