Bouland and Aaronson, Equation (24)

Percentage Accurate: 72.9% → 99.1%

Time: 8.4s

Alternatives: 10

Speedup: 6.2×

• 61.3% 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: 72.9% 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: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+67}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a - a \cdot a\right) \cdot a\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 7.5e+67)
   (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* (- a (* a a)) a))) 1.0)
   (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 7.5e+67) {
		tmp = (pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a - (a * a)) * a))) - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	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 (a <= 7.5d+67) then
        tmp = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * ((a - (a * a)) * a))) - 1.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 7.5e+67) {
		tmp = (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a - (a * a)) * a))) - 1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 7.5e+67:
		tmp = (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a - (a * a)) * a))) - 1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 7.5e+67)
		tmp = Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(a - Float64(a * a)) * a))) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 7.5e+67)
		tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * ((a - (a * a)) * a))) - 1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 7.5e+67], N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(a - N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.5000000000000005e67

   1. Initial program 88.7%

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

   3. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lft-identity N/A

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

      10. mul-1-neg N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. unpow2 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 7.5000000000000005e67 < a

   1. Initial program 16.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in a around inf

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

       1. lower-pow.f64 100.0

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

   12. Applied rewrites 100.0%

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

Alternative 2: 94.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, -4, 4 + a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 0.105:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(\frac{\frac{4}{a} + -4}{a} + 1\right) \cdot \left(a \cdot a\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.6e-7)
   (- (* (* (fma a -4.0 (+ 4.0 (* a a))) a) a) 1.0)
   (if (<= a 0.105)
     (- (fma (* b b) 12.0 (pow b 4.0)) 1.0)
     (- (* (* a a) (* (+ (/ (+ (/ 4.0 a) -4.0) a) 1.0) (* a a))) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.6e-7) {
		tmp = ((fma(a, -4.0, (4.0 + (a * a))) * a) * a) - 1.0;
	} else if (a <= 0.105) {
		tmp = fma((b * b), 12.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = ((a * a) * (((((4.0 / a) + -4.0) / a) + 1.0) * (a * a))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.6e-7)
		tmp = Float64(Float64(Float64(fma(a, -4.0, Float64(4.0 + Float64(a * a))) * a) * a) - 1.0);
	elseif (a <= 0.105)
		tmp = Float64(fma(Float64(b * b), 12.0, (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(Float64(Float64(Float64(4.0 / a) + -4.0) / a) + 1.0) * Float64(a * a))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.6e-7], N[(N[(N[(N[(a * -4.0 + N[(4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 0.105], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(N[(N[(N[(4.0 / a), $MachinePrecision] + -4.0), $MachinePrecision] / a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.59999999999999914e-7

   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 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.8%

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

   9. Applied rewrites 91.8%

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

   if -9.59999999999999914e-7 < a < 0.104999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.104999999999999996 < a

   1. Initial program 33.7%

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 93.8%

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

Alternative 3: 94.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, -4, 4 + a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 0.105:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(\frac{\frac{4}{a} + -4}{a} + 1\right) \cdot \left(a \cdot a\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.6e-7)
   (- (* (* (fma a -4.0 (+ 4.0 (* a a))) a) a) 1.0)
   (if (<= a 0.105)
     (- (* (* (fma b b 12.0) b) b) 1.0)
     (- (* (* a a) (* (+ (/ (+ (/ 4.0 a) -4.0) a) 1.0) (* a a))) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.6e-7) {
		tmp = ((fma(a, -4.0, (4.0 + (a * a))) * a) * a) - 1.0;
	} else if (a <= 0.105) {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	} else {
		tmp = ((a * a) * (((((4.0 / a) + -4.0) / a) + 1.0) * (a * a))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.6e-7)
		tmp = Float64(Float64(Float64(fma(a, -4.0, Float64(4.0 + Float64(a * a))) * a) * a) - 1.0);
	elseif (a <= 0.105)
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(Float64(Float64(Float64(4.0 / a) + -4.0) / a) + 1.0) * Float64(a * a))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.6e-7], N[(N[(N[(N[(a * -4.0 + N[(4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 0.105], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(N[(N[(N[(4.0 / a), $MachinePrecision] + -4.0), $MachinePrecision] / a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.59999999999999914e-7

   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 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.8%

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

   9. Applied rewrites 91.8%

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

   if -9.59999999999999914e-7 < a < 0.104999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.9%

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

   if 0.104999999999999996 < a

   1. Initial program 33.7%

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 93.8%

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

Alternative 4: 94.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-7} \lor \neg \left(a \leq 0.105\right):\\ \;\;\;\;\left(\mathsf{fma}\left(a, a + -4, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -9.6e-7) (not (<= a 0.105)))
   (- (* (* (fma a (+ a -4.0) 4.0) a) a) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -9.6e-7) || !(a <= 0.105)) {
		tmp = ((fma(a, (a + -4.0), 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -9.6e-7) || !(a <= 0.105))
		tmp = Float64(Float64(Float64(fma(a, Float64(a + -4.0), 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -9.6e-7], N[Not[LessEqual[a, 0.105]], $MachinePrecision]], N[(N[(N[(N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.59999999999999914e-7 or 0.104999999999999996 < a

   1. Initial program 51.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 92.7

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 92.7%

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

   if -9.59999999999999914e-7 < a < 0.104999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-7} \lor \neg \left(a \leq 0.105\right):\\ \;\;\;\;\left(\mathsf{fma}\left(a, a + -4, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, -4, 4 + a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 0.105:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, a + -4, 4\right) \cdot a\right) \cdot a - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.6e-7)
   (- (* (* (fma a -4.0 (+ 4.0 (* a a))) a) a) 1.0)
   (if (<= a 0.105)
     (- (* (* (fma b b 12.0) b) b) 1.0)
     (- (* (* (fma a (+ a -4.0) 4.0) a) a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.6e-7) {
		tmp = ((fma(a, -4.0, (4.0 + (a * a))) * a) * a) - 1.0;
	} else if (a <= 0.105) {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	} else {
		tmp = ((fma(a, (a + -4.0), 4.0) * a) * a) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.6e-7)
		tmp = Float64(Float64(Float64(fma(a, -4.0, Float64(4.0 + Float64(a * a))) * a) * a) - 1.0);
	elseif (a <= 0.105)
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(a, Float64(a + -4.0), 4.0) * a) * a) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.6e-7], N[(N[(N[(N[(a * -4.0 + N[(4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 0.105], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.59999999999999914e-7

   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 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.8%

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

   9. Applied rewrites 91.8%

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

   if -9.59999999999999914e-7 < a < 0.104999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.9%

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

   if 0.104999999999999996 < a

   1. Initial program 33.7%

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 93.7

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

   5. Applied rewrites 93.7%

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

   7. Applied rewrites 93.8%

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

Alternative 6: 93.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+32} \lor \neg \left(a \leq 0.22\right):\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -8.2e+32) (not (<= a 0.22)))
   (- (* (* (* a a) a) a) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -8.2e+32) || !(a <= 0.22)) {
		tmp = (((a * a) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -8.2e+32) || !(a <= 0.22))
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -8.2e+32], N[Not[LessEqual[a, 0.22]], $MachinePrecision]], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.19999999999999961e32 or 0.220000000000000001 < a

   1. Initial program 50.7%

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 95.2

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

   5. Applied rewrites 95.2%

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

   7. Applied rewrites 95.3%

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

   9. Applied rewrites 95.3%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 94.0%

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

   if -8.19999999999999961e32 < a < 0.220000000000000001

   1. Initial program 96.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+32} \lor \neg \left(a \leq 0.22\right):\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(12 \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e+153) (- (* (* (* a a) a) a) 1.0) (- (* (* 12.0 b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.1e+153) {
		tmp = (((a * a) * a) * a) - 1.0;
	} else {
		tmp = ((12.0 * b) * b) - 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 <= 2.1d+153) then
        tmp = (((a * a) * a) * a) - 1.0d0
    else
        tmp = ((12.0d0 * b) * b) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.1e+153) {
		tmp = (((a * a) * a) * a) - 1.0;
	} else {
		tmp = ((12.0 * b) * b) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.1e+153:
		tmp = (((a * a) * a) * a) - 1.0
	else:
		tmp = ((12.0 * b) * b) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.1e+153)
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(12.0 * b) * b) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.1e+153)
		tmp = (((a * a) * a) * a) - 1.0;
	else
		tmp = ((12.0 * b) * b) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.1e+153], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(12.0 * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000017e153

   1. Initial program 77.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 77.8

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.8%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 76.1%

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

   if 2.10000000000000017e153 < b

   1. Initial program 60.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 97.5%

       \[\leadsto \left(12 \cdot b\right) \cdot b - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(12 \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(12 \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e+153) (- (* (* a a) (* a a)) 1.0) (- (* (* 12.0 b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.1e+153) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((12.0 * b) * b) - 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 <= 2.1d+153) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = ((12.0d0 * b) * b) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.1e+153) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((12.0 * b) * b) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.1e+153:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = ((12.0 * b) * b) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.1e+153)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(12.0 * b) * b) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.1e+153)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = ((12.0 * b) * b) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.1e+153], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(12.0 * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000017e153

   1. Initial program 77.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 76.1%

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

   if 2.10000000000000017e153 < b

   1. Initial program 60.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 97.5%

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

Alternative 9: 60.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+134}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(12 \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.6e+134) (- (* (* a a) 4.0) 1.0) (- (* (* 12.0 b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.6e+134) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((12.0 * b) * b) - 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 <= 6.6d+134) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((12.0d0 * b) * b) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.6e+134) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((12.0 * b) * b) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.6e+134:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((12.0 * b) * b) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.6e+134)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(12.0 * b) * b) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.6e+134)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((12.0 * b) * b) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.6e+134], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(12.0 * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.6e134

   1. Initial program 77.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 54.7%

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

   if 6.6e134 < b

   1. Initial program 60.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 92.5%

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

   10. Applied rewrites 92.5%

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

Alternative 10: 51.3% accurate, 11.1× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return ((12.0 * b) * b) - 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 = ((12.0d0 * b) * b) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((12.0 * b) * b) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 65.5

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

5. Applied rewrites 65.5%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 48.3%

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

10. Applied rewrites 48.3%

    \[\leadsto \left(12 \cdot b\right) \cdot b - 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024363 
(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))