Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.3% → 98.9%

Time: 6.3s

Alternatives: 8

Speedup: 6.2×

• 61.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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: 73.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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: 98.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot a + b \cdot b\\ \left(t\_0 \cdot t\_0 + \left(b \cdot b\right) \cdot 12\right) - 1 \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* a a) (* b b)))) (- (+ (* t_0 t_0) (* (* b b) 12.0)) 1.0)))

C

double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return ((t_0 * t_0) + ((b * b) * 12.0)) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    t_0 = (a * a) + (b * b)
    code = ((t_0 * t_0) + ((b * b) * 12.0d0)) - 1.0d0
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return ((t_0 * t_0) + ((b * b) * 12.0)) - 1.0;
}

Python

def code(a, b):
	t_0 = (a * a) + (b * b)
	return ((t_0 * t_0) + ((b * b) * 12.0)) - 1.0

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b))
	return Float64(Float64(Float64(t_0 * t_0) + Float64(Float64(b * b) * 12.0)) - 1.0)
end

MATLAB

function tmp = code(a, b)
	t_0 = (a * a) + (b * b);
	tmp = ((t_0 * t_0) + ((b * b) * 12.0)) - 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 * t$95$0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $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. lift-pow.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-+.f64 75.7

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

4. Applied rewrites 75.7%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

      \[\leadsto \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right) - 1 \]

   4. lift-*.f64 98.8

      \[\leadsto \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right) - 1 \]

7. Applied rewrites 98.8%

   \[\leadsto \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \color{blue}{\left(b \cdot b\right) \cdot 12}\right) - 1 \]
8. Add Preprocessing

Alternative 2: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \leq -0.5:\\ \;\;\;\;\left(4 \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b b)) 2.0)
        (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
       1.0)
      -0.5)
   (- (* (* 4.0 a) a) 1.0)
   (* (* a a) (* a a))))

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) * (3.0 + a))))) - 1.0) <= -0.5) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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) * (3.0d0 + a))))) - 1.0d0) <= (-0.5d0)) then
        tmp = ((4.0d0 * a) * a) - 1.0d0
    else
        tmp = (a * a) * (a * a)
    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) * (3.0 + a))))) - 1.0) <= -0.5) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	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) * (3.0 + a))))) - 1.0) <= -0.5:
		tmp = ((4.0 * a) * a) - 1.0
	else:
		tmp = (a * a) * (a * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (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) <= -0.5)
		tmp = Float64(Float64(Float64(4.0 * a) * a) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	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) * (3.0 + a))))) - 1.0) <= -0.5)
		tmp = ((4.0 * a) * a) - 1.0;
	else
		tmp = (a * a) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.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))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 100.0%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. pow2 N/A

         \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

      2. associate-*r* N/A

         \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

      4. lower-*.f64 99.2

         \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

   8. Applied rewrites 99.2%

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

   if -0.5 < (-.f64 (+.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))))) #s(literal 1 binary64))

   1. Initial program 67.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 61.2

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

   5. Applied rewrites 61.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 61.2

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

   7. Applied rewrites 61.2%

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

Alternative 3: 83.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 116000000000:\\ \;\;\;\;\left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a + b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 116000000000.0)
   (- (* (* (+ (* (- a 4.0) a) 4.0) a) a) 1.0)
   (- (+ (* (+ (* a a) (* b b)) (* b b)) (* (* b b) 12.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 116000000000.0) {
		tmp = (((((a - 4.0) * a) + 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((((a * a) + (b * b)) * (b * b)) + ((b * b) * 12.0)) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 116000000000.0d0) then
        tmp = (((((a - 4.0d0) * a) + 4.0d0) * a) * a) - 1.0d0
    else
        tmp = ((((a * a) + (b * b)) * (b * b)) + ((b * b) * 12.0d0)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 116000000000.0) {
		tmp = (((((a - 4.0) * a) + 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((((a * a) + (b * b)) * (b * b)) + ((b * b) * 12.0)) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 116000000000.0:
		tmp = (((((a - 4.0) * a) + 4.0) * a) * a) - 1.0
	else:
		tmp = ((((a * a) + (b * b)) * (b * b)) + ((b * b) * 12.0)) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 116000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(a - 4.0) * a) + 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) + Float64(b * b)) * Float64(b * b)) + Float64(Float64(b * b) * 12.0)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 116000000000.0)
		tmp = (((((a - 4.0) * a) + 4.0) * a) * a) - 1.0;
	else
		tmp = ((((a * a) + (b * b)) * (b * b)) + ((b * b) * 12.0)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 116000000000.0], N[(N[(N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.16e11

   1. Initial program 78.5%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-pow.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      9. lift-*.f64 80.9

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

   8. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      10. lift--.f64 N/A

          \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      11. lift-+.f64 80.9

          \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

   10. Applied rewrites 80.9%

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

   if 1.16e11 < b

   1. Initial program 67.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      4. lift-*.f64 99.9

         \[\leadsto \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right) - 1 \]

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 97.0

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

   10. Applied rewrites 97.0%

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

Alternative 4: 81.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 122000000000:\\ \;\;\;\;\left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 122000000000.0)
   (- (* (* (+ (* (- a 4.0) a) 4.0) a) a) 1.0)
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 122000000000.0) {
		tmp = (((((a - 4.0) * a) + 4.0) * a) * a) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 122000000000.0d0) then
        tmp = (((((a - 4.0d0) * a) + 4.0d0) * 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 <= 122000000000.0) {
		tmp = (((((a - 4.0) * a) + 4.0) * a) * a) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 122000000000.0:
		tmp = (((((a - 4.0) * a) + 4.0) * a) * a) - 1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 122000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(a - 4.0) * a) + 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 122000000000.0)
		tmp = (((((a - 4.0) * a) + 4.0) * a) * a) - 1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 122000000000.0], N[(N[(N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.22e11

   1. Initial program 78.5%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-pow.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      9. lift-*.f64 80.9

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

   8. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      10. lift--.f64 N/A

          \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

      11. lift-+.f64 80.9

          \[\leadsto \left(\left(\left(a - 4\right) \cdot a + 4\right) \cdot a\right) \cdot a - 1 \]

   10. Applied rewrites 80.9%

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

   if 1.22e11 < b

   1. Initial program 67.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 90.9

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

   5. Applied rewrites 90.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

Alternative 5: 80.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 122000000000:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 122000000000.0) (- (* (* (* a a) a) a) 1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 122000000000.0) {
		tmp = (((a * a) * a) * a) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 122000000000.0d0) 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 <= 122000000000.0) {
		tmp = (((a * a) * a) * a) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 122000000000.0:
		tmp = (((a * a) * a) * a) - 1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 122000000000.0)
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 122000000000.0)
		tmp = (((a * a) * a) * a) - 1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 122000000000.0], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.22e11

   1. Initial program 78.5%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-pow.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      9. lift-*.f64 80.9

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

   8. Applied rewrites 80.9%

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

   9. Taylor expanded in a around inf

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

       1. pow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

       2. lift-*.f64 79.5

          \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

   11. Applied rewrites 79.5%

       \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]

       5. lower-*.f64 79.5

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]

   13. Applied rewrites 79.5%

       \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]

   if 1.22e11 < b

   1. Initial program 67.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 90.9

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

   5. Applied rewrites 90.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

Alternative 6: 80.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 122000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 122000000000.0) (- (* (* a a) (* a a)) 1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 122000000000.0) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 122000000000.0d0) 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 <= 122000000000.0) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 122000000000.0:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 122000000000.0)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 122000000000.0)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 122000000000.0], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.22e11

   1. Initial program 78.5%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-pow.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

      9. lift-*.f64 80.9

         \[\leadsto \left(\left(a - 4\right) \cdot a + 4\right) \cdot \left(a \cdot a\right) - 1 \]

   8. Applied rewrites 80.9%

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

   9. Taylor expanded in a around inf

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

       1. pow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

       2. lift-*.f64 79.5

          \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

   11. Applied rewrites 79.5%

       \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

   if 1.22e11 < b

   1. Initial program 67.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 90.9

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

   5. Applied rewrites 90.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

Alternative 7: 66.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 70000000000:\\ \;\;\;\;\left(4 \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 70000000000.0) (- (* (* 4.0 a) a) 1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 70000000000.0) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 70000000000.0d0) then
        tmp = ((4.0d0 * 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 <= 70000000000.0) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 70000000000.0:
		tmp = ((4.0 * a) * a) - 1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 70000000000.0)
		tmp = Float64(Float64(Float64(4.0 * a) * a) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 70000000000.0)
		tmp = ((4.0 * a) * a) - 1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 70000000000.0], N[(N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7e10

   1. Initial program 78.5%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-pow.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in a around 0

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

      1. pow2 N/A

         \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

      2. associate-*r* N/A

         \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

      4. lower-*.f64 55.9

         \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

   8. Applied rewrites 55.9%

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

   if 7e10 < b

   1. Initial program 67.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 90.9

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

   5. Applied rewrites 90.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

Alternative 8: 50.8% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \left(4 \cdot a\right) \cdot a - 1 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((4.0d0 * a) * a) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return ((4.0 * a) * a) - 1.0;
}

Python

def code(a, b):
	return ((4.0 * a) * a) - 1.0

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((4.0 * a) * a) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(4.0 * a), $MachinePrecision] * a), $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. 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. lower-+.f64 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lower-pow.f64 54.8

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

5. Applied rewrites 54.8%

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

6. Taylor expanded in a around 0

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

   1. pow2 N/A

      \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

   2. associate-*r* N/A

      \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

   3. lower-*.f64 N/A

      \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

   4. lower-*.f64 47.9

      \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

8. Applied rewrites 47.9%

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

Reproduce

herbie shell --seed 2025065 
(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))