Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.7% → 98.8%

Time: 12.3s

Alternatives: 8

Speedup: 9.1×

• 60.9% 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: 73.7% accurate, 1.0× speedup

Math

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

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: 98.8% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    t_0 = (a * a) + (b * b)
    code = ((t_0 / (1.0d0 / t_0)) + (4.0d0 * ((b * b) * 3.0d0))) + (-1.0d0)
end function

Java

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

Python

def code(a, b):
	t_0 = (a * a) + (b * b)
	return ((t_0 / (1.0 / t_0)) + (4.0 * ((b * b) * 3.0))) + -1.0

Julia

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

MATLAB

function tmp = code(a, b)
	t_0 = (a * a) + (b * b);
	tmp = ((t_0 / (1.0 / t_0)) + (4.0 * ((b * b) * 3.0))) + -1.0;
end

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \frac{{\left(a \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \frac{1}{\frac{\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{{\left(a \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}}\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{a \cdot a + b \cdot b}{\frac{\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{{\left(a \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}}\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a + b \cdot b\right), \left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{{\left(a \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}\right)\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{{\left(a \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}\right)\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{{\left(a \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}\right)\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{{\left(a \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}\right)\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{1}{\frac{{\left(a \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}}\right)\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

   10. flip3-+ N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{1}{a \cdot a + b \cdot b}\right)\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \left(a \cdot a + b \cdot b\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{\_.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(3, a\right)\right)\right)\right)\right), 1\right) \]

4. Applied egg-rr 75.7%

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

5. Taylor expanded in a around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(4, \color{blue}{\left(3 \cdot {b}^{2}\right)}\right)\right), 1\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(4, \left({b}^{2} \cdot 3\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\left({b}^{2}\right), 3\right)\right)\right), 1\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\left(b \cdot b\right), 3\right)\right)\right), 1\right) \]

   4. *-lowering-*.f64 99.3%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 3\right)\right)\right), 1\right) \]

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 46.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;b \leq 1.2 \cdot 10^{-272}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-191}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-25}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= b 1.2e-272)
     -1.0
     (if (<= b 4.2e-191)
       t_0
       (if (<= b 9.5e-25) -1.0 (if (<= b 4.6e+46) t_0 (* b (* b (* b b)))))))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (b <= 1.2e-272) {
		tmp = -1.0;
	} else if (b <= 4.2e-191) {
		tmp = t_0;
	} else if (b <= 9.5e-25) {
		tmp = -1.0;
	} else if (b <= 4.6e+46) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    if (b <= 1.2d-272) then
        tmp = -1.0d0
    else if (b <= 4.2d-191) then
        tmp = t_0
    else if (b <= 9.5d-25) then
        tmp = -1.0d0
    else if (b <= 4.6d+46) then
        tmp = t_0
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (b <= 1.2e-272) {
		tmp = -1.0;
	} else if (b <= 4.2e-191) {
		tmp = t_0;
	} else if (b <= 9.5e-25) {
		tmp = -1.0;
	} else if (b <= 4.6e+46) {
		tmp = t_0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if b <= 1.2e-272:
		tmp = -1.0
	elif b <= 4.2e-191:
		tmp = t_0
	elif b <= 9.5e-25:
		tmp = -1.0
	elif b <= 4.6e+46:
		tmp = t_0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (b <= 1.2e-272)
		tmp = -1.0;
	elseif (b <= 4.2e-191)
		tmp = t_0;
	elseif (b <= 9.5e-25)
		tmp = -1.0;
	elseif (b <= 4.6e+46)
		tmp = t_0;
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	tmp = 0.0;
	if (b <= 1.2e-272)
		tmp = -1.0;
	elseif (b <= 4.2e-191)
		tmp = t_0;
	elseif (b <= 9.5e-25)
		tmp = -1.0;
	elseif (b <= 4.6e+46)
		tmp = t_0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.2e-272], -1.0, If[LessEqual[b, 4.2e-191], t$95$0, If[LessEqual[b, 9.5e-25], -1.0, If[LessEqual[b, 4.6e+46], t$95$0, N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.19999999999999995e-272 or 4.19999999999999971e-191 < b < 9.50000000000000065e-25

   1. Initial program 76.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(3 + 1\right)}\right), 1\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{3} \cdot a\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 79.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), 1\right) \]

   5. Simplified 79.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 39.1%

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

   if 1.19999999999999995e-272 < b < 4.19999999999999971e-191 or 9.50000000000000065e-25 < b < 4.6000000000000001e46

   1. Initial program 69.2%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      9. *-lowering-*.f64 51.4%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   5. Simplified 51.4%

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

   if 4.6000000000000001e46 < b

   1. Initial program 76.0%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 98.1%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   5. Simplified 98.1%

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

Alternative 3: 93.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2.5 \cdot 10^{+93}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + -4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2.5e+93)
   (+ (* (* a a) (+ 4.0 (* a (+ a -4.0)))) -1.0)
   (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2.5e+93) {
		tmp = ((a * a) * (4.0 + (a * (a + -4.0)))) + -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) <= 2.5d+93) then
        tmp = ((a * a) * (4.0d0 + (a * (a + (-4.0d0))))) + (-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) <= 2.5e+93) {
		tmp = ((a * a) * (4.0 + (a * (a + -4.0)))) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2.5e+93:
		tmp = ((a * a) * (4.0 + (a * (a + -4.0)))) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2.5e+93)
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + -4.0)))) + -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) <= 2.5e+93)
		tmp = ((a * a) * (4.0 + (a * (a + -4.0)))) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2.5e+93], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + -4.0), $MachinePrecision]), $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) < 2.5000000000000001e93

   1. Initial program 81.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left({a}^{2}\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \left(1 - a\right)\right)\right)\right), 1\right) \]

      14. --lowering--.f64 97.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(1, a\right)\right)\right)\right), 1\right) \]

   5. Simplified 97.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \color{blue}{\left(4 + a \cdot \left(a - 4\right)\right)}\right), 1\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + a \cdot \left(a + \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + a \cdot \left(a + -4\right)\right)\right), 1\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left(a \cdot a + -4 \cdot a\right)\right)\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} + -4 \cdot a\right)\right)\right), 1\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + -4 \cdot a\right)\right)\right), 1\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + -4 \cdot \left(a \cdot 1\right)\right)\right)\right), 1\right) \]

      7. rgt-mult-inverse N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + -4 \cdot \left(a \cdot \left(a \cdot \frac{1}{a}\right)\right)\right)\right)\right), 1\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + -4 \cdot \left(a \cdot \frac{a \cdot 1}{a}\right)\right)\right)\right), 1\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + -4 \cdot \left(a \cdot \frac{a}{a}\right)\right)\right)\right), 1\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + -4 \cdot \frac{a \cdot a}{a}\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + -4 \cdot \frac{{a}^{2}}{a}\right)\right)\right), 1\right) \]

      12. *-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + -4 \cdot \frac{1 \cdot {a}^{2}}{a}\right)\right)\right), 1\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + -4 \cdot \left(\frac{1}{a} \cdot {a}^{2}\right)\right)\right)\right), 1\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + \left(-4 \cdot \frac{1}{a}\right) \cdot {a}^{2}\right)\right)\right), 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{a}\right) \cdot {a}^{2}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right) \cdot {a}^{2}\right)\right)\right), 1\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + \left({a}^{2} \cdot 1 + {a}^{2} \cdot \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right)\right)\right)\right), 1\right) \]

      18. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + {a}^{2} \cdot \left(1 + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right)\right)\right)\right), 1\right) \]

      19. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + {a}^{2} \cdot \left(1 - 4 \cdot \frac{1}{a}\right)\right)\right), 1\right) \]

   8. Simplified 97.1%

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

   if 2.5000000000000001e93 < (*.f64 b b)

   1. Initial program 67.0%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 97.3%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   5. Simplified 97.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2.5 \cdot 10^{+93}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + -4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -0.41:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.4:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -0.41) t_0 (if (<= a 2.4) -1.0 t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -0.41) {
		tmp = t_0;
	} else if (a <= 2.4) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    if (a <= (-0.41d0)) then
        tmp = t_0
    else if (a <= 2.4d0) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -0.41) {
		tmp = t_0;
	} else if (a <= 2.4) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -0.41:
		tmp = t_0
	elif a <= 2.4:
		tmp = -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 <= -0.41)
		tmp = t_0;
	elseif (a <= 2.4)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	tmp = 0.0;
	if (a <= -0.41)
		tmp = t_0;
	elseif (a <= 2.4)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.41], t$95$0, If[LessEqual[a, 2.4], -1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.409999999999999976 or 2.39999999999999991 < a

   1. Initial program 47.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      9. *-lowering-*.f64 84.9%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   5. Simplified 84.9%

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

   if -0.409999999999999976 < a < 2.39999999999999991

   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 inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(3 + 1\right)}\right), 1\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{3} \cdot a\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 59.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), 1\right) \]

   5. Simplified 59.8%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 59.8%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2.5 \cdot 10^{+93}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2.5e+93) (+ (* a (* a (* a a))) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2.5e+93) {
		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) <= 2.5d+93) 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) <= 2.5e+93) {
		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) <= 2.5e+93:
		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) <= 2.5e+93)
		tmp = Float64(Float64(a * 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) <= 2.5e+93)
		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], 2.5e+93], N[(N[(a * 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) < 2.5000000000000001e93

   1. Initial program 81.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(3 + 1\right)}\right), 1\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{3} \cdot a\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 95.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), 1\right) \]

   5. Simplified 95.9%

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

   if 2.5000000000000001e93 < (*.f64 b b)

   1. Initial program 67.0%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 97.3%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   5. Simplified 97.3%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2.5 \cdot 10^{+93}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+16}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+16) (+ (* (* a a) 4.0) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+16) {
		tmp = ((a * a) * 4.0) + -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) <= 1d+16) then
        tmp = ((a * a) * 4.0d0) + (-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) <= 1e+16) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+16:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+16)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -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) <= 1e+16)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+16], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e16

   1. Initial program 81.5%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left({a}^{2}\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \left(1 - a\right)\right)\right)\right), 1\right) \]

      14. --lowering--.f64 99.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(1, a\right)\right)\right)\right), 1\right) \]

   5. Simplified 99.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \color{blue}{4}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 82.1%

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

   if 1e16 < (*.f64 b b)

   1. Initial program 68.2%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   5. Simplified 94.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+16}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.29:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.29) -1.0 (* b (* b 12.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.29) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.29d0) 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 (b <= 0.29) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.29:
		tmp = -1.0
	else:
		tmp = b * (b * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.29)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.29)
		tmp = -1.0;
	else
		tmp = b * (b * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.29], -1.0, N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.28999999999999998

   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 inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(3 + 1\right)}\right), 1\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{3} \cdot a\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 81.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), 1\right) \]

   5. Simplified 81.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 40.8%

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

   if 0.28999999999999998 < b

   1. Initial program 75.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{2} \cdot {b}^{2} + 12 \cdot {b}^{2}\right), 1\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 12\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 12\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 12\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 12\right)\right), 1\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 12\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 12\right)\right), 1\right) \]

      10. *-lowering-*.f64 93.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 12\right)\right), 1\right) \]

   5. Simplified 93.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2}\right)}, 1\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{2} \cdot 12\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), 12\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 12\right), 1\right) \]

      4. *-lowering-*.f64 58.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 12\right), 1\right) \]

   8. Simplified 58.5%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(12 \cdot b\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{12}\right)\right) \]

       6. *-lowering-*.f64 58.5%

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{12}\right)\right) \]

   11. Simplified 58.5%

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

Alternative 8: 24.2% accurate, 128.0× 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 inf

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(3 + 1\right)}\right), 1\right) \]

   2. pow-plus N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({a}^{3} \cdot a\right), 1\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

   5. cube-mult N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), 1\right) \]

   9. *-lowering-*.f64 71.5%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), 1\right) \]

5. Simplified 71.5%

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

6. Taylor expanded in a around 0

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

8. Simplified 32.3%

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

Reproduce

herbie shell --seed 2024191 
(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))