Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.7% → 99.1%

Time: 13.4s

Alternatives: 13

Speedup: 5.7×

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

Specification

Math

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

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: 73.7% 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: 99.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

4. Applied egg-rr 74.9%

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

5. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 99.2

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

7. Simplified 99.2%

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

Alternative 2: 51.0% accurate, 0.9× speedup

Math

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

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))))))
      5e-5)
   -1.0
   (* b (* b 4.0))))

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)))))) <= 5e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((b * b) + (a * a)) ** 2.0d0) + (4.0d0 * (((a * a) * (a + 1.0d0)) + ((b * b) * (1.0d0 - (a * 3.0d0)))))) <= 5d-5) then
        tmp = -1.0d0
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 5e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	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)))))) <= 5e-5:
		tmp = -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= 5e-5)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((b * b) + (a * a)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 5e-5)
		tmp = -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-5], -1.0, N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < 5.00000000000000024e-5

   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. lower-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. lower-*.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. lower-fma.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 96.7%

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

   if 5.00000000000000024e-5 < (+.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 65.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-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. lower-*.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. lower-fma.f64 57.5

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

   5. Simplified 57.5%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. cube-unmult N/A

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

      10. lower-fma.f64 N/A

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

      11. cube-unmult N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. sub-neg N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 23.2

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

   7. Applied egg-rr 23.2%

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

   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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. unpow2 N/A

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

      18. lower-fma.f64 58.1

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

   10. Simplified 58.1%

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

   11. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 37.0

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

   13. Simplified 37.0%

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

4. Final simplification 52.0%

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

Alternative 3: 97.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \mathsf{fma}\left(b, b \cdot 2, a \cdot a\right)\\ \mathbf{if}\;a \leq -49000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 520:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (fma b (* b 2.0) (* a a)))))
   (if (<= a -49000.0)
     t_0
     (if (<= a 520.0) (fma (* b b) (fma b b 4.0) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * fma(b, (b * 2.0), (a * a));
	double tmp;
	if (a <= -49000.0) {
		tmp = t_0;
	} else if (a <= 520.0) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * fma(b, Float64(b * 2.0), Float64(a * a)))
	tmp = 0.0
	if (a <= -49000.0)
		tmp = t_0;
	elseif (a <= 520.0)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(b * N[(b * 2.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -49000.0], t$95$0, If[LessEqual[a, 520.0], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -49000 or 520 < a

   1. Initial program 50.6%

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

   4. Applied egg-rr 52.1%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*l* N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   10. Simplified 97.9%

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

   if -49000 < a < 520

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-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. lower-*.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. lower-fma.f64 98.9

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

   5. Simplified 98.9%

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

Alternative 4: 96.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), a \cdot a, -1\right)\\ \mathbf{if}\;a \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma a a (* b b)) (* a a) -1.0)))
   (if (<= a -3.6e-5)
     t_0
     (if (<= a 6.8e-53) (fma (* b b) (fma b b 4.0) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma(fma(a, a, (b * b)), (a * a), -1.0);
	double tmp;
	if (a <= -3.6e-5) {
		tmp = t_0;
	} else if (a <= 6.8e-53) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(a, a, Float64(b * b)), Float64(a * a), -1.0)
	tmp = 0.0
	if (a <= -3.6e-5)
		tmp = t_0;
	elseif (a <= 6.8e-53)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -3.6e-5], t$95$0, If[LessEqual[a, 6.8e-53], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.60000000000000009e-5 or 6.8e-53 < a

   1. Initial program 56.2%

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

   4. Applied egg-rr 57.5%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.7

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

   7. Simplified 98.7%

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

   8. Taylor expanded in b around 0

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

   10. Simplified 98.7%

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

   11. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 97.1

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

   13. Simplified 97.1%

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

   if -3.60000000000000009e-5 < a < 6.8e-53

   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. lower-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. lower-*.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. lower-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%

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

Alternative 5: 53.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-178}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+291}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-178)
   -1.0
   (if (<= (* b b) 5e+291) (* 4.0 (* a a)) (* b (* b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-178) {
		tmp = -1.0;
	} else if ((b * b) <= 5e+291) {
		tmp = 4.0 * (a * a);
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d-178) then
        tmp = -1.0d0
    else if ((b * b) <= 5d+291) then
        tmp = 4.0d0 * (a * a)
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-178) {
		tmp = -1.0;
	} else if ((b * b) <= 5e+291) {
		tmp = 4.0 * (a * a);
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-178:
		tmp = -1.0
	elif (b * b) <= 5e+291:
		tmp = 4.0 * (a * a)
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-178)
		tmp = -1.0;
	elseif (Float64(b * b) <= 5e+291)
		tmp = Float64(4.0 * Float64(a * a));
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-178)
		tmp = -1.0;
	elseif ((b * b) <= 5e+291)
		tmp = 4.0 * (a * a);
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-178], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 5e+291], N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 1.9999999999999999e-178

   1. Initial program 85.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-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. lower-*.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. lower-fma.f64 54.7

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

   5. Simplified 54.7%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 54.7%

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

   if 1.9999999999999999e-178 < (*.f64 b b) < 5.0000000000000001e291

   1. Initial program 72.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 76.2%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 71.2

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

   8. Simplified 71.2%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 58.2%

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

   12. Taylor expanded in a around 0

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

   14. Simplified 40.5%

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

   if 5.0000000000000001e291 < (*.f64 b b)

   1. Initial program 61.8%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. cube-unmult N/A

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

      10. lower-fma.f64 N/A

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

      11. cube-unmult N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. sub-neg N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 2.9

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

   7. Applied egg-rr 2.9%

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

   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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. unpow2 N/A

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

      18. lower-fma.f64 100.0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 97.5

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

   13. Simplified 97.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-178}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+291}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 84.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 95.3

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

   8. Simplified 95.3%

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

   if 1e86 < (*.f64 b b)

   1. Initial program 60.5%

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

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 95.0

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

   5. Simplified 95.0%

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

4. Final simplification 95.2%

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

Alternative 7: 93.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{+74}:\\ \;\;\;\;a \cdot t\_0\\ \mathbf{elif}\;a \leq 13500000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + 4\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a a))))
   (if (<= a -2.9e+74)
     (* a t_0)
     (if (<= a 13500000000000.0)
       (fma (* b b) (fma b b 4.0) -1.0)
       (* (+ a 4.0) t_0)))))

C

double code(double a, double b) {
	double t_0 = a * (a * a);
	double tmp;
	if (a <= -2.9e+74) {
		tmp = a * t_0;
	} else if (a <= 13500000000000.0) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = (a + 4.0) * t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * a))
	tmp = 0.0
	if (a <= -2.9e+74)
		tmp = Float64(a * t_0);
	elseif (a <= 13500000000000.0)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = Float64(Float64(a + 4.0) * t_0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.9e+74], N[(a * t$95$0), $MachinePrecision], If[LessEqual[a, 13500000000000.0], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a + 4.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.9000000000000002e74

   1. Initial program 13.3%

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -2.9000000000000002e74 < a < 1.35e13

   1. Initial program 97.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-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. lower-*.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. lower-fma.f64 95.4

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

   5. Simplified 95.4%

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

   if 1.35e13 < a

   1. Initial program 67.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 85.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 94.0

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

   8. Simplified 94.0%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. distribute-rgt-in N/A

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

       10. associate-*l* N/A

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

       11. lft-mult-inverse N/A

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

       12. metadata-eval N/A

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

       13. *-lft-identity N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 94.0

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

   11. Simplified 94.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+74}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 13500000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + 4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+74}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 13500000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e+74)
   (* a (* a (* a a)))
   (if (<= a 13500000000000.0)
     (fma (* b b) (fma b b 4.0) -1.0)
     (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.9e+74) {
		tmp = a * (a * (a * a));
	} else if (a <= 13500000000000.0) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.9e+74)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	elseif (a <= 13500000000000.0)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.9e+74], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 13500000000000.0], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.9000000000000002e74

   1. Initial program 13.3%

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -2.9000000000000002e74 < a < 1.35e13

   1. Initial program 97.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-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. lower-*.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. lower-fma.f64 95.4

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

   5. Simplified 95.4%

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

   if 1.35e13 < a

   1. Initial program 67.4%

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 94.0

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

   5. Simplified 94.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 94.0

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

   7. Applied egg-rr 94.0%

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

Alternative 9: 98.5% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

4. Applied egg-rr 74.9%

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

5. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 99.2

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

7. Simplified 99.2%

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

8. Taylor expanded in b around 0

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

10. Simplified 98.8%

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

Alternative 10: 82.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+26}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 12000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.2e+26)
   (* a (* a (* a a)))
   (if (<= a 12000000000000.0) (fma (* b b) 4.0 -1.0) (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.2e+26) {
		tmp = a * (a * (a * a));
	} else if (a <= 12000000000000.0) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.2e+26)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	elseif (a <= 12000000000000.0)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.2e+26], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 12000000000000.0], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.20000000000000004e26

   1. Initial program 27.7%

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

   if -5.20000000000000004e26 < a < 1.2e13

   1. Initial program 97.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-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. lower-*.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. lower-fma.f64 98.2

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

   5. Simplified 98.2%

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

   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 80.6%

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

   if 1.2e13 < a

   1. Initial program 67.4%

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 94.0

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

   5. Simplified 94.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 94.0

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

   7. Applied egg-rr 94.0%

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

Alternative 11: 82.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -5.2 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 12000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -5.2e+26)
     t_0
     (if (<= a 12000000000000.0) (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 <= -5.2e+26) {
		tmp = t_0;
	} else if (a <= 12000000000000.0) {
		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 <= -5.2e+26)
		tmp = t_0;
	elseif (a <= 12000000000000.0)
		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, -5.2e+26], t$95$0, If[LessEqual[a, 12000000000000.0], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.20000000000000004e26 or 1.2e13 < a

   1. Initial program 50.6%

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.9

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

   5. Simplified 92.9%

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

   if -5.20000000000000004e26 < a < 1.2e13

   1. Initial program 97.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-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. lower-*.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. lower-fma.f64 98.2

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

   5. Simplified 98.2%

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

   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 80.6%

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

Alternative 12: 68.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 78.6%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 85.1

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

   8. Simplified 85.1%

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

   9. Taylor expanded in a around 0

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

   11. Simplified 66.0%

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

   if 5.0000000000000001e291 < (*.f64 b b)

   1. Initial program 61.8%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. cube-unmult N/A

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

      10. lower-fma.f64 N/A

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

      11. cube-unmult N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. sub-neg N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 2.9

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

   7. Applied egg-rr 2.9%

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

   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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. unpow2 N/A

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

      18. lower-fma.f64 100.0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 97.5

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

   13. Simplified 97.5%

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

Alternative 13: 24.1% 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 74.1%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
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. lower-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. lower-*.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. lower-fma.f64 67.7

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

5. Simplified 67.7%

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

6. Taylor expanded in b around 0

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

8. Simplified 24.7%

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

Reproduce

herbie shell --seed 2024210 
(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))