Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.1% → 98.5%

Time: 11.9s

Alternatives: 10

Speedup: 5.5×

• 60.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% 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: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 89.1

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

   5. Simplified 89.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 90.8

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

   8. Simplified 90.8%

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

4. Final simplification 97.6%

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

Alternative 2: 50.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
      2e-10)
   -1.0
   (* (* b b) 12.0)))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 2e-10) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (a + 3.0d0))))) <= 2d-10) then
        tmp = -1.0d0
    else
        tmp = (b * b) * 12.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 2e-10) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 2e-10:
		tmp = -1.0
	else:
		tmp = (b * b) * 12.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (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(a + 3.0))))) <= 2e-10)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * 12.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 2e-10)
		tmp = -1.0;
	else
		tmp = (b * b) * 12.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-10], -1.0, N[(N[(b * b), $MachinePrecision] * 12.0), $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))))) < 2.00000000000000007e-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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 99.5%

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

   if 2.00000000000000007e-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 66.3%

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

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 60.6

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

   5. Simplified 60.6%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 32.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 32.4

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

   11. Simplified 32.4%

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

4. Final simplification 51.0%

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

Alternative 3: 82.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-17)
   (fma 4.0 (* a a) -1.0)
   (if (<= (* b b) 5e+15) (* a (* a (* a a))) (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-17) {
		tmp = fma(4.0, (a * a), -1.0);
	} else if ((b * b) <= 5e+15) {
		tmp = a * (a * (a * a));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-17)
		tmp = fma(4.0, Float64(a * a), -1.0);
	elseif (Float64(b * b) <= 5e+15)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-17], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 5e+15], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 2.00000000000000014e-17

   1. Initial program 86.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 72.5

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

   8. Simplified 72.5%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 81.3

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

   11. Simplified 81.3%

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

   if 2.00000000000000014e-17 < (*.f64 b b) < 5e15

   1. Initial program 69.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 90.0

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

   5. Simplified 90.0%

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

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   7. 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 90.4

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

   8. Simplified 90.4%

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

   if 5e15 < (*.f64 b b)

   1. Initial program 65.2%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 91.3

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

   5. Simplified 91.3%

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

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\color{blue}{\left(3 + 1\right)}} \]

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

Alternative 4: 94.5% accurate, 4.6× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+15)
   (+ (* (* 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) <= 5e+15) {
		tmp = ((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) <= 5e+15)
		tmp = Float64(Float64(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], 5e+15], N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $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) < 5e15

   1. Initial program 85.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 99.2

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

   8. Simplified 99.2%

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

   if 5e15 < (*.f64 b b)

   1. Initial program 65.2%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 91.3

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

   5. Simplified 91.3%

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

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\color{blue}{\left(3 + 1\right)}} \]

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

4. Final simplification 95.5%

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

Alternative 5: 93.9% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+15) then
        tmp = ((a + (-4.0d0)) * (a * (a * a))) + (-1.0d0)
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

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

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+15:
		tmp = ((a + -4.0) * (a * (a * a))) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+15)
		tmp = ((a + -4.0) * (a * (a * a))) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+15], N[(N[(N[(a + -4.0), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $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) < 5e15

   1. Initial program 85.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in a around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*l* N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

      20. lower-*.f64 N/A

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

   8. Simplified 98.6%

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

   if 5e15 < (*.f64 b b)

   1. Initial program 65.2%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 91.3

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

   5. Simplified 91.3%

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

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\color{blue}{\left(3 + 1\right)}} \]

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.6% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+15) then
        tmp = ((a * a) * (a * a)) + (-1.0d0)
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+15) {
		tmp = ((a * a) * (a * a)) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+15:
		tmp = ((a * a) * (a * a)) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+15)
		tmp = ((a * a) * (a * a)) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+15], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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) < 5e15

   1. Initial program 85.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.1

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

   8. Simplified 98.1%

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

   if 5e15 < (*.f64 b b)

   1. Initial program 65.2%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 91.3

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

   5. Simplified 91.3%

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

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\color{blue}{\left(3 + 1\right)}} \]

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -34000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -34000000000000.0)
     t_0
     (if (<= a 2.7e+51) (fma 12.0 (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -34000000000000.0) {
		tmp = t_0;
	} else if (a <= 2.7e+51) {
		tmp = fma(12.0, (b * b), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (a <= -34000000000000.0)
		tmp = t_0;
	elseif (a <= 2.7e+51)
		tmp = fma(12.0, Float64(b * b), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -34000000000000.0], t$95$0, If[LessEqual[a, 2.7e+51], N[(12.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4e13 or 2.69999999999999992e51 < a

   1. Initial program 44.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   7. 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 92.1

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

   8. Simplified 92.1%

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

   if -3.4e13 < a < 2.69999999999999992e51

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 73.9

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

   8. Simplified 73.9%

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

Alternative 8: 93.6% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+15], N[(a * N[(a * N[(a * a), $MachinePrecision]), $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) < 5e15

   1. Initial program 85.1%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.1

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

   5. Simplified 98.1%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} + -1 \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} + -1 \]

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.1

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

   8. Simplified 98.1%

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

   if 5e15 < (*.f64 b b)

   1. Initial program 65.2%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 91.3

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

   5. Simplified 91.3%

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

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\color{blue}{\left(3 + 1\right)}} \]

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

Alternative 9: 69.6% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 80.0

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

   5. Simplified 80.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 51.5

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

   8. Simplified 51.5%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 62.9

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

   11. Simplified 62.9%

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

   if 1e292 < (*.f64 b b)

   1. Initial program 62.7%

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

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 92.6%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 92.6

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

   11. Simplified 92.6%

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

4. Final simplification 69.7%

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

Alternative 10: 25.1% accurate, 155.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 75.7%

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

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

   2. pow-sqr N/A

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

   3. distribute-rgt-out N/A

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

   4. +-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 71.4

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

5. Simplified 71.4%

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

6. Taylor expanded in b around 0

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

8. Simplified 28.1%

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

Reproduce

herbie shell --seed 2024215 
(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))