Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.4% → 98.3%

Time: 8.0s

Alternatives: 13

Speedup: 5.7×

• 61.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(3 + a\right) \cdot \left(b \cdot b\right) + \left(1 - a\right) \cdot \left(a \cdot a\right)\right) \cdot 4 + {\left(b \cdot b + a \cdot a\right)}^{2}\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ (* (+ 3.0 a) (* b b)) (* (- 1.0 a) (* a a))) 4.0)
          (pow (+ (* b b) (* a a)) 2.0))))
   (if (<= t_0 INFINITY) (- t_0 1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double t_0 = ((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + pow(((b * b) + (a * a)), 2.0);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + Math.pow(((b * b) + (a * a)), 2.0);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 - 1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + math.pow(((b * b) + (a * a)), 2.0)
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 - 1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(Float64(Float64(3.0 + a) * Float64(b * b)) + Float64(Float64(1.0 - a) * Float64(a * a))) * 4.0) + (Float64(Float64(b * b) + Float64(a * a)) ^ 2.0))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + (((b * b) + (a * a)) ^ 2.0);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 - 1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(N[(3.0 + a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < +inf.0

   1. Initial program 99.9%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.1

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 98.8

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

   8. Applied rewrites 98.8%

      \[\leadsto \color{blue}{{a}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

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

Alternative 2: 51.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (* (+ (* (+ 3.0 a) (* b b)) (* (- 1.0 a) (* a a))) 4.0)
       (pow (+ (* b b) (* a a)) 2.0))
      1e-10)
   -1.0
   (* 12.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + pow(((b * b) + (a * a)), 2.0)) <= 1e-10) {
		tmp = -1.0;
	} else {
		tmp = 12.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((3.0d0 + a) * (b * b)) + ((1.0d0 - a) * (a * a))) * 4.0d0) + (((b * b) + (a * a)) ** 2.0d0)) <= 1d-10) then
        tmp = -1.0d0
    else
        tmp = 12.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + Math.pow(((b * b) + (a * a)), 2.0)) <= 1e-10) {
		tmp = -1.0;
	} else {
		tmp = 12.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + math.pow(((b * b) + (a * a)), 2.0)) <= 1e-10:
		tmp = -1.0
	else:
		tmp = 12.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(3.0 + a) * Float64(b * b)) + Float64(Float64(1.0 - a) * Float64(a * a))) * 4.0) + (Float64(Float64(b * b) + Float64(a * a)) ^ 2.0)) <= 1e-10)
		tmp = -1.0;
	else
		tmp = Float64(12.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + (((b * b) + (a * a)) ^ 2.0)) <= 1e-10)
		tmp = -1.0;
	else
		tmp = 12.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[(N[(N[(3.0 + a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1e-10], -1.0, N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 1.00000000000000004e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 99.5%

       \[\leadsto -1 \]

   if 1.00000000000000004e-10 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 60.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 62.9

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 63.0%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 38.7%

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

4. Final simplification 54.6%

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

Alternative 3: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, {b}^{4}\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.5)
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (- (fma 12.0 (* b b) (pow b 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma(12.0, (b * b), pow(b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.5)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = Float64(fma(12.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(12.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.5

   1. Initial program 77.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 100.0%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 64.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 92.6

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

   5. Applied rewrites 92.6%

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

Alternative 4: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+27) (fma (* (fma (- a 4.0) a 4.0) a) a -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+27) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+27)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+27], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000001e27

   1. Initial program 76.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 97.1%

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

   if 4.0000000000000001e27 < (*.f64 b b)

   1. Initial program 64.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 95.3

         \[\leadsto \color{blue}{{b}^{4}} \]

   8. Applied rewrites 95.3%

      \[\leadsto \color{blue}{{b}^{4}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.5)
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.5)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.5

   1. Initial program 77.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 100.0%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 64.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 92.5

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

   8. Applied rewrites 92.5%

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

Alternative 6: 93.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a - 4\right) \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.5)
   (fma (* (* (- a 4.0) a) a) a -1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = fma((((a - 4.0) * a) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.5)
		tmp = fma(Float64(Float64(Float64(a - 4.0) * a) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.5

   1. Initial program 77.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 99.2%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 64.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 92.5

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

   8. Applied rewrites 92.5%

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

Alternative 7: 93.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.5)
   (fma (* (* a a) a) a -1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = fma(((a * a) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.5)
		tmp = fma(Float64(Float64(a * a) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.5

   1. Initial program 77.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.9%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 64.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 92.5

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

   8. Applied rewrites 92.5%

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

Alternative 8: 93.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2000000.0)
   (fma (* (* a a) a) a -1.0)
   (* (* (fma b b 12.0) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2000000.0) {
		tmp = fma(((a * a) * a), a, -1.0);
	} else {
		tmp = (fma(b, b, 12.0) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2000000.0)
		tmp = fma(Float64(Float64(a * a) * a), a, -1.0);
	else
		tmp = Float64(Float64(fma(b, b, 12.0) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2000000.0], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e6

   1. Initial program 76.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.2%

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

   if 2e6 < (*.f64 b b)

   1. Initial program 65.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 93.1

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 93.1%

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

Alternative 9: 83.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2000000.0) (fma (* 4.0 a) a -1.0) (* (* (fma b b 12.0) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2000000.0) {
		tmp = fma((4.0 * a), a, -1.0);
	} else {
		tmp = (fma(b, b, 12.0) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2000000.0)
		tmp = fma(Float64(4.0 * a), a, -1.0);
	else
		tmp = Float64(Float64(fma(b, b, 12.0) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2000000.0], N[(N[(4.0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e6

   1. Initial program 76.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 76.3%

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

   if 2e6 < (*.f64 b b)

   1. Initial program 65.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 93.1

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 93.1%

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

Alternative 10: 83.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2000000.0) (fma (* 4.0 a) a -1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2000000.0) {
		tmp = fma((4.0 * a), a, -1.0);
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2000000.0)
		tmp = fma(Float64(4.0 * a), a, -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2000000.0], N[(N[(4.0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e6

   1. Initial program 76.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 76.3%

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

   if 2e6 < (*.f64 b b)

   1. Initial program 65.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 93.1

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 93.1%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 92.5%

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

Alternative 11: 70.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+287) (fma (* 4.0 a) a -1.0) (* 12.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+287) {
		tmp = fma((4.0 * a), a, -1.0);
	} else {
		tmp = 12.0 * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+287)
		tmp = fma(Float64(4.0 * a), a, -1.0);
	else
		tmp = Float64(12.0 * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+287], N[(N[(4.0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e287

   1. Initial program 74.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 62.9%

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

   if 5e287 < (*.f64 b b)

   1. Initial program 62.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 97.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.2% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 72.5

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

5. Applied rewrites 72.5%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-in N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. unpow2 N/A

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

   14. lower-fma.f64 72.5

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

8. Applied rewrites 72.5%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 54.3%

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

Alternative 13: 24.8% accurate, 155.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -1.0d0
end function

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0;
end

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 72.5

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

5. Applied rewrites 72.5%

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

6. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

8. Applied rewrites 69.6%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 26.6%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))