Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 99.0%

Time: 12.4s

Alternatives: 16

Speedup: 5.9×

• 61.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 76.0%

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

   1. unpow2 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. clear-num N/A

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

   9. flip-+ N/A

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

   10. /-lowering-/.f64 N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 76.0

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

4. Applied egg-rr 76.0%

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

5. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 99.2

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 2: 51.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* b b) (* a a)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
      1e-9)
   -1.0
   (* 4.0 (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 1e-9) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (a * a);
	}
	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) * (a + 1.0d0)) + ((b * b) * (1.0d0 - (a * 3.0d0)))))) <= 1d-9) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (a * a)
    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) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 1e-9) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (a * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 1e-9:
		tmp = -1.0
	else:
		tmp = 4.0 * (a * a)
	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(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= 1e-9)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(a * a));
	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) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 1e-9)
		tmp = -1.0;
	else
		tmp = 4.0 * (a * a);
	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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-9], -1.0, N[(4.0 * N[(a * a), $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 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < 1.00000000000000006e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 99.2%

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

   if 1.00000000000000006e-9 < (+.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 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 68.2%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. accelerator-lowering-fma.f64 55.1

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

   5. Simplified 55.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 31.5%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{4 \cdot {a}^{2}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{4 \cdot {a}^{2}} \]

       2. unpow2 N/A

          \[\leadsto 4 \cdot \color{blue}{\left(a \cdot a\right)} \]

       3. *-lowering-*.f64 32.0

          \[\leadsto 4 \cdot \color{blue}{\left(a \cdot a\right)} \]

   11. Simplified 32.0%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000016e-5

   1. Initial program 88.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-+r+ N/A

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

      6. distribute-rgt-out N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-lowering-+.f64 99.5

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

   7. Applied egg-rr 99.5%

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

   if 2.00000000000000016e-5 < (*.f64 b b)

   1. Initial program 65.8%

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

      1. unpow2 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 65.8

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

   4. Applied egg-rr 65.8%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 96.4

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

   10. Simplified 96.4%

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

4. Final simplification 97.8%

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

Alternative 4: 97.7% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-5)
   (fma (* a (fma a (+ a 4.0) 4.0)) a -1.0)
   (* (* b b) (fma b b (fma 2.0 (* a a) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-5) {
		tmp = fma((a * fma(a, (a + 4.0), 4.0)), a, -1.0);
	} else {
		tmp = (b * b) * fma(b, b, fma(2.0, (a * a), 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-5)
		tmp = fma(Float64(a * fma(a, Float64(a + 4.0), 4.0)), a, -1.0);
	else
		tmp = Float64(Float64(b * b) * fma(b, b, fma(2.0, Float64(a * a), 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-5], N[(N[(a * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(2.0 * N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000016e-5

   1. Initial program 88.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-+r+ N/A

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

      6. distribute-rgt-out N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-lowering-+.f64 99.5

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

   7. Applied egg-rr 99.5%

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

   if 2.00000000000000016e-5 < (*.f64 b b)

   1. Initial program 65.8%

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

      1. unpow2 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 65.8

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

   4. Applied egg-rr 65.8%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in b around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

   10. Simplified 96.3%

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

Alternative 5: 97.6% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000016e-5

   1. Initial program 88.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-+r+ N/A

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

      6. distribute-rgt-out N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-lowering-+.f64 99.5

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

   7. Applied egg-rr 99.5%

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

   if 2.00000000000000016e-5 < (*.f64 b b)

   1. Initial program 65.8%

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

      1. unpow2 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 65.8

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

   4. Applied egg-rr 65.8%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 96.4

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

   10. Simplified 96.4%

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

   11. Taylor expanded in b around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*l* N/A

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

       5. associate-*l/ N/A

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

       6. *-lft-identity N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

       9. distribute-rgt-out N/A

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

       10. metadata-eval N/A

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

       11. pow-sqr N/A

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

       12. associate-/l* N/A

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

       13. *-inverses N/A

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

       14. *-rgt-identity N/A

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

       15. *-rgt-identity N/A

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

       16. metadata-eval N/A

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

   13. Simplified 96.1%

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

Alternative 6: 94.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(a, a + 4, 4\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, b, 4\right), b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0004)
   (fma (* a (fma a (+ a 4.0) 4.0)) a -1.0)
   (fma (* b (fma b b 4.0)) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0004) {
		tmp = fma((a * fma(a, (a + 4.0), 4.0)), a, -1.0);
	} else {
		tmp = fma((b * fma(b, b, 4.0)), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0004)
		tmp = fma(Float64(a * fma(a, Float64(a + 4.0), 4.0)), a, -1.0);
	else
		tmp = fma(Float64(b * fma(b, b, 4.0)), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0004], N[(N[(a * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000019e-4

   1. Initial program 88.1%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-+r+ N/A

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

      6. distribute-rgt-out N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-lowering-+.f64 99.5

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

   7. Applied egg-rr 99.5%

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

   if 4.00000000000000019e-4 < (*.f64 b b)

   1. Initial program 65.6%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 92.8

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

   5. Simplified 92.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. accelerator-lowering-fma.f64 92.8

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

   7. Applied egg-rr 92.8%

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

Alternative 7: 93.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -8.8e+15) t_0 (if (<= a 5e+35) (fma (* b b) (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -8.8e+15) {
		tmp = t_0;
	} else if (a <= 5e+35) {
		tmp = fma((b * b), (b * 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 <= -8.8e+15)
		tmp = t_0;
	elseif (a <= 5e+35)
		tmp = fma(Float64(b * b), Float64(b * 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, -8.8e+15], t$95$0, If[LessEqual[a, 5e+35], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.8e15 or 5.00000000000000021e35 < a

   1. Initial program 46.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 92.1

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

   5. Simplified 92.1%

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

   if -8.8e15 < a < 5.00000000000000021e35

   1. Initial program 97.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 95.3

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

   5. Simplified 95.3%

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

   6. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 94.1

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

   8. Simplified 94.1%

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

Alternative 8: 93.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, b, 4\right), b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0004)
   (fma (* a a) (fma a a 4.0) -1.0)
   (fma (* b (fma b b 4.0)) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0004) {
		tmp = fma((a * a), fma(a, a, 4.0), -1.0);
	} else {
		tmp = fma((b * fma(b, b, 4.0)), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0004)
		tmp = fma(Float64(a * a), fma(a, a, 4.0), -1.0);
	else
		tmp = fma(Float64(b * fma(b, b, 4.0)), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0004], N[(N[(a * a), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000019e-4

   1. Initial program 88.1%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 98.1%

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

   if 4.00000000000000019e-4 < (*.f64 b b)

   1. Initial program 65.6%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 92.8

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

   5. Simplified 92.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. accelerator-lowering-fma.f64 92.8

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

   7. Applied egg-rr 92.8%

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

Alternative 9: 93.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0004)
   (fma (* a a) (fma a a 4.0) -1.0)
   (fma (* b b) (fma b b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0004) {
		tmp = fma((a * a), fma(a, a, 4.0), -1.0);
	} else {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0004)
		tmp = fma(Float64(a * a), fma(a, a, 4.0), -1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0004], N[(N[(a * a), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000019e-4

   1. Initial program 88.1%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 98.1%

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

   if 4.00000000000000019e-4 < (*.f64 b b)

   1. Initial program 65.6%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 92.8

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

   5. Simplified 92.8%

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

Alternative 10: 93.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0004)
   (fma (* a a) (fma a a 4.0) -1.0)
   (* (* b b) (fma b b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0004) {
		tmp = fma((a * a), fma(a, a, 4.0), -1.0);
	} else {
		tmp = (b * b) * fma(b, b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0004)
		tmp = fma(Float64(a * a), fma(a, a, 4.0), -1.0);
	else
		tmp = Float64(Float64(b * b) * fma(b, b, 4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0004], N[(N[(a * a), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000019e-4

   1. Initial program 88.1%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 98.1%

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

   if 4.00000000000000019e-4 < (*.f64 b b)

   1. Initial program 65.6%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 92.8

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

   5. Simplified 92.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-+r- N/A

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

      6. --lowering--.f64 N/A

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

      7. associate-*l* N/A

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

      8. cube-unmult N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. cube-unmult N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, \frac{1}{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), \color{blue}{16}\right) - \left(b \cdot b\right) \cdot 4}{{\left(b \cdot b\right)}^{3} + {4}^{3}}}, -1\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, \frac{1}{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), 16\right) - \color{blue}{\left(b \cdot b\right) \cdot 4}}{{\left(b \cdot b\right)}^{3} + {4}^{3}}}, -1\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, \frac{1}{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), 16\right) - \color{blue}{\left(b \cdot b\right)} \cdot 4}{{\left(b \cdot b\right)}^{3} + {4}^{3}}}, -1\right) \]

   7. Applied egg-rr 12.3%

      \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), 16\right) - \left(b \cdot b\right) \cdot 4}{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), b \cdot \left(b \cdot b\right), 64\right)}}}, -1\right) \]

   8. Taylor expanded in b around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

         \[\leadsto 4 \cdot \color{blue}{{b}^{2}} + 1 \cdot {b}^{4} \]

      9. *-lft-identity N/A

         \[\leadsto 4 \cdot {b}^{2} + \color{blue}{{b}^{4}} \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

          \[\leadsto 4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      12. distribute-rgt-in N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-commutative N/A

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

      17. unpow2 N/A

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

      18. accelerator-lowering-fma.f64 92.4

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

   10. Simplified 92.4%

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

Alternative 11: 82.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -66000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 320:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -66000000.0) t_0 (if (<= a 320.0) (fma (* b b) 4.0 -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -66000000.0) {
		tmp = t_0;
	} else if (a <= 320.0) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (a <= -66000000.0)
		tmp = t_0;
	elseif (a <= 320.0)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -66000000.0], t$95$0, If[LessEqual[a, 320.0], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.6e7 or 320 < a

   1. Initial program 50.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 86.1

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

   5. Simplified 86.1%

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

   if -6.6e7 < a < 320

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 77.3%

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

Alternative 12: 82.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0004) (fma 4.0 (* a a) -1.0) (* (* b b) (fma b b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0004) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = (b * b) * fma(b, b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0004)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(b * b) * fma(b, b, 4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0004], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000019e-4

   1. Initial program 88.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(3 + 1\right)}} + 4 \cdot {b}^{2}\right) + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) - 1\right) \]

      6. pow-plus N/A

         \[\leadsto \left(\color{blue}{{b}^{3} \cdot b} + 4 \cdot {b}^{2}\right) + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) - 1\right) \]

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 75.1%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.7

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

   8. Simplified 74.7%

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

   if 4.00000000000000019e-4 < (*.f64 b b)

   1. Initial program 65.6%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 92.8

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

   5. Simplified 92.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-+r- N/A

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

      6. --lowering--.f64 N/A

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

      7. associate-*l* N/A

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

      8. cube-unmult N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. cube-unmult N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, \frac{1}{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), \color{blue}{16}\right) - \left(b \cdot b\right) \cdot 4}{{\left(b \cdot b\right)}^{3} + {4}^{3}}}, -1\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, \frac{1}{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), 16\right) - \color{blue}{\left(b \cdot b\right) \cdot 4}}{{\left(b \cdot b\right)}^{3} + {4}^{3}}}, -1\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, \frac{1}{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), 16\right) - \color{blue}{\left(b \cdot b\right)} \cdot 4}{{\left(b \cdot b\right)}^{3} + {4}^{3}}}, -1\right) \]

   7. Applied egg-rr 12.3%

      \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), 16\right) - \left(b \cdot b\right) \cdot 4}{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), b \cdot \left(b \cdot b\right), 64\right)}}}, -1\right) \]

   8. Taylor expanded in b around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

         \[\leadsto 4 \cdot \color{blue}{{b}^{2}} + 1 \cdot {b}^{4} \]

      9. *-lft-identity N/A

         \[\leadsto 4 \cdot {b}^{2} + \color{blue}{{b}^{4}} \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

          \[\leadsto 4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      12. distribute-rgt-in N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-commutative N/A

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

      17. unpow2 N/A

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

      18. accelerator-lowering-fma.f64 92.4

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

   10. Simplified 92.4%

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

Alternative 13: 82.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0004) (fma 4.0 (* a a) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0004) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0004)
		tmp = fma(4.0, Float64(a * a), -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], 0.0004], N[(4.0 * N[(a * a), $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) < 4.00000000000000019e-4

   1. Initial program 88.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(3 + 1\right)}} + 4 \cdot {b}^{2}\right) + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) - 1\right) \]

      6. pow-plus N/A

         \[\leadsto \left(\color{blue}{{b}^{3} \cdot b} + 4 \cdot {b}^{2}\right) + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) - 1\right) \]

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 75.1%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.7

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

   8. Simplified 74.7%

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

   if 4.00000000000000019e-4 < (*.f64 b b)

   1. Initial program 65.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. *-lowering-*.f64 N/A

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

      6. *-lowering-*.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. *-lowering-*.f64 91.9

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

   5. Simplified 91.9%

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

Alternative 14: 69.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.4 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4.4e+307) (fma 4.0 (* a a) -1.0) (fma (* b b) 4.0 -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.4e+307) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = fma((b * b), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4.4e+307)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = fma(Float64(b * b), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4.4e+307], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.3999999999999999e307

   1. Initial program 85.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(3 + 1\right)}} + 4 \cdot {b}^{2}\right) + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) - 1\right) \]

      6. pow-plus N/A

         \[\leadsto \left(\color{blue}{{b}^{3} \cdot b} + 4 \cdot {b}^{2}\right) + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) - 1\right) \]

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 80.8%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 56.2

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

   8. Simplified 56.2%

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

   if 4.3999999999999999e307 < (*.f64 b b)

   1. Initial program 53.8%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 100.0%

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

Alternative 15: 51.3% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

      \[\leadsto \left({b}^{\color{blue}{\left(3 + 1\right)}} + 4 \cdot {b}^{2}\right) + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) - 1\right) \]

   6. pow-plus N/A

      \[\leadsto \left(\color{blue}{{b}^{3} \cdot b} + 4 \cdot {b}^{2}\right) + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) - 1\right) \]

   7. cube-unmult N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt-out N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

5. Simplified 80.0%

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

6. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 48.1

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

8. Simplified 48.1%

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

Alternative 16: 24.8% accurate, 160.0× 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 76.0%

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

3. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-out N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. accelerator-lowering-fma.f64 74.4

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

5. Simplified 74.4%

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

6. Taylor expanded in b around 0

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

8. Simplified 25.0%

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

Reproduce

herbie shell --seed 2024201 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))