Q^3 (Cubic Equation Discriminant Part)

Percentage Accurate: 81.1% → 99.3%

Time: 5.3s

Alternatives: 17

Speedup: 2.6×

• 55.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(\left(-1000000000 \leq a \land a \leq 1000000000\right) \land \left(-1000000000 \leq b \land b \leq 1000000000\right)\right) \land \left(-1000000000 \leq c \land c \leq 1000000000\right)\]

Math

\[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (pow (/ (- (* 3.0 (* a c)) (pow b 2.0)) (* 9.0 (pow a 2.0))) 3.0))

C

double code(double a, double b, double c) {
	return pow((((3.0 * (a * c)) - pow(b, 2.0)) / (9.0 * pow(a, 2.0))), 3.0);
}

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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((3.0d0 * (a * c)) - (b ** 2.0d0)) / (9.0d0 * (a ** 2.0d0))) ** 3.0d0
end function

Java

public static double code(double a, double b, double c) {
	return Math.pow((((3.0 * (a * c)) - Math.pow(b, 2.0)) / (9.0 * Math.pow(a, 2.0))), 3.0);
}

Python

def code(a, b, c):
	return math.pow((((3.0 * (a * c)) - math.pow(b, 2.0)) / (9.0 * math.pow(a, 2.0))), 3.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(3.0 * Float64(a * c)) - (b ^ 2.0)) / Float64(9.0 * (a ^ 2.0))) ^ 3.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((3.0 * (a * c)) - (b ^ 2.0)) / (9.0 * (a ^ 2.0))) ^ 3.0;
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(9.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Initial Program: 81.1% accurate, 1.0× speedup

Math

\[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (pow (/ (- (* 3.0 (* a c)) (pow b 2.0)) (* 9.0 (pow a 2.0))) 3.0))

C

double code(double a, double b, double c) {
	return pow((((3.0 * (a * c)) - pow(b, 2.0)) / (9.0 * pow(a, 2.0))), 3.0);
}

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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((3.0d0 * (a * c)) - (b ** 2.0d0)) / (9.0d0 * (a ** 2.0d0))) ** 3.0d0
end function

Java

public static double code(double a, double b, double c) {
	return Math.pow((((3.0 * (a * c)) - Math.pow(b, 2.0)) / (9.0 * Math.pow(a, 2.0))), 3.0);
}

Python

def code(a, b, c):
	return math.pow((((3.0 * (a * c)) - math.pow(b, 2.0)) / (9.0 * math.pow(a, 2.0))), 3.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(3.0 * Float64(a * c)) - (b ^ 2.0)) / Float64(9.0 * (a ^ 2.0))) ^ 3.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((3.0 * (a * c)) - (b ^ 2.0)) / (9.0 * (a ^ 2.0))) ^ 3.0;
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(9.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Alternative 1: 99.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\\ \left(t\_0 \cdot t\_0\right) \cdot t\_0 \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0
        (/
         (-
          (* -0.1111111111111111 (* (/ b a) b))
          (* -0.3333333333333333 c))
         a)))
  (* (* t_0 t_0) t_0)))

C

double code(double a, double b, double c) {
	double t_0 = ((-0.1111111111111111 * ((b / a) * b)) - (-0.3333333333333333 * c)) / a;
	return (t_0 * t_0) * t_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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = (((-0.1111111111111111d0) * ((b / a) * b)) - ((-0.3333333333333333d0) * c)) / a
    code = (t_0 * t_0) * t_0
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = ((-0.1111111111111111 * ((b / a) * b)) - (-0.3333333333333333 * c)) / a;
	return (t_0 * t_0) * t_0;
}

Python

def code(a, b, c):
	t_0 = ((-0.1111111111111111 * ((b / a) * b)) - (-0.3333333333333333 * c)) / a
	return (t_0 * t_0) * t_0

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(-0.1111111111111111 * Float64(Float64(b / a) * b)) - Float64(-0.3333333333333333 * c)) / a)
	return Float64(Float64(t_0 * t_0) * t_0)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = ((-0.1111111111111111 * ((b / a) * b)) - (-0.3333333333333333 * c)) / a;
	tmp = (t_0 * t_0) * t_0;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(-0.1111111111111111 * N[(N[(b / a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(-0.3333333333333333 * c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 81.1%

   \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

2. Taylor expanded in a around inf

   \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

   2. lower-+.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   3. lower-*.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   4. lower-/.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   5. lower-pow.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   6. lower-*.f64 94.0%

      \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

4. Applied rewrites 94.0%

   \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   2. +-commutative N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c + \frac{-1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   3. lift-*.f64 N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c + \frac{-1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

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

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   5. lower--.f64 N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   7. lower-*.f64 94.0%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   8. lift-/.f64 N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   9. lift-pow.f64 N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   10. pow2 N/A

       \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{b \cdot b}{a}}{a}\right)}^{3} \]

   11. associate-/l* N/A

       \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]

   12. lower-*.f64 N/A

       \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]

   13. lower-/.f64 99.3%

       \[\leadsto {\left(\frac{0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]

6. Applied rewrites 99.3%

   \[\leadsto {\left(\frac{0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3}} \]

   2. unpow3 N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a} \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right) \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a} \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right) \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}} \]

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\left(\frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}} \]
9. Add Preprocessing

Alternative 2: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(c \cdot a\right) \cdot 0.3333333333333333 - 0.1111111111111111 \cdot \left(b \cdot b\right)}{a \cdot a}\\ \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{-305}:\\ \;\;\;\;{\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0
        (/
         (-
          (* (* c a) 0.3333333333333333)
          (* 0.1111111111111111 (* b b)))
         (* a a))))
  (if (<= (pow b 2.0) 2e-305)
    (pow (/ (* 0.3333333333333333 c) a) 3.0)
    (* (* t_0 t_0) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * (b * b))) / (a * a);
	double tmp;
	if (pow(b, 2.0) <= 2e-305) {
		tmp = pow(((0.3333333333333333 * c) / a), 3.0);
	} else {
		tmp = (t_0 * t_0) * t_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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((c * a) * 0.3333333333333333d0) - (0.1111111111111111d0 * (b * b))) / (a * a)
    if ((b ** 2.0d0) <= 2d-305) then
        tmp = ((0.3333333333333333d0 * c) / a) ** 3.0d0
    else
        tmp = (t_0 * t_0) * t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * (b * b))) / (a * a);
	double tmp;
	if (Math.pow(b, 2.0) <= 2e-305) {
		tmp = Math.pow(((0.3333333333333333 * c) / a), 3.0);
	} else {
		tmp = (t_0 * t_0) * t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * (b * b))) / (a * a)
	tmp = 0
	if math.pow(b, 2.0) <= 2e-305:
		tmp = math.pow(((0.3333333333333333 * c) / a), 3.0)
	else:
		tmp = (t_0 * t_0) * t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(Float64(c * a) * 0.3333333333333333) - Float64(0.1111111111111111 * Float64(b * b))) / Float64(a * a))
	tmp = 0.0
	if ((b ^ 2.0) <= 2e-305)
		tmp = Float64(Float64(0.3333333333333333 * c) / a) ^ 3.0;
	else
		tmp = Float64(Float64(t_0 * t_0) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * (b * b))) / (a * a);
	tmp = 0.0;
	if ((b ^ 2.0) <= 2e-305)
		tmp = ((0.3333333333333333 * c) / a) ^ 3.0;
	else
		tmp = (t_0 * t_0) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(N[(c * a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(0.1111111111111111 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 2e-305], N[Power[N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 2e-305

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   if 2e-305 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3}} \]

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a} \cdot \frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a} \cdot \frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}} \]

   6. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\left(\frac{\left(c \cdot a\right) \cdot 0.3333333333333333 - 0.1111111111111111 \cdot \left(b \cdot b\right)}{a \cdot a} \cdot \frac{\left(c \cdot a\right) \cdot 0.3333333333333333 - 0.1111111111111111 \cdot \left(b \cdot b\right)}{a \cdot a}\right) \cdot \frac{\left(c \cdot a\right) \cdot 0.3333333333333333 - 0.1111111111111111 \cdot \left(b \cdot b\right)}{a \cdot a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 4.5 \cdot 10^{-153}:\\ \;\;\;\;{\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(c \cdot a\right) \cdot 3 - t\_0\right) \cdot \frac{\frac{0.012345679012345678 \cdot \left(\left(c \cdot 3\right) \cdot a - t\_0\right)}{a \cdot a}}{a \cdot a}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - t\_0 \cdot 0.1111111111111111\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<= (fabs b) 4.5e-153)
    (pow (/ (* 0.3333333333333333 c) a) 3.0)
    (*
     (/
      (*
       (- (* (* c a) 3.0) t_0)
       (/
        (/ (* 0.012345679012345678 (- (* (* c 3.0) a) t_0)) (* a a))
        (* a a)))
      (* a a))
     (- (* (* c a) 0.3333333333333333) (* t_0 0.1111111111111111))))))

C

double code(double a, double b, double c) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 4.5e-153) {
		tmp = pow(((0.3333333333333333 * c) / a), 3.0);
	} else {
		tmp = (((((c * a) * 3.0) - t_0) * (((0.012345679012345678 * (((c * 3.0) * a) - t_0)) / (a * a)) / (a * a))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	}
	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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(b) * abs(b)
    if (abs(b) <= 4.5d-153) then
        tmp = ((0.3333333333333333d0 * c) / a) ** 3.0d0
    else
        tmp = (((((c * a) * 3.0d0) - t_0) * (((0.012345679012345678d0 * (((c * 3.0d0) * a) - t_0)) / (a * a)) / (a * a))) / (a * a)) * (((c * a) * 0.3333333333333333d0) - (t_0 * 0.1111111111111111d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 4.5e-153) {
		tmp = Math.pow(((0.3333333333333333 * c) / a), 3.0);
	} else {
		tmp = (((((c * a) * 3.0) - t_0) * (((0.012345679012345678 * (((c * 3.0) * a) - t_0)) / (a * a)) / (a * a))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.fabs(b) * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 4.5e-153:
		tmp = math.pow(((0.3333333333333333 * c) / a), 3.0)
	else:
		tmp = (((((c * a) * 3.0) - t_0) * (((0.012345679012345678 * (((c * 3.0) * a) - t_0)) / (a * a)) / (a * a))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 4.5e-153)
		tmp = Float64(Float64(0.3333333333333333 * c) / a) ^ 3.0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(c * a) * 3.0) - t_0) * Float64(Float64(Float64(0.012345679012345678 * Float64(Float64(Float64(c * 3.0) * a) - t_0)) / Float64(a * a)) / Float64(a * a))) / Float64(a * a)) * Float64(Float64(Float64(c * a) * 0.3333333333333333) - Float64(t_0 * 0.1111111111111111)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = abs(b) * abs(b);
	tmp = 0.0;
	if (abs(b) <= 4.5e-153)
		tmp = ((0.3333333333333333 * c) / a) ^ 3.0;
	else
		tmp = (((((c * a) * 3.0) - t_0) * (((0.012345679012345678 * (((c * 3.0) * a) - t_0)) / (a * a)) / (a * a))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.5e-153], N[Power[N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[(N[(0.012345679012345678 * N[(N[(N[(c * 3.0), $MachinePrecision] * a), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(t$95$0 * 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5e-153

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   if 4.5e-153 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{a \cdot a}}{a \cdot a}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{a \cdot a}}{a \cdot a}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

   5. Applied rewrites 76.2%

      \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\frac{0.012345679012345678 \cdot \left(\left(c \cdot 3\right) \cdot a - b \cdot b\right)}{a \cdot a}}{a \cdot a}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 4.5 \cdot 10^{-153}:\\ \;\;\;\;{\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(c \cdot a\right) \cdot 3 - t\_0\right) \cdot \left(\left(c \cdot 3 - \frac{\left|b\right|}{a} \cdot \left|b\right|\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot a}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - t\_0 \cdot 0.1111111111111111\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<= (fabs b) 4.5e-153)
    (pow (/ (* 0.3333333333333333 c) a) 3.0)
    (*
     (/
      (*
       (- (* (* c a) 3.0) t_0)
       (*
        (- (* c 3.0) (* (/ (fabs b) a) (fabs b)))
        (/ 0.012345679012345678 (* (* a a) a))))
      (* a a))
     (- (* (* c a) 0.3333333333333333) (* t_0 0.1111111111111111))))))

C

double code(double a, double b, double c) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 4.5e-153) {
		tmp = pow(((0.3333333333333333 * c) / a), 3.0);
	} else {
		tmp = (((((c * a) * 3.0) - t_0) * (((c * 3.0) - ((fabs(b) / a) * fabs(b))) * (0.012345679012345678 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	}
	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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(b) * abs(b)
    if (abs(b) <= 4.5d-153) then
        tmp = ((0.3333333333333333d0 * c) / a) ** 3.0d0
    else
        tmp = (((((c * a) * 3.0d0) - t_0) * (((c * 3.0d0) - ((abs(b) / a) * abs(b))) * (0.012345679012345678d0 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333d0) - (t_0 * 0.1111111111111111d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 4.5e-153) {
		tmp = Math.pow(((0.3333333333333333 * c) / a), 3.0);
	} else {
		tmp = (((((c * a) * 3.0) - t_0) * (((c * 3.0) - ((Math.abs(b) / a) * Math.abs(b))) * (0.012345679012345678 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.fabs(b) * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 4.5e-153:
		tmp = math.pow(((0.3333333333333333 * c) / a), 3.0)
	else:
		tmp = (((((c * a) * 3.0) - t_0) * (((c * 3.0) - ((math.fabs(b) / a) * math.fabs(b))) * (0.012345679012345678 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 4.5e-153)
		tmp = Float64(Float64(0.3333333333333333 * c) / a) ^ 3.0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(c * a) * 3.0) - t_0) * Float64(Float64(Float64(c * 3.0) - Float64(Float64(abs(b) / a) * abs(b))) * Float64(0.012345679012345678 / Float64(Float64(a * a) * a)))) / Float64(a * a)) * Float64(Float64(Float64(c * a) * 0.3333333333333333) - Float64(t_0 * 0.1111111111111111)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = abs(b) * abs(b);
	tmp = 0.0;
	if (abs(b) <= 4.5e-153)
		tmp = ((0.3333333333333333 * c) / a) ^ 3.0;
	else
		tmp = (((((c * a) * 3.0) - t_0) * (((c * 3.0) - ((abs(b) / a) * abs(b))) * (0.012345679012345678 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.5e-153], N[Power[N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[(N[(c * 3.0), $MachinePrecision] - N[(N[(N[Abs[b], $MachinePrecision] / a), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.012345679012345678 / N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(t$95$0 * 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5e-153

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   if 4.5e-153 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot a}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\color{blue}{a \cdot \left(\left(a \cdot a\right) \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      9. times-frac N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a} \cdot \frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot a}\right)}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a} \cdot \frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot a}\right)}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

   5. Applied rewrites 71.6%

      \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\left(c \cdot 3 - \frac{b}{a} \cdot b\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot a}\right)}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 88.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ t_1 := \frac{0.3333333333333333 \cdot c}{a}\\ \mathbf{if}\;\left|b\right| \leq 4.5 \cdot 10^{-153}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(c \cdot a\right) \cdot 3 - t\_0\right) \cdot \left(\left(c \cdot 3 - \frac{\left|b\right|}{a} \cdot \left|b\right|\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot a}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - t\_0 \cdot 0.1111111111111111\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b)))
       (t_1 (/ (* 0.3333333333333333 c) a)))
  (if (<= (fabs b) 4.5e-153)
    (* (* t_1 t_1) t_1)
    (*
     (/
      (*
       (- (* (* c a) 3.0) t_0)
       (*
        (- (* c 3.0) (* (/ (fabs b) a) (fabs b)))
        (/ 0.012345679012345678 (* (* a a) a))))
      (* a a))
     (- (* (* c a) 0.3333333333333333) (* t_0 0.1111111111111111))))))

C

double code(double a, double b, double c) {
	double t_0 = fabs(b) * fabs(b);
	double t_1 = (0.3333333333333333 * c) / a;
	double tmp;
	if (fabs(b) <= 4.5e-153) {
		tmp = (t_1 * t_1) * t_1;
	} else {
		tmp = (((((c * a) * 3.0) - t_0) * (((c * 3.0) - ((fabs(b) / a) * fabs(b))) * (0.012345679012345678 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	}
	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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(b) * abs(b)
    t_1 = (0.3333333333333333d0 * c) / a
    if (abs(b) <= 4.5d-153) then
        tmp = (t_1 * t_1) * t_1
    else
        tmp = (((((c * a) * 3.0d0) - t_0) * (((c * 3.0d0) - ((abs(b) / a) * abs(b))) * (0.012345679012345678d0 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333d0) - (t_0 * 0.1111111111111111d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double t_1 = (0.3333333333333333 * c) / a;
	double tmp;
	if (Math.abs(b) <= 4.5e-153) {
		tmp = (t_1 * t_1) * t_1;
	} else {
		tmp = (((((c * a) * 3.0) - t_0) * (((c * 3.0) - ((Math.abs(b) / a) * Math.abs(b))) * (0.012345679012345678 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.fabs(b) * math.fabs(b)
	t_1 = (0.3333333333333333 * c) / a
	tmp = 0
	if math.fabs(b) <= 4.5e-153:
		tmp = (t_1 * t_1) * t_1
	else:
		tmp = (((((c * a) * 3.0) - t_0) * (((c * 3.0) - ((math.fabs(b) / a) * math.fabs(b))) * (0.012345679012345678 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(abs(b) * abs(b))
	t_1 = Float64(Float64(0.3333333333333333 * c) / a)
	tmp = 0.0
	if (abs(b) <= 4.5e-153)
		tmp = Float64(Float64(t_1 * t_1) * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(c * a) * 3.0) - t_0) * Float64(Float64(Float64(c * 3.0) - Float64(Float64(abs(b) / a) * abs(b))) * Float64(0.012345679012345678 / Float64(Float64(a * a) * a)))) / Float64(a * a)) * Float64(Float64(Float64(c * a) * 0.3333333333333333) - Float64(t_0 * 0.1111111111111111)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = abs(b) * abs(b);
	t_1 = (0.3333333333333333 * c) / a;
	tmp = 0.0;
	if (abs(b) <= 4.5e-153)
		tmp = (t_1 * t_1) * t_1;
	else
		tmp = (((((c * a) * 3.0) - t_0) * (((c * 3.0) - ((abs(b) / a) * abs(b))) * (0.012345679012345678 / ((a * a) * a)))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.5e-153], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[(N[(c * 3.0), $MachinePrecision] - N[(N[(N[Abs[b], $MachinePrecision] / a), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.012345679012345678 / N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(t$95$0 * 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5e-153

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 4.5e-153 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot a}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{81}}{\color{blue}{a \cdot \left(\left(a \cdot a\right) \cdot a\right)}}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      9. times-frac N/A

         \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a} \cdot \frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot a}\right)}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a} \cdot \frac{\frac{1}{81}}{\left(a \cdot a\right) \cdot a}\right)}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot \frac{1}{3} - \left(b \cdot b\right) \cdot \frac{1}{9}\right) \]

   5. Applied rewrites 71.6%

      \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\left(c \cdot 3 - \frac{b}{a} \cdot b\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot a}\right)}}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot a\right) \cdot 3\\ t_1 := \left|b\right| \cdot \left|b\right|\\ t_2 := t\_0 - t\_1\\ t_3 := \frac{0.3333333333333333 \cdot c}{a}\\ \mathbf{if}\;\left|b\right| \leq 2.2 \cdot 10^{-131}:\\ \;\;\;\;\left(t\_3 \cdot t\_3\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{81 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \cdot \left(\left(t\_1 - t\_0\right) \cdot \left(\frac{-0.1111111111111111}{a \cdot a} \cdot t\_2\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* c a) 3.0))
       (t_1 (* (fabs b) (fabs b)))
       (t_2 (- t_0 t_1))
       (t_3 (/ (* 0.3333333333333333 c) a)))
  (if (<= (fabs b) 2.2e-131)
    (* (* t_3 t_3) t_3)
    (*
     (/ t_2 (* 81.0 (* (* a a) (* a a))))
     (* (- t_1 t_0) (* (/ -0.1111111111111111 (* a a)) t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * 3.0;
	double t_1 = fabs(b) * fabs(b);
	double t_2 = t_0 - t_1;
	double t_3 = (0.3333333333333333 * c) / a;
	double tmp;
	if (fabs(b) <= 2.2e-131) {
		tmp = (t_3 * t_3) * t_3;
	} else {
		tmp = (t_2 / (81.0 * ((a * a) * (a * a)))) * ((t_1 - t_0) * ((-0.1111111111111111 / (a * a)) * t_2));
	}
	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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (c * a) * 3.0d0
    t_1 = abs(b) * abs(b)
    t_2 = t_0 - t_1
    t_3 = (0.3333333333333333d0 * c) / a
    if (abs(b) <= 2.2d-131) then
        tmp = (t_3 * t_3) * t_3
    else
        tmp = (t_2 / (81.0d0 * ((a * a) * (a * a)))) * ((t_1 - t_0) * (((-0.1111111111111111d0) / (a * a)) * t_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * a) * 3.0;
	double t_1 = Math.abs(b) * Math.abs(b);
	double t_2 = t_0 - t_1;
	double t_3 = (0.3333333333333333 * c) / a;
	double tmp;
	if (Math.abs(b) <= 2.2e-131) {
		tmp = (t_3 * t_3) * t_3;
	} else {
		tmp = (t_2 / (81.0 * ((a * a) * (a * a)))) * ((t_1 - t_0) * ((-0.1111111111111111 / (a * a)) * t_2));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c * a) * 3.0
	t_1 = math.fabs(b) * math.fabs(b)
	t_2 = t_0 - t_1
	t_3 = (0.3333333333333333 * c) / a
	tmp = 0
	if math.fabs(b) <= 2.2e-131:
		tmp = (t_3 * t_3) * t_3
	else:
		tmp = (t_2 / (81.0 * ((a * a) * (a * a)))) * ((t_1 - t_0) * ((-0.1111111111111111 / (a * a)) * t_2))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * 3.0)
	t_1 = Float64(abs(b) * abs(b))
	t_2 = Float64(t_0 - t_1)
	t_3 = Float64(Float64(0.3333333333333333 * c) / a)
	tmp = 0.0
	if (abs(b) <= 2.2e-131)
		tmp = Float64(Float64(t_3 * t_3) * t_3);
	else
		tmp = Float64(Float64(t_2 / Float64(81.0 * Float64(Float64(a * a) * Float64(a * a)))) * Float64(Float64(t_1 - t_0) * Float64(Float64(-0.1111111111111111 / Float64(a * a)) * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c * a) * 3.0;
	t_1 = abs(b) * abs(b);
	t_2 = t_0 - t_1;
	t_3 = (0.3333333333333333 * c) / a;
	tmp = 0.0;
	if (abs(b) <= 2.2e-131)
		tmp = (t_3 * t_3) * t_3;
	else
		tmp = (t_2 / (81.0 * ((a * a) * (a * a)))) * ((t_1 - t_0) * ((-0.1111111111111111 / (a * a)) * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.2e-131], N[(N[(t$95$3 * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[(t$95$2 / N[(81.0 * N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 - t$95$0), $MachinePrecision] * N[(N[(-0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2e-131

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 2.2e-131 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{81 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \cdot \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(\frac{-0.1111111111111111}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ t_1 := \left(c \cdot a\right) \cdot 3 - t\_0\\ t_2 := \frac{0.3333333333333333 \cdot c}{a}\\ \mathbf{if}\;\left|b\right| \leq 4.5 \cdot 10^{-153}:\\ \;\;\;\;\left(t\_2 \cdot t\_2\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot \left(t\_1 \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - t\_0 \cdot 0.1111111111111111\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b)))
       (t_1 (- (* (* c a) 3.0) t_0))
       (t_2 (/ (* 0.3333333333333333 c) a)))
  (if (<= (fabs b) 4.5e-153)
    (* (* t_2 t_2) t_2)
    (*
     (/
      (* t_1 (* t_1 (/ 0.012345679012345678 (* (* a a) (* a a)))))
      (* a a))
     (- (* (* c a) 0.3333333333333333) (* t_0 0.1111111111111111))))))

C

double code(double a, double b, double c) {
	double t_0 = fabs(b) * fabs(b);
	double t_1 = ((c * a) * 3.0) - t_0;
	double t_2 = (0.3333333333333333 * c) / a;
	double tmp;
	if (fabs(b) <= 4.5e-153) {
		tmp = (t_2 * t_2) * t_2;
	} else {
		tmp = ((t_1 * (t_1 * (0.012345679012345678 / ((a * a) * (a * a))))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	}
	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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = abs(b) * abs(b)
    t_1 = ((c * a) * 3.0d0) - t_0
    t_2 = (0.3333333333333333d0 * c) / a
    if (abs(b) <= 4.5d-153) then
        tmp = (t_2 * t_2) * t_2
    else
        tmp = ((t_1 * (t_1 * (0.012345679012345678d0 / ((a * a) * (a * a))))) / (a * a)) * (((c * a) * 0.3333333333333333d0) - (t_0 * 0.1111111111111111d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double t_1 = ((c * a) * 3.0) - t_0;
	double t_2 = (0.3333333333333333 * c) / a;
	double tmp;
	if (Math.abs(b) <= 4.5e-153) {
		tmp = (t_2 * t_2) * t_2;
	} else {
		tmp = ((t_1 * (t_1 * (0.012345679012345678 / ((a * a) * (a * a))))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.fabs(b) * math.fabs(b)
	t_1 = ((c * a) * 3.0) - t_0
	t_2 = (0.3333333333333333 * c) / a
	tmp = 0
	if math.fabs(b) <= 4.5e-153:
		tmp = (t_2 * t_2) * t_2
	else:
		tmp = ((t_1 * (t_1 * (0.012345679012345678 / ((a * a) * (a * a))))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(abs(b) * abs(b))
	t_1 = Float64(Float64(Float64(c * a) * 3.0) - t_0)
	t_2 = Float64(Float64(0.3333333333333333 * c) / a)
	tmp = 0.0
	if (abs(b) <= 4.5e-153)
		tmp = Float64(Float64(t_2 * t_2) * t_2);
	else
		tmp = Float64(Float64(Float64(t_1 * Float64(t_1 * Float64(0.012345679012345678 / Float64(Float64(a * a) * Float64(a * a))))) / Float64(a * a)) * Float64(Float64(Float64(c * a) * 0.3333333333333333) - Float64(t_0 * 0.1111111111111111)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = abs(b) * abs(b);
	t_1 = ((c * a) * 3.0) - t_0;
	t_2 = (0.3333333333333333 * c) / a;
	tmp = 0.0;
	if (abs(b) <= 4.5e-153)
		tmp = (t_2 * t_2) * t_2;
	else
		tmp = ((t_1 * (t_1 * (0.012345679012345678 / ((a * a) * (a * a))))) / (a * a)) * (((c * a) * 0.3333333333333333) - (t_0 * 0.1111111111111111));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.5e-153], N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(t$95$1 * N[(t$95$1 * N[(0.012345679012345678 / N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(t$95$0 * 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5e-153

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 4.5e-153 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot 3\right) \cdot a - \left|b\right| \cdot \left|b\right|\\ t_1 := \frac{0.3333333333333333 \cdot c}{a}\\ \mathbf{if}\;\left|b\right| \leq 2.2 \cdot 10^{-131}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \frac{0.012345679012345678 \cdot t\_0}{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot a}\right) \cdot \left(0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \left(\frac{\left|b\right|}{a} \cdot \left|b\right|\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (- (* (* c 3.0) a) (* (fabs b) (fabs b))))
       (t_1 (/ (* 0.3333333333333333 c) a)))
  (if (<= (fabs b) 2.2e-131)
    (* (* t_1 t_1) t_1)
    (*
     (*
      t_0
      (/ (* 0.012345679012345678 t_0) (* (* (* (* a a) a) a) a)))
     (-
      (* 0.3333333333333333 c)
      (* 0.1111111111111111 (* (/ (fabs b) a) (fabs b))))))))

C

double code(double a, double b, double c) {
	double t_0 = ((c * 3.0) * a) - (fabs(b) * fabs(b));
	double t_1 = (0.3333333333333333 * c) / a;
	double tmp;
	if (fabs(b) <= 2.2e-131) {
		tmp = (t_1 * t_1) * t_1;
	} else {
		tmp = (t_0 * ((0.012345679012345678 * t_0) / ((((a * a) * a) * a) * a))) * ((0.3333333333333333 * c) - (0.1111111111111111 * ((fabs(b) / a) * fabs(b))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((c * 3.0d0) * a) - (abs(b) * abs(b))
    t_1 = (0.3333333333333333d0 * c) / a
    if (abs(b) <= 2.2d-131) then
        tmp = (t_1 * t_1) * t_1
    else
        tmp = (t_0 * ((0.012345679012345678d0 * t_0) / ((((a * a) * a) * a) * a))) * ((0.3333333333333333d0 * c) - (0.1111111111111111d0 * ((abs(b) / a) * abs(b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = ((c * 3.0) * a) - (Math.abs(b) * Math.abs(b));
	double t_1 = (0.3333333333333333 * c) / a;
	double tmp;
	if (Math.abs(b) <= 2.2e-131) {
		tmp = (t_1 * t_1) * t_1;
	} else {
		tmp = (t_0 * ((0.012345679012345678 * t_0) / ((((a * a) * a) * a) * a))) * ((0.3333333333333333 * c) - (0.1111111111111111 * ((Math.abs(b) / a) * Math.abs(b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = ((c * 3.0) * a) - (math.fabs(b) * math.fabs(b))
	t_1 = (0.3333333333333333 * c) / a
	tmp = 0
	if math.fabs(b) <= 2.2e-131:
		tmp = (t_1 * t_1) * t_1
	else:
		tmp = (t_0 * ((0.012345679012345678 * t_0) / ((((a * a) * a) * a) * a))) * ((0.3333333333333333 * c) - (0.1111111111111111 * ((math.fabs(b) / a) * math.fabs(b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(c * 3.0) * a) - Float64(abs(b) * abs(b)))
	t_1 = Float64(Float64(0.3333333333333333 * c) / a)
	tmp = 0.0
	if (abs(b) <= 2.2e-131)
		tmp = Float64(Float64(t_1 * t_1) * t_1);
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(0.012345679012345678 * t_0) / Float64(Float64(Float64(Float64(a * a) * a) * a) * a))) * Float64(Float64(0.3333333333333333 * c) - Float64(0.1111111111111111 * Float64(Float64(abs(b) / a) * abs(b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = ((c * 3.0) * a) - (abs(b) * abs(b));
	t_1 = (0.3333333333333333 * c) / a;
	tmp = 0.0;
	if (abs(b) <= 2.2e-131)
		tmp = (t_1 * t_1) * t_1;
	else
		tmp = (t_0 * ((0.012345679012345678 * t_0) / ((((a * a) * a) * a) * a))) * ((0.3333333333333333 * c) - (0.1111111111111111 * ((abs(b) / a) * abs(b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(c * 3.0), $MachinePrecision] * a), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.2e-131], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t$95$0 * N[(N[(0.012345679012345678 * t$95$0), $MachinePrecision] / N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.3333333333333333 * c), $MachinePrecision] - N[(0.1111111111111111 * N[(N[(N[Abs[b], $MachinePrecision] / a), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2e-131

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 2.2e-131 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot 3\right) \cdot a - b \cdot b\right) \cdot \frac{0.012345679012345678 \cdot \left(\left(c \cdot 3\right) \cdot a - b \cdot b\right)}{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot a}\right) \cdot \left(0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot a\\ t_1 := \frac{0.3333333333333333 \cdot c}{a}\\ t_2 := \left|b\right| \cdot \left|b\right|\\ t_3 := \left(c \cdot 3\right) \cdot a - t\_2\\ t_4 := \left(c \cdot a\right) \cdot 3\\ t_5 := t\_2 - t\_4\\ \mathbf{if}\;\left|b\right| \leq 2.6 \cdot 10^{-131}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot t\_1\\ \mathbf{elif}\;\left|b\right| \leq 1.25 \cdot 10^{-81}:\\ \;\;\;\;\left(\left(c \cdot a\right) \cdot 0.3333333333333333 - 0.1111111111111111 \cdot t\_2\right) \cdot \left(\frac{0.012345679012345678 \cdot t\_3}{\left(t\_0 \cdot a\right) \cdot \left(a \cdot a\right)} \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_5 \cdot t\_5\right) \cdot \left(\frac{t\_4 - t\_2}{t\_0} \cdot \frac{0.0013717421124828531}{t\_0}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* a a) a))
       (t_1 (/ (* 0.3333333333333333 c) a))
       (t_2 (* (fabs b) (fabs b)))
       (t_3 (- (* (* c 3.0) a) t_2))
       (t_4 (* (* c a) 3.0))
       (t_5 (- t_2 t_4)))
  (if (<= (fabs b) 2.6e-131)
    (* (* t_1 t_1) t_1)
    (if (<= (fabs b) 1.25e-81)
      (*
       (- (* (* c a) 0.3333333333333333) (* 0.1111111111111111 t_2))
       (* (/ (* 0.012345679012345678 t_3) (* (* t_0 a) (* a a))) t_3))
      (*
       (* t_5 t_5)
       (* (/ (- t_4 t_2) t_0) (/ 0.0013717421124828531 t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (a * a) * a;
	double t_1 = (0.3333333333333333 * c) / a;
	double t_2 = fabs(b) * fabs(b);
	double t_3 = ((c * 3.0) * a) - t_2;
	double t_4 = (c * a) * 3.0;
	double t_5 = t_2 - t_4;
	double tmp;
	if (fabs(b) <= 2.6e-131) {
		tmp = (t_1 * t_1) * t_1;
	} else if (fabs(b) <= 1.25e-81) {
		tmp = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * t_2)) * (((0.012345679012345678 * t_3) / ((t_0 * a) * (a * a))) * t_3);
	} else {
		tmp = (t_5 * t_5) * (((t_4 - t_2) / t_0) * (0.0013717421124828531 / t_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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (a * a) * a
    t_1 = (0.3333333333333333d0 * c) / a
    t_2 = abs(b) * abs(b)
    t_3 = ((c * 3.0d0) * a) - t_2
    t_4 = (c * a) * 3.0d0
    t_5 = t_2 - t_4
    if (abs(b) <= 2.6d-131) then
        tmp = (t_1 * t_1) * t_1
    else if (abs(b) <= 1.25d-81) then
        tmp = (((c * a) * 0.3333333333333333d0) - (0.1111111111111111d0 * t_2)) * (((0.012345679012345678d0 * t_3) / ((t_0 * a) * (a * a))) * t_3)
    else
        tmp = (t_5 * t_5) * (((t_4 - t_2) / t_0) * (0.0013717421124828531d0 / t_0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * a) * a;
	double t_1 = (0.3333333333333333 * c) / a;
	double t_2 = Math.abs(b) * Math.abs(b);
	double t_3 = ((c * 3.0) * a) - t_2;
	double t_4 = (c * a) * 3.0;
	double t_5 = t_2 - t_4;
	double tmp;
	if (Math.abs(b) <= 2.6e-131) {
		tmp = (t_1 * t_1) * t_1;
	} else if (Math.abs(b) <= 1.25e-81) {
		tmp = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * t_2)) * (((0.012345679012345678 * t_3) / ((t_0 * a) * (a * a))) * t_3);
	} else {
		tmp = (t_5 * t_5) * (((t_4 - t_2) / t_0) * (0.0013717421124828531 / t_0));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (a * a) * a
	t_1 = (0.3333333333333333 * c) / a
	t_2 = math.fabs(b) * math.fabs(b)
	t_3 = ((c * 3.0) * a) - t_2
	t_4 = (c * a) * 3.0
	t_5 = t_2 - t_4
	tmp = 0
	if math.fabs(b) <= 2.6e-131:
		tmp = (t_1 * t_1) * t_1
	elif math.fabs(b) <= 1.25e-81:
		tmp = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * t_2)) * (((0.012345679012345678 * t_3) / ((t_0 * a) * (a * a))) * t_3)
	else:
		tmp = (t_5 * t_5) * (((t_4 - t_2) / t_0) * (0.0013717421124828531 / t_0))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * a) * a)
	t_1 = Float64(Float64(0.3333333333333333 * c) / a)
	t_2 = Float64(abs(b) * abs(b))
	t_3 = Float64(Float64(Float64(c * 3.0) * a) - t_2)
	t_4 = Float64(Float64(c * a) * 3.0)
	t_5 = Float64(t_2 - t_4)
	tmp = 0.0
	if (abs(b) <= 2.6e-131)
		tmp = Float64(Float64(t_1 * t_1) * t_1);
	elseif (abs(b) <= 1.25e-81)
		tmp = Float64(Float64(Float64(Float64(c * a) * 0.3333333333333333) - Float64(0.1111111111111111 * t_2)) * Float64(Float64(Float64(0.012345679012345678 * t_3) / Float64(Float64(t_0 * a) * Float64(a * a))) * t_3));
	else
		tmp = Float64(Float64(t_5 * t_5) * Float64(Float64(Float64(t_4 - t_2) / t_0) * Float64(0.0013717421124828531 / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (a * a) * a;
	t_1 = (0.3333333333333333 * c) / a;
	t_2 = abs(b) * abs(b);
	t_3 = ((c * 3.0) * a) - t_2;
	t_4 = (c * a) * 3.0;
	t_5 = t_2 - t_4;
	tmp = 0.0;
	if (abs(b) <= 2.6e-131)
		tmp = (t_1 * t_1) * t_1;
	elseif (abs(b) <= 1.25e-81)
		tmp = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * t_2)) * (((0.012345679012345678 * t_3) / ((t_0 * a) * (a * a))) * t_3);
	else
		tmp = (t_5 * t_5) * (((t_4 - t_2) / t_0) * (0.0013717421124828531 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(c * 3.0), $MachinePrecision] * a), $MachinePrecision] - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 - t$95$4), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.6e-131], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.25e-81], N[(N[(N[(N[(c * a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(0.1111111111111111 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.012345679012345678 * t$95$3), $MachinePrecision] / N[(N[(t$95$0 * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$5 * t$95$5), $MachinePrecision] * N[(N[(N[(t$95$4 - t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(0.0013717421124828531 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.6e-131

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 2.6e-131 < b < 1.25e-81

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)} \]

   5. Applied rewrites 59.3%

      \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 0.3333333333333333 - 0.1111111111111111 \cdot \left(b \cdot b\right)\right) \cdot \left(\frac{0.012345679012345678 \cdot \left(\left(c \cdot 3\right) \cdot a - b \cdot b\right)}{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \left(\left(c \cdot 3\right) \cdot a - b \cdot b\right)\right)} \]

   if 1.25e-81 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}}^{3} \]

      3. mult-flip N/A

         \[\leadsto {\color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \frac{1}{9 \cdot {a}^{2}}\right)}}^{3} \]

      4. cube-prod N/A

         \[\leadsto \color{blue}{{\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)}^{3} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}} \]

      5. unpow3 N/A

         \[\leadsto \color{blue}{\left(\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right)} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

   3. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \color{blue}{\left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(a \cdot a\right)} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\left(a \cdot a\right) \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      8. unswap-sqr N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)}} \]

      11. times-frac N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \color{blue}{\left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a \cdot \left(a \cdot a\right)} \cdot \frac{\frac{1}{729}}{a \cdot \left(a \cdot a\right)}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a \cdot \left(a \cdot a\right)} \cdot \frac{\color{blue}{{\frac{1}{9}}^{3}}}{a \cdot \left(a \cdot a\right)}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a \cdot \left(a \cdot a\right)} \cdot \frac{{\frac{1}{9}}^{3}}{\color{blue}{a \cdot \left(a \cdot a\right)}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a \cdot \left(a \cdot a\right)} \cdot \frac{{\frac{1}{9}}^{3}}{a \cdot \color{blue}{\left(a \cdot a\right)}}\right) \]

      15. cube-unmult N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a \cdot \left(a \cdot a\right)} \cdot \frac{{\frac{1}{9}}^{3}}{\color{blue}{{a}^{3}}}\right) \]

      16. cube-div N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a \cdot \left(a \cdot a\right)} \cdot \color{blue}{{\left(\frac{\frac{1}{9}}{a}\right)}^{3}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{a \cdot \left(a \cdot a\right)} \cdot {\color{blue}{\left(\frac{\frac{1}{9}}{a}\right)}}^{3}\right) \]

   5. Applied rewrites 46.5%

      \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \color{blue}{\left(\frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{\left(a \cdot a\right) \cdot a} \cdot \frac{0.0013717421124828531}{\left(a \cdot a\right) \cdot a}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ t_1 := \frac{0.3333333333333333 \cdot c}{a}\\ t_2 := \left(c \cdot 3\right) \cdot a - t\_0\\ \mathbf{if}\;\left|b\right| \leq 2.6 \cdot 10^{-131}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot a\right) \cdot 0.3333333333333333 - 0.1111111111111111 \cdot t\_0\right) \cdot \left(\frac{0.012345679012345678 \cdot t\_2}{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right)} \cdot t\_2\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b)))
       (t_1 (/ (* 0.3333333333333333 c) a))
       (t_2 (- (* (* c 3.0) a) t_0)))
  (if (<= (fabs b) 2.6e-131)
    (* (* t_1 t_1) t_1)
    (*
     (- (* (* c a) 0.3333333333333333) (* 0.1111111111111111 t_0))
     (*
      (/ (* 0.012345679012345678 t_2) (* (* (* (* a a) a) a) (* a a)))
      t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs(b) * fabs(b);
	double t_1 = (0.3333333333333333 * c) / a;
	double t_2 = ((c * 3.0) * a) - t_0;
	double tmp;
	if (fabs(b) <= 2.6e-131) {
		tmp = (t_1 * t_1) * t_1;
	} else {
		tmp = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * t_0)) * (((0.012345679012345678 * t_2) / ((((a * a) * a) * a) * (a * a))) * t_2);
	}
	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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = abs(b) * abs(b)
    t_1 = (0.3333333333333333d0 * c) / a
    t_2 = ((c * 3.0d0) * a) - t_0
    if (abs(b) <= 2.6d-131) then
        tmp = (t_1 * t_1) * t_1
    else
        tmp = (((c * a) * 0.3333333333333333d0) - (0.1111111111111111d0 * t_0)) * (((0.012345679012345678d0 * t_2) / ((((a * a) * a) * a) * (a * a))) * t_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double t_1 = (0.3333333333333333 * c) / a;
	double t_2 = ((c * 3.0) * a) - t_0;
	double tmp;
	if (Math.abs(b) <= 2.6e-131) {
		tmp = (t_1 * t_1) * t_1;
	} else {
		tmp = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * t_0)) * (((0.012345679012345678 * t_2) / ((((a * a) * a) * a) * (a * a))) * t_2);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.fabs(b) * math.fabs(b)
	t_1 = (0.3333333333333333 * c) / a
	t_2 = ((c * 3.0) * a) - t_0
	tmp = 0
	if math.fabs(b) <= 2.6e-131:
		tmp = (t_1 * t_1) * t_1
	else:
		tmp = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * t_0)) * (((0.012345679012345678 * t_2) / ((((a * a) * a) * a) * (a * a))) * t_2)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(abs(b) * abs(b))
	t_1 = Float64(Float64(0.3333333333333333 * c) / a)
	t_2 = Float64(Float64(Float64(c * 3.0) * a) - t_0)
	tmp = 0.0
	if (abs(b) <= 2.6e-131)
		tmp = Float64(Float64(t_1 * t_1) * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(c * a) * 0.3333333333333333) - Float64(0.1111111111111111 * t_0)) * Float64(Float64(Float64(0.012345679012345678 * t_2) / Float64(Float64(Float64(Float64(a * a) * a) * a) * Float64(a * a))) * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = abs(b) * abs(b);
	t_1 = (0.3333333333333333 * c) / a;
	t_2 = ((c * 3.0) * a) - t_0;
	tmp = 0.0;
	if (abs(b) <= 2.6e-131)
		tmp = (t_1 * t_1) * t_1;
	else
		tmp = (((c * a) * 0.3333333333333333) - (0.1111111111111111 * t_0)) * (((0.012345679012345678 * t_2) / ((((a * a) * a) * a) * (a * a))) * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(c * 3.0), $MachinePrecision] * a), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.6e-131], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(c * a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(0.1111111111111111 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.012345679012345678 * t$95$2), $MachinePrecision] / N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-131

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 2.6e-131 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.012345679012345678}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)} \]

   5. Applied rewrites 59.3%

      \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 0.3333333333333333 - 0.1111111111111111 \cdot \left(b \cdot b\right)\right) \cdot \left(\frac{0.012345679012345678 \cdot \left(\left(c \cdot 3\right) \cdot a - b \cdot b\right)}{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \left(\left(c \cdot 3\right) \cdot a - b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot a\right) \cdot 3 - \left|b\right| \cdot \left|b\right|\\ t_1 := \frac{0.3333333333333333 \cdot c}{a}\\ t_2 := \left(a \cdot a\right) \cdot a\\ \mathbf{if}\;\left|b\right| \leq 2.6 \cdot 10^{-131}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_0 \cdot \left(\frac{0.0013717421124828531}{t\_2 \cdot t\_2} \cdot t\_0\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (- (* (* c a) 3.0) (* (fabs b) (fabs b))))
       (t_1 (/ (* 0.3333333333333333 c) a))
       (t_2 (* (* a a) a)))
  (if (<= (fabs b) 2.6e-131)
    (* (* t_1 t_1) t_1)
    (* t_0 (* t_0 (* (/ 0.0013717421124828531 (* t_2 t_2)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = ((c * a) * 3.0) - (fabs(b) * fabs(b));
	double t_1 = (0.3333333333333333 * c) / a;
	double t_2 = (a * a) * a;
	double tmp;
	if (fabs(b) <= 2.6e-131) {
		tmp = (t_1 * t_1) * t_1;
	} else {
		tmp = t_0 * (t_0 * ((0.0013717421124828531 / (t_2 * t_2)) * t_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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((c * a) * 3.0d0) - (abs(b) * abs(b))
    t_1 = (0.3333333333333333d0 * c) / a
    t_2 = (a * a) * a
    if (abs(b) <= 2.6d-131) then
        tmp = (t_1 * t_1) * t_1
    else
        tmp = t_0 * (t_0 * ((0.0013717421124828531d0 / (t_2 * t_2)) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = ((c * a) * 3.0) - (Math.abs(b) * Math.abs(b));
	double t_1 = (0.3333333333333333 * c) / a;
	double t_2 = (a * a) * a;
	double tmp;
	if (Math.abs(b) <= 2.6e-131) {
		tmp = (t_1 * t_1) * t_1;
	} else {
		tmp = t_0 * (t_0 * ((0.0013717421124828531 / (t_2 * t_2)) * t_0));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = ((c * a) * 3.0) - (math.fabs(b) * math.fabs(b))
	t_1 = (0.3333333333333333 * c) / a
	t_2 = (a * a) * a
	tmp = 0
	if math.fabs(b) <= 2.6e-131:
		tmp = (t_1 * t_1) * t_1
	else:
		tmp = t_0 * (t_0 * ((0.0013717421124828531 / (t_2 * t_2)) * t_0))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(c * a) * 3.0) - Float64(abs(b) * abs(b)))
	t_1 = Float64(Float64(0.3333333333333333 * c) / a)
	t_2 = Float64(Float64(a * a) * a)
	tmp = 0.0
	if (abs(b) <= 2.6e-131)
		tmp = Float64(Float64(t_1 * t_1) * t_1);
	else
		tmp = Float64(t_0 * Float64(t_0 * Float64(Float64(0.0013717421124828531 / Float64(t_2 * t_2)) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = ((c * a) * 3.0) - (abs(b) * abs(b));
	t_1 = (0.3333333333333333 * c) / a;
	t_2 = (a * a) * a;
	tmp = 0.0;
	if (abs(b) <= 2.6e-131)
		tmp = (t_1 * t_1) * t_1;
	else
		tmp = t_0 * (t_0 * ((0.0013717421124828531 / (t_2 * t_2)) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.6e-131], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$0 * N[(t$95$0 * N[(N[(0.0013717421124828531 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-131

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 2.6e-131 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}}^{3} \]

      3. mult-flip N/A

         \[\leadsto {\color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \frac{1}{9 \cdot {a}^{2}}\right)}}^{3} \]

      4. cube-prod N/A

         \[\leadsto \color{blue}{{\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)}^{3} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}} \]

      5. unpow3 N/A

         \[\leadsto \color{blue}{\left(\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right)} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

   3. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \color{blue}{\left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \color{blue}{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(a \cdot a\right)} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\left(a \cdot a\right) \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      8. unswap-sqr N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)}} \]

      11. associate-/r* N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \color{blue}{\frac{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{a \cdot \left(a \cdot a\right)}}{a \cdot \left(a \cdot a\right)}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \color{blue}{\frac{\frac{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{1}{729}}{a \cdot \left(a \cdot a\right)}}{a \cdot \left(a \cdot a\right)}} \]

   5. Applied rewrites 46.8%

      \[\leadsto \left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \color{blue}{\frac{\frac{0.0013717421124828531 \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right)} \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a} \]

      3. sqr-neg-rev N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right)\right)\right)} \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a} \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right)\right)\right) \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a} \]

      5. sub-negate-rev N/A

         \[\leadsto \left(\color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)} \cdot \left(\mathsf{neg}\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right)\right)\right) \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a} \]

      6. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)} \cdot \left(\mathsf{neg}\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right)\right)\right) \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a} \]

      7. lift--.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)}\right)\right)\right) \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a} \]

      8. sub-negate-rev N/A

         \[\leadsto \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}\right) \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a} \]

      9. lift--.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}\right) \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a} \]

      10. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{\frac{1}{729} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)}{\left(a \cdot a\right) \cdot a}}{\left(a \cdot a\right) \cdot a}\right)} \]

   7. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 80.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot a\right) \cdot 3\\ t_1 := \left|b\right| \cdot \left|b\right|\\ t_2 := t\_0 - t\_1\\ t_3 := \frac{0.3333333333333333 \cdot c}{a}\\ t_4 := t\_1 - t\_0\\ \mathbf{if}\;\left|b\right| \leq 2.25 \cdot 10^{-138}:\\ \;\;\;\;\left(t\_3 \cdot t\_3\right) \cdot t\_3\\ \mathbf{elif}\;\left|b\right| \leq 2.45 \cdot 10^{-80}:\\ \;\;\;\;t\_2 \cdot \left(\left(-0.037037037037037035 \cdot \frac{c}{a}\right) \cdot \frac{t\_4 \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 \cdot t\_4\right) \cdot \frac{t\_2}{729 \cdot \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)\right)}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* c a) 3.0))
       (t_1 (* (fabs b) (fabs b)))
       (t_2 (- t_0 t_1))
       (t_3 (/ (* 0.3333333333333333 c) a))
       (t_4 (- t_1 t_0)))
  (if (<= (fabs b) 2.25e-138)
    (* (* t_3 t_3) t_3)
    (if (<= (fabs b) 2.45e-80)
      (*
       t_2
       (*
        (* -0.037037037037037035 (/ c a))
        (/ (* t_4 (/ 0.1111111111111111 (* a a))) (* a a))))
      (*
       (* t_4 t_4)
       (/ t_2 (* 729.0 (* (* (* a a) (* a a)) (* a a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * 3.0;
	double t_1 = fabs(b) * fabs(b);
	double t_2 = t_0 - t_1;
	double t_3 = (0.3333333333333333 * c) / a;
	double t_4 = t_1 - t_0;
	double tmp;
	if (fabs(b) <= 2.25e-138) {
		tmp = (t_3 * t_3) * t_3;
	} else if (fabs(b) <= 2.45e-80) {
		tmp = t_2 * ((-0.037037037037037035 * (c / a)) * ((t_4 * (0.1111111111111111 / (a * a))) / (a * a)));
	} else {
		tmp = (t_4 * t_4) * (t_2 / (729.0 * (((a * a) * (a * a)) * (a * a))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (c * a) * 3.0d0
    t_1 = abs(b) * abs(b)
    t_2 = t_0 - t_1
    t_3 = (0.3333333333333333d0 * c) / a
    t_4 = t_1 - t_0
    if (abs(b) <= 2.25d-138) then
        tmp = (t_3 * t_3) * t_3
    else if (abs(b) <= 2.45d-80) then
        tmp = t_2 * (((-0.037037037037037035d0) * (c / a)) * ((t_4 * (0.1111111111111111d0 / (a * a))) / (a * a)))
    else
        tmp = (t_4 * t_4) * (t_2 / (729.0d0 * (((a * a) * (a * a)) * (a * a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * a) * 3.0;
	double t_1 = Math.abs(b) * Math.abs(b);
	double t_2 = t_0 - t_1;
	double t_3 = (0.3333333333333333 * c) / a;
	double t_4 = t_1 - t_0;
	double tmp;
	if (Math.abs(b) <= 2.25e-138) {
		tmp = (t_3 * t_3) * t_3;
	} else if (Math.abs(b) <= 2.45e-80) {
		tmp = t_2 * ((-0.037037037037037035 * (c / a)) * ((t_4 * (0.1111111111111111 / (a * a))) / (a * a)));
	} else {
		tmp = (t_4 * t_4) * (t_2 / (729.0 * (((a * a) * (a * a)) * (a * a))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c * a) * 3.0
	t_1 = math.fabs(b) * math.fabs(b)
	t_2 = t_0 - t_1
	t_3 = (0.3333333333333333 * c) / a
	t_4 = t_1 - t_0
	tmp = 0
	if math.fabs(b) <= 2.25e-138:
		tmp = (t_3 * t_3) * t_3
	elif math.fabs(b) <= 2.45e-80:
		tmp = t_2 * ((-0.037037037037037035 * (c / a)) * ((t_4 * (0.1111111111111111 / (a * a))) / (a * a)))
	else:
		tmp = (t_4 * t_4) * (t_2 / (729.0 * (((a * a) * (a * a)) * (a * a))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * 3.0)
	t_1 = Float64(abs(b) * abs(b))
	t_2 = Float64(t_0 - t_1)
	t_3 = Float64(Float64(0.3333333333333333 * c) / a)
	t_4 = Float64(t_1 - t_0)
	tmp = 0.0
	if (abs(b) <= 2.25e-138)
		tmp = Float64(Float64(t_3 * t_3) * t_3);
	elseif (abs(b) <= 2.45e-80)
		tmp = Float64(t_2 * Float64(Float64(-0.037037037037037035 * Float64(c / a)) * Float64(Float64(t_4 * Float64(0.1111111111111111 / Float64(a * a))) / Float64(a * a))));
	else
		tmp = Float64(Float64(t_4 * t_4) * Float64(t_2 / Float64(729.0 * Float64(Float64(Float64(a * a) * Float64(a * a)) * Float64(a * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c * a) * 3.0;
	t_1 = abs(b) * abs(b);
	t_2 = t_0 - t_1;
	t_3 = (0.3333333333333333 * c) / a;
	t_4 = t_1 - t_0;
	tmp = 0.0;
	if (abs(b) <= 2.25e-138)
		tmp = (t_3 * t_3) * t_3;
	elseif (abs(b) <= 2.45e-80)
		tmp = t_2 * ((-0.037037037037037035 * (c / a)) * ((t_4 * (0.1111111111111111 / (a * a))) / (a * a)));
	else
		tmp = (t_4 * t_4) * (t_2 / (729.0 * (((a * a) * (a * a)) * (a * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 - t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.25e-138], N[(N[(t$95$3 * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 2.45e-80], N[(t$95$2 * N[(N[(-0.037037037037037035 * N[(c / a), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 * N[(0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(t$95$2 / N[(729.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.25e-138

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 2.25e-138 < b < 2.45e-80

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\frac{0.1111111111111111 \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \left(\frac{0.1111111111111111}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)\right)\right)} \]

   5. Applied rewrites 76.0%

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\frac{\left(0.1111111111111111 \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot 0.3333333333333333\right) \cdot 0.1111111111111111}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\color{blue}{\left(\frac{-1}{27} \cdot \frac{c}{a}\right)} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\frac{-1}{27} \cdot \color{blue}{\frac{c}{a}}\right) \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      2. lower-/.f64 52.6%

         \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(-0.037037037037037035 \cdot \frac{c}{\color{blue}{a}}\right) \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   8. Applied rewrites 52.6%

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\color{blue}{\left(-0.037037037037037035 \cdot \frac{c}{a}\right)} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   if 2.45e-80 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}}^{3} \]

      3. cube-div N/A

         \[\leadsto \color{blue}{\frac{{\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)}^{3}}{{\left(9 \cdot {a}^{2}\right)}^{3}}} \]

      4. unpow3 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)}}{{\left(9 \cdot {a}^{2}\right)}^{3}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{{\left(9 \cdot {a}^{2}\right)}^{3}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{{\left(9 \cdot {a}^{2}\right)}^{3}}} \]

   3. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \frac{\left(c \cdot a\right) \cdot 3 - b \cdot b}{729 \cdot \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left(3 \cdot c\right) \cdot a\\ t_1 := \left|b\right| \cdot \left|b\right|\\ t_2 := \frac{0.3333333333333333 \cdot c}{a}\\ t_3 := \left(c \cdot a\right) \cdot 3\\ t_4 := t\_1 - t\_0\\ \mathbf{if}\;\left|b\right| \leq 2.25 \cdot 10^{-138}:\\ \;\;\;\;\left(t\_2 \cdot t\_2\right) \cdot t\_2\\ \mathbf{elif}\;\left|b\right| \leq 2.45 \cdot 10^{-80}:\\ \;\;\;\;\left(t\_3 - t\_1\right) \cdot \left(\left(-0.037037037037037035 \cdot \frac{c}{a}\right) \cdot \frac{\left(t\_1 - t\_3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 \cdot t\_4\right) \cdot \left(\left(t\_0 - t\_1\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* 3.0 c) a))
       (t_1 (* (fabs b) (fabs b)))
       (t_2 (/ (* 0.3333333333333333 c) a))
       (t_3 (* (* c a) 3.0))
       (t_4 (- t_1 t_0)))
  (if (<= (fabs b) 2.25e-138)
    (* (* t_2 t_2) t_2)
    (if (<= (fabs b) 2.45e-80)
      (*
       (- t_3 t_1)
       (*
        (* -0.037037037037037035 (/ c a))
        (/ (* (- t_1 t_3) (/ 0.1111111111111111 (* a a))) (* a a))))
      (*
       (* t_4 t_4)
       (*
        (- t_0 t_1)
        (/ 0.0013717421124828531 (* (* (* a a) (* a a)) (* a a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (3.0 * c) * a;
	double t_1 = fabs(b) * fabs(b);
	double t_2 = (0.3333333333333333 * c) / a;
	double t_3 = (c * a) * 3.0;
	double t_4 = t_1 - t_0;
	double tmp;
	if (fabs(b) <= 2.25e-138) {
		tmp = (t_2 * t_2) * t_2;
	} else if (fabs(b) <= 2.45e-80) {
		tmp = (t_3 - t_1) * ((-0.037037037037037035 * (c / a)) * (((t_1 - t_3) * (0.1111111111111111 / (a * a))) / (a * a)));
	} else {
		tmp = (t_4 * t_4) * ((t_0 - t_1) * (0.0013717421124828531 / (((a * a) * (a * a)) * (a * a))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (3.0d0 * c) * a
    t_1 = abs(b) * abs(b)
    t_2 = (0.3333333333333333d0 * c) / a
    t_3 = (c * a) * 3.0d0
    t_4 = t_1 - t_0
    if (abs(b) <= 2.25d-138) then
        tmp = (t_2 * t_2) * t_2
    else if (abs(b) <= 2.45d-80) then
        tmp = (t_3 - t_1) * (((-0.037037037037037035d0) * (c / a)) * (((t_1 - t_3) * (0.1111111111111111d0 / (a * a))) / (a * a)))
    else
        tmp = (t_4 * t_4) * ((t_0 - t_1) * (0.0013717421124828531d0 / (((a * a) * (a * a)) * (a * a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (3.0 * c) * a;
	double t_1 = Math.abs(b) * Math.abs(b);
	double t_2 = (0.3333333333333333 * c) / a;
	double t_3 = (c * a) * 3.0;
	double t_4 = t_1 - t_0;
	double tmp;
	if (Math.abs(b) <= 2.25e-138) {
		tmp = (t_2 * t_2) * t_2;
	} else if (Math.abs(b) <= 2.45e-80) {
		tmp = (t_3 - t_1) * ((-0.037037037037037035 * (c / a)) * (((t_1 - t_3) * (0.1111111111111111 / (a * a))) / (a * a)));
	} else {
		tmp = (t_4 * t_4) * ((t_0 - t_1) * (0.0013717421124828531 / (((a * a) * (a * a)) * (a * a))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (3.0 * c) * a
	t_1 = math.fabs(b) * math.fabs(b)
	t_2 = (0.3333333333333333 * c) / a
	t_3 = (c * a) * 3.0
	t_4 = t_1 - t_0
	tmp = 0
	if math.fabs(b) <= 2.25e-138:
		tmp = (t_2 * t_2) * t_2
	elif math.fabs(b) <= 2.45e-80:
		tmp = (t_3 - t_1) * ((-0.037037037037037035 * (c / a)) * (((t_1 - t_3) * (0.1111111111111111 / (a * a))) / (a * a)))
	else:
		tmp = (t_4 * t_4) * ((t_0 - t_1) * (0.0013717421124828531 / (((a * a) * (a * a)) * (a * a))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(3.0 * c) * a)
	t_1 = Float64(abs(b) * abs(b))
	t_2 = Float64(Float64(0.3333333333333333 * c) / a)
	t_3 = Float64(Float64(c * a) * 3.0)
	t_4 = Float64(t_1 - t_0)
	tmp = 0.0
	if (abs(b) <= 2.25e-138)
		tmp = Float64(Float64(t_2 * t_2) * t_2);
	elseif (abs(b) <= 2.45e-80)
		tmp = Float64(Float64(t_3 - t_1) * Float64(Float64(-0.037037037037037035 * Float64(c / a)) * Float64(Float64(Float64(t_1 - t_3) * Float64(0.1111111111111111 / Float64(a * a))) / Float64(a * a))));
	else
		tmp = Float64(Float64(t_4 * t_4) * Float64(Float64(t_0 - t_1) * Float64(0.0013717421124828531 / Float64(Float64(Float64(a * a) * Float64(a * a)) * Float64(a * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (3.0 * c) * a;
	t_1 = abs(b) * abs(b);
	t_2 = (0.3333333333333333 * c) / a;
	t_3 = (c * a) * 3.0;
	t_4 = t_1 - t_0;
	tmp = 0.0;
	if (abs(b) <= 2.25e-138)
		tmp = (t_2 * t_2) * t_2;
	elseif (abs(b) <= 2.45e-80)
		tmp = (t_3 - t_1) * ((-0.037037037037037035 * (c / a)) * (((t_1 - t_3) * (0.1111111111111111 / (a * a))) / (a * a)));
	else
		tmp = (t_4 * t_4) * ((t_0 - t_1) * (0.0013717421124828531 / (((a * a) * (a * a)) * (a * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(3.0 * c), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 - t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.25e-138], N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 2.45e-80], N[(N[(t$95$3 - t$95$1), $MachinePrecision] * N[(N[(-0.037037037037037035 * N[(c / a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$1 - t$95$3), $MachinePrecision] * N[(0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(N[(t$95$0 - t$95$1), $MachinePrecision] * N[(0.0013717421124828531 / N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.25e-138

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 2.25e-138 < b < 2.45e-80

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\frac{0.1111111111111111 \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \left(\frac{0.1111111111111111}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)\right)\right)} \]

   5. Applied rewrites 76.0%

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\frac{\left(0.1111111111111111 \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot 0.3333333333333333\right) \cdot 0.1111111111111111}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\color{blue}{\left(\frac{-1}{27} \cdot \frac{c}{a}\right)} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\frac{-1}{27} \cdot \color{blue}{\frac{c}{a}}\right) \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      2. lower-/.f64 52.6%

         \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(-0.037037037037037035 \cdot \frac{c}{\color{blue}{a}}\right) \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   8. Applied rewrites 52.6%

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\color{blue}{\left(-0.037037037037037035 \cdot \frac{c}{a}\right)} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   if 2.45e-80 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}}^{3} \]

      3. mult-flip N/A

         \[\leadsto {\color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \frac{1}{9 \cdot {a}^{2}}\right)}}^{3} \]

      4. cube-prod N/A

         \[\leadsto \color{blue}{{\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)}^{3} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}} \]

      5. unpow3 N/A

         \[\leadsto \color{blue}{\left(\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right)} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

   3. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \color{blue}{\left(c \cdot a\right) \cdot 3}\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - \color{blue}{3 \cdot \left(c \cdot a\right)}\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(b \cdot b - \color{blue}{\left(3 \cdot c\right) \cdot a}\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \color{blue}{\left(3 \cdot c\right) \cdot a}\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      6. lower-*.f64 40.3%

         \[\leadsto \left(\left(b \cdot b - \color{blue}{\left(3 \cdot c\right)} \cdot a\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   5. Applied rewrites 40.3%

      \[\leadsto \left(\left(b \cdot b - \color{blue}{\left(3 \cdot c\right) \cdot a}\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \color{blue}{\left(c \cdot a\right) \cdot 3}\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \color{blue}{3 \cdot \left(c \cdot a\right)}\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \color{blue}{\left(3 \cdot c\right) \cdot a}\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \color{blue}{\left(3 \cdot c\right) \cdot a}\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      6. lower-*.f64 40.3%

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \color{blue}{\left(3 \cdot c\right)} \cdot a\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   7. Applied rewrites 40.3%

      \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \color{blue}{\left(3 \cdot c\right) \cdot a}\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \left(3 \cdot c\right) \cdot a\right)\right) \cdot \left(\left(\color{blue}{\left(c \cdot a\right) \cdot 3} - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \left(3 \cdot c\right) \cdot a\right)\right) \cdot \left(\left(\color{blue}{3 \cdot \left(c \cdot a\right)} - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \left(3 \cdot c\right) \cdot a\right)\right) \cdot \left(\left(3 \cdot \color{blue}{\left(c \cdot a\right)} - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \left(3 \cdot c\right) \cdot a\right)\right) \cdot \left(\left(\color{blue}{\left(3 \cdot c\right) \cdot a} - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \left(3 \cdot c\right) \cdot a\right)\right) \cdot \left(\left(\color{blue}{\left(3 \cdot c\right) \cdot a} - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      6. lower-*.f64 40.3%

         \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \left(3 \cdot c\right) \cdot a\right)\right) \cdot \left(\left(\color{blue}{\left(3 \cdot c\right)} \cdot a - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   9. Applied rewrites 40.3%

      \[\leadsto \left(\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \left(b \cdot b - \left(3 \cdot c\right) \cdot a\right)\right) \cdot \left(\left(\color{blue}{\left(3 \cdot c\right) \cdot a} - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot a\right) \cdot 3\\ t_1 := \left(c \cdot a\right) \cdot -3\\ t_2 := \left|b\right| \cdot \left|b\right|\\ t_3 := t\_0 - t\_2\\ t_4 := \frac{0.3333333333333333 \cdot c}{a}\\ t_5 := \left(a \cdot a\right) \cdot a\\ \mathbf{if}\;\left|b\right| \leq 2.25 \cdot 10^{-138}:\\ \;\;\;\;\left(t\_4 \cdot t\_4\right) \cdot t\_4\\ \mathbf{elif}\;\left|b\right| \leq 1.55 \cdot 10^{-77}:\\ \;\;\;\;t\_3 \cdot \left(\left(-0.037037037037037035 \cdot \frac{c}{a}\right) \cdot \frac{\left(t\_2 - t\_0\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot \frac{0.0013717421124828531}{t\_5 \cdot t\_5}\right) \cdot t\_3\right) \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* c a) 3.0))
       (t_1 (* (* c a) -3.0))
       (t_2 (* (fabs b) (fabs b)))
       (t_3 (- t_0 t_2))
       (t_4 (/ (* 0.3333333333333333 c) a))
       (t_5 (* (* a a) a)))
  (if (<= (fabs b) 2.25e-138)
    (* (* t_4 t_4) t_4)
    (if (<= (fabs b) 1.55e-77)
      (*
       t_3
       (*
        (* -0.037037037037037035 (/ c a))
        (/ (* (- t_2 t_0) (/ 0.1111111111111111 (* a a))) (* a a))))
      (* (* (* t_1 (/ 0.0013717421124828531 (* t_5 t_5))) t_3) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * 3.0;
	double t_1 = (c * a) * -3.0;
	double t_2 = fabs(b) * fabs(b);
	double t_3 = t_0 - t_2;
	double t_4 = (0.3333333333333333 * c) / a;
	double t_5 = (a * a) * a;
	double tmp;
	if (fabs(b) <= 2.25e-138) {
		tmp = (t_4 * t_4) * t_4;
	} else if (fabs(b) <= 1.55e-77) {
		tmp = t_3 * ((-0.037037037037037035 * (c / a)) * (((t_2 - t_0) * (0.1111111111111111 / (a * a))) / (a * a)));
	} else {
		tmp = ((t_1 * (0.0013717421124828531 / (t_5 * t_5))) * t_3) * t_1;
	}
	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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (c * a) * 3.0d0
    t_1 = (c * a) * (-3.0d0)
    t_2 = abs(b) * abs(b)
    t_3 = t_0 - t_2
    t_4 = (0.3333333333333333d0 * c) / a
    t_5 = (a * a) * a
    if (abs(b) <= 2.25d-138) then
        tmp = (t_4 * t_4) * t_4
    else if (abs(b) <= 1.55d-77) then
        tmp = t_3 * (((-0.037037037037037035d0) * (c / a)) * (((t_2 - t_0) * (0.1111111111111111d0 / (a * a))) / (a * a)))
    else
        tmp = ((t_1 * (0.0013717421124828531d0 / (t_5 * t_5))) * t_3) * t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * a) * 3.0;
	double t_1 = (c * a) * -3.0;
	double t_2 = Math.abs(b) * Math.abs(b);
	double t_3 = t_0 - t_2;
	double t_4 = (0.3333333333333333 * c) / a;
	double t_5 = (a * a) * a;
	double tmp;
	if (Math.abs(b) <= 2.25e-138) {
		tmp = (t_4 * t_4) * t_4;
	} else if (Math.abs(b) <= 1.55e-77) {
		tmp = t_3 * ((-0.037037037037037035 * (c / a)) * (((t_2 - t_0) * (0.1111111111111111 / (a * a))) / (a * a)));
	} else {
		tmp = ((t_1 * (0.0013717421124828531 / (t_5 * t_5))) * t_3) * t_1;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c * a) * 3.0
	t_1 = (c * a) * -3.0
	t_2 = math.fabs(b) * math.fabs(b)
	t_3 = t_0 - t_2
	t_4 = (0.3333333333333333 * c) / a
	t_5 = (a * a) * a
	tmp = 0
	if math.fabs(b) <= 2.25e-138:
		tmp = (t_4 * t_4) * t_4
	elif math.fabs(b) <= 1.55e-77:
		tmp = t_3 * ((-0.037037037037037035 * (c / a)) * (((t_2 - t_0) * (0.1111111111111111 / (a * a))) / (a * a)))
	else:
		tmp = ((t_1 * (0.0013717421124828531 / (t_5 * t_5))) * t_3) * t_1
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * 3.0)
	t_1 = Float64(Float64(c * a) * -3.0)
	t_2 = Float64(abs(b) * abs(b))
	t_3 = Float64(t_0 - t_2)
	t_4 = Float64(Float64(0.3333333333333333 * c) / a)
	t_5 = Float64(Float64(a * a) * a)
	tmp = 0.0
	if (abs(b) <= 2.25e-138)
		tmp = Float64(Float64(t_4 * t_4) * t_4);
	elseif (abs(b) <= 1.55e-77)
		tmp = Float64(t_3 * Float64(Float64(-0.037037037037037035 * Float64(c / a)) * Float64(Float64(Float64(t_2 - t_0) * Float64(0.1111111111111111 / Float64(a * a))) / Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(t_1 * Float64(0.0013717421124828531 / Float64(t_5 * t_5))) * t_3) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c * a) * 3.0;
	t_1 = (c * a) * -3.0;
	t_2 = abs(b) * abs(b);
	t_3 = t_0 - t_2;
	t_4 = (0.3333333333333333 * c) / a;
	t_5 = (a * a) * a;
	tmp = 0.0;
	if (abs(b) <= 2.25e-138)
		tmp = (t_4 * t_4) * t_4;
	elseif (abs(b) <= 1.55e-77)
		tmp = t_3 * ((-0.037037037037037035 * (c / a)) * (((t_2 - t_0) * (0.1111111111111111 / (a * a))) / (a * a)));
	else
		tmp = ((t_1 * (0.0013717421124828531 / (t_5 * t_5))) * t_3) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$5 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.25e-138], N[(N[(t$95$4 * t$95$4), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.55e-77], N[(t$95$3 * N[(N[(-0.037037037037037035 * N[(c / a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$2 - t$95$0), $MachinePrecision] * N[(0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(0.0013717421124828531 / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.25e-138

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 2.25e-138 < b < 1.55e-77

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\frac{0.1111111111111111 \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \left(\frac{0.1111111111111111}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)\right)\right)} \]

   5. Applied rewrites 76.0%

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\frac{\left(0.1111111111111111 \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot 0.3333333333333333\right) \cdot 0.1111111111111111}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\color{blue}{\left(\frac{-1}{27} \cdot \frac{c}{a}\right)} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(\frac{-1}{27} \cdot \color{blue}{\frac{c}{a}}\right) \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      2. lower-/.f64 52.6%

         \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\left(-0.037037037037037035 \cdot \frac{c}{\color{blue}{a}}\right) \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   8. Applied rewrites 52.6%

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\color{blue}{\left(-0.037037037037037035 \cdot \frac{c}{a}\right)} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   if 1.55e-77 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}}^{3} \]

      3. mult-flip N/A

         \[\leadsto {\color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \frac{1}{9 \cdot {a}^{2}}\right)}}^{3} \]

      4. cube-prod N/A

         \[\leadsto \color{blue}{{\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)}^{3} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}} \]

      5. unpow3 N/A

         \[\leadsto \color{blue}{\left(\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right)} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

   3. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)} \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-3 \cdot \color{blue}{\left(a \cdot c\right)}\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      2. lower-*.f64 18.9%

         \[\leadsto \left(\left(-3 \cdot \left(a \cdot \color{blue}{c}\right)\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   6. Applied rewrites 18.9%

      \[\leadsto \left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)} \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   7. Taylor expanded in a around inf

      \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \color{blue}{\left(a \cdot c\right)}\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      2. lower-*.f64 17.4%

         \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot \color{blue}{c}\right)\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   9. Applied rewrites 17.4%

      \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)} \]

   11. Applied rewrites 35.8%

       \[\leadsto \color{blue}{\left(\left(\left(\left(c \cdot a\right) \cdot -3\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}\right) \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_0 := \left(3 \cdot c\right) \cdot a\\ t_1 := \left(c \cdot a\right) \cdot -3\\ t_2 := \left|b\right| \cdot \left|b\right|\\ t_3 := \frac{0.3333333333333333 \cdot c}{a}\\ t_4 := \left(a \cdot a\right) \cdot a\\ \mathbf{if}\;\left|b\right| \leq 2.25 \cdot 10^{-138}:\\ \;\;\;\;\left(t\_3 \cdot t\_3\right) \cdot t\_3\\ \mathbf{elif}\;\left|b\right| \leq 1.55 \cdot 10^{-77}:\\ \;\;\;\;\left(t\_0 - t\_2\right) \cdot \left(\left(-0.037037037037037035 \cdot \frac{c}{a}\right) \cdot \frac{\left(t\_2 - t\_0\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot \frac{0.0013717421124828531}{t\_4 \cdot t\_4}\right) \cdot \left(\left(c \cdot a\right) \cdot 3 - t\_2\right)\right) \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* 3.0 c) a))
       (t_1 (* (* c a) -3.0))
       (t_2 (* (fabs b) (fabs b)))
       (t_3 (/ (* 0.3333333333333333 c) a))
       (t_4 (* (* a a) a)))
  (if (<= (fabs b) 2.25e-138)
    (* (* t_3 t_3) t_3)
    (if (<= (fabs b) 1.55e-77)
      (*
       (- t_0 t_2)
       (*
        (* -0.037037037037037035 (/ c a))
        (/ (* (- t_2 t_0) (/ 0.1111111111111111 (* a a))) (* a a))))
      (*
       (*
        (* t_1 (/ 0.0013717421124828531 (* t_4 t_4)))
        (- (* (* c a) 3.0) t_2))
       t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = (3.0 * c) * a;
	double t_1 = (c * a) * -3.0;
	double t_2 = fabs(b) * fabs(b);
	double t_3 = (0.3333333333333333 * c) / a;
	double t_4 = (a * a) * a;
	double tmp;
	if (fabs(b) <= 2.25e-138) {
		tmp = (t_3 * t_3) * t_3;
	} else if (fabs(b) <= 1.55e-77) {
		tmp = (t_0 - t_2) * ((-0.037037037037037035 * (c / a)) * (((t_2 - t_0) * (0.1111111111111111 / (a * a))) / (a * a)));
	} else {
		tmp = ((t_1 * (0.0013717421124828531 / (t_4 * t_4))) * (((c * a) * 3.0) - t_2)) * t_1;
	}
	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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (3.0d0 * c) * a
    t_1 = (c * a) * (-3.0d0)
    t_2 = abs(b) * abs(b)
    t_3 = (0.3333333333333333d0 * c) / a
    t_4 = (a * a) * a
    if (abs(b) <= 2.25d-138) then
        tmp = (t_3 * t_3) * t_3
    else if (abs(b) <= 1.55d-77) then
        tmp = (t_0 - t_2) * (((-0.037037037037037035d0) * (c / a)) * (((t_2 - t_0) * (0.1111111111111111d0 / (a * a))) / (a * a)))
    else
        tmp = ((t_1 * (0.0013717421124828531d0 / (t_4 * t_4))) * (((c * a) * 3.0d0) - t_2)) * t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (3.0 * c) * a;
	double t_1 = (c * a) * -3.0;
	double t_2 = Math.abs(b) * Math.abs(b);
	double t_3 = (0.3333333333333333 * c) / a;
	double t_4 = (a * a) * a;
	double tmp;
	if (Math.abs(b) <= 2.25e-138) {
		tmp = (t_3 * t_3) * t_3;
	} else if (Math.abs(b) <= 1.55e-77) {
		tmp = (t_0 - t_2) * ((-0.037037037037037035 * (c / a)) * (((t_2 - t_0) * (0.1111111111111111 / (a * a))) / (a * a)));
	} else {
		tmp = ((t_1 * (0.0013717421124828531 / (t_4 * t_4))) * (((c * a) * 3.0) - t_2)) * t_1;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (3.0 * c) * a
	t_1 = (c * a) * -3.0
	t_2 = math.fabs(b) * math.fabs(b)
	t_3 = (0.3333333333333333 * c) / a
	t_4 = (a * a) * a
	tmp = 0
	if math.fabs(b) <= 2.25e-138:
		tmp = (t_3 * t_3) * t_3
	elif math.fabs(b) <= 1.55e-77:
		tmp = (t_0 - t_2) * ((-0.037037037037037035 * (c / a)) * (((t_2 - t_0) * (0.1111111111111111 / (a * a))) / (a * a)))
	else:
		tmp = ((t_1 * (0.0013717421124828531 / (t_4 * t_4))) * (((c * a) * 3.0) - t_2)) * t_1
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(3.0 * c) * a)
	t_1 = Float64(Float64(c * a) * -3.0)
	t_2 = Float64(abs(b) * abs(b))
	t_3 = Float64(Float64(0.3333333333333333 * c) / a)
	t_4 = Float64(Float64(a * a) * a)
	tmp = 0.0
	if (abs(b) <= 2.25e-138)
		tmp = Float64(Float64(t_3 * t_3) * t_3);
	elseif (abs(b) <= 1.55e-77)
		tmp = Float64(Float64(t_0 - t_2) * Float64(Float64(-0.037037037037037035 * Float64(c / a)) * Float64(Float64(Float64(t_2 - t_0) * Float64(0.1111111111111111 / Float64(a * a))) / Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(t_1 * Float64(0.0013717421124828531 / Float64(t_4 * t_4))) * Float64(Float64(Float64(c * a) * 3.0) - t_2)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (3.0 * c) * a;
	t_1 = (c * a) * -3.0;
	t_2 = abs(b) * abs(b);
	t_3 = (0.3333333333333333 * c) / a;
	t_4 = (a * a) * a;
	tmp = 0.0;
	if (abs(b) <= 2.25e-138)
		tmp = (t_3 * t_3) * t_3;
	elseif (abs(b) <= 1.55e-77)
		tmp = (t_0 - t_2) * ((-0.037037037037037035 * (c / a)) * (((t_2 - t_0) * (0.1111111111111111 / (a * a))) / (a * a)));
	else
		tmp = ((t_1 * (0.0013717421124828531 / (t_4 * t_4))) * (((c * a) * 3.0) - t_2)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(3.0 * c), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.3333333333333333 * c), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$4 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.25e-138], N[(N[(t$95$3 * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.55e-77], N[(N[(t$95$0 - t$95$2), $MachinePrecision] * N[(N[(-0.037037037037037035 * N[(c / a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$2 - t$95$0), $MachinePrecision] * N[(0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(0.0013717421124828531 / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.25e-138

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]

   5. Taylor expanded in a around inf

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c}{a}\right)}^{3} \]
   6. Step-by-step derivation

      1. lower-*.f64 58.9%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   7. Applied rewrites 58.9%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c}{a}\right)}^{3} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c}{a} \cdot \frac{\frac{1}{3} \cdot c}{a}\right) \cdot \frac{\frac{1}{3} \cdot c}{a}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333 \cdot c}{a} \cdot \frac{0.3333333333333333 \cdot c}{a}\right) \cdot \frac{0.3333333333333333 \cdot c}{a}} \]

   if 2.25e-138 < b < 1.55e-77

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   3. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \left(\frac{0.1111111111111111 \cdot \left(\left(c \cdot a\right) \cdot 0.3333333333333333 - \left(b \cdot b\right) \cdot 0.1111111111111111\right)}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \left(\frac{0.1111111111111111}{a \cdot a} \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)\right)\right)} \]

   5. Applied rewrites 76.0%

      \[\leadsto \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \color{blue}{\left(\frac{\left(0.1111111111111111 \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot 0.3333333333333333\right) \cdot 0.1111111111111111}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(c \cdot a\right) \cdot 3} - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{3 \cdot \left(c \cdot a\right)} - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(3 \cdot \color{blue}{\left(c \cdot a\right)} - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(3 \cdot c\right) \cdot a} - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(3 \cdot c\right) \cdot a} - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      6. lower-*.f64 76.1%

         \[\leadsto \left(\color{blue}{\left(3 \cdot c\right)} \cdot a - b \cdot b\right) \cdot \left(\frac{\left(0.1111111111111111 \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot 0.3333333333333333\right) \cdot 0.1111111111111111}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   7. Applied rewrites 76.1%

      \[\leadsto \left(\color{blue}{\left(3 \cdot c\right) \cdot a} - b \cdot b\right) \cdot \left(\frac{\left(0.1111111111111111 \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot 0.3333333333333333\right) \cdot 0.1111111111111111}{a \cdot a} \cdot \frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - \color{blue}{\left(c \cdot a\right) \cdot 3}\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - \color{blue}{3 \cdot \left(c \cdot a\right)}\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - \color{blue}{\left(3 \cdot c\right) \cdot a}\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\frac{\left(\frac{1}{9} \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot \frac{1}{3}\right) \cdot \frac{1}{9}}{a \cdot a} \cdot \frac{\left(b \cdot b - \color{blue}{\left(3 \cdot c\right) \cdot a}\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

      6. lower-*.f64 76.1%

         \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\frac{\left(0.1111111111111111 \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot 0.3333333333333333\right) \cdot 0.1111111111111111}{a \cdot a} \cdot \frac{\left(b \cdot b - \color{blue}{\left(3 \cdot c\right)} \cdot a\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   9. Applied rewrites 76.1%

      \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\frac{\left(0.1111111111111111 \cdot \left(b \cdot b\right) - \left(c \cdot a\right) \cdot 0.3333333333333333\right) \cdot 0.1111111111111111}{a \cdot a} \cdot \frac{\left(b \cdot b - \color{blue}{\left(3 \cdot c\right) \cdot a}\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   10. Taylor expanded in a around inf

       \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\color{blue}{\left(\frac{-1}{27} \cdot \frac{c}{a}\right)} \cdot \frac{\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\left(\frac{-1}{27} \cdot \color{blue}{\frac{c}{a}}\right) \cdot \frac{\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \frac{\frac{1}{9}}{a \cdot a}}{a \cdot a}\right) \]

       2. lower-/.f64 52.6%

          \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\left(-0.037037037037037035 \cdot \frac{c}{\color{blue}{a}}\right) \cdot \frac{\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   12. Applied rewrites 52.6%

       \[\leadsto \left(\left(3 \cdot c\right) \cdot a - b \cdot b\right) \cdot \left(\color{blue}{\left(-0.037037037037037035 \cdot \frac{c}{a}\right)} \cdot \frac{\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \frac{0.1111111111111111}{a \cdot a}}{a \cdot a}\right) \]

   if 1.55e-77 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}}^{3} \]

      3. mult-flip N/A

         \[\leadsto {\color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \frac{1}{9 \cdot {a}^{2}}\right)}}^{3} \]

      4. cube-prod N/A

         \[\leadsto \color{blue}{{\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)}^{3} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}} \]

      5. unpow3 N/A

         \[\leadsto \color{blue}{\left(\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right)} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

   3. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)} \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-3 \cdot \color{blue}{\left(a \cdot c\right)}\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      2. lower-*.f64 18.9%

         \[\leadsto \left(\left(-3 \cdot \left(a \cdot \color{blue}{c}\right)\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   6. Applied rewrites 18.9%

      \[\leadsto \left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)} \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   7. Taylor expanded in a around inf

      \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \color{blue}{\left(a \cdot c\right)}\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      2. lower-*.f64 17.4%

         \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot \color{blue}{c}\right)\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   9. Applied rewrites 17.4%

      \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)} \]

   11. Applied rewrites 35.8%

       \[\leadsto \color{blue}{\left(\left(\left(\left(c \cdot a\right) \cdot -3\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}\right) \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 63.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot a\right) \cdot -3\\ t_1 := \left(a \cdot a\right) \cdot a\\ t_2 := 0.3333333333333333 \cdot \frac{c}{a}\\ \mathbf{if}\;\left|b\right| \leq 1.25 \cdot 10^{-76}:\\ \;\;\;\;\left(t\_2 \cdot t\_2\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot \frac{0.0013717421124828531}{t\_1 \cdot t\_1}\right) \cdot \left(\left(c \cdot a\right) \cdot 3 - \left|b\right| \cdot \left|b\right|\right)\right) \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* c a) -3.0))
       (t_1 (* (* a a) a))
       (t_2 (* 0.3333333333333333 (/ c a))))
  (if (<= (fabs b) 1.25e-76)
    (* (* t_2 t_2) t_2)
    (*
     (*
      (* t_0 (/ 0.0013717421124828531 (* t_1 t_1)))
      (- (* (* c a) 3.0) (* (fabs b) (fabs b))))
     t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * -3.0;
	double t_1 = (a * a) * a;
	double t_2 = 0.3333333333333333 * (c / a);
	double tmp;
	if (fabs(b) <= 1.25e-76) {
		tmp = (t_2 * t_2) * t_2;
	} else {
		tmp = ((t_0 * (0.0013717421124828531 / (t_1 * t_1))) * (((c * a) * 3.0) - (fabs(b) * fabs(b)))) * t_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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (c * a) * (-3.0d0)
    t_1 = (a * a) * a
    t_2 = 0.3333333333333333d0 * (c / a)
    if (abs(b) <= 1.25d-76) then
        tmp = (t_2 * t_2) * t_2
    else
        tmp = ((t_0 * (0.0013717421124828531d0 / (t_1 * t_1))) * (((c * a) * 3.0d0) - (abs(b) * abs(b)))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * a) * -3.0;
	double t_1 = (a * a) * a;
	double t_2 = 0.3333333333333333 * (c / a);
	double tmp;
	if (Math.abs(b) <= 1.25e-76) {
		tmp = (t_2 * t_2) * t_2;
	} else {
		tmp = ((t_0 * (0.0013717421124828531 / (t_1 * t_1))) * (((c * a) * 3.0) - (Math.abs(b) * Math.abs(b)))) * t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c * a) * -3.0
	t_1 = (a * a) * a
	t_2 = 0.3333333333333333 * (c / a)
	tmp = 0
	if math.fabs(b) <= 1.25e-76:
		tmp = (t_2 * t_2) * t_2
	else:
		tmp = ((t_0 * (0.0013717421124828531 / (t_1 * t_1))) * (((c * a) * 3.0) - (math.fabs(b) * math.fabs(b)))) * t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * -3.0)
	t_1 = Float64(Float64(a * a) * a)
	t_2 = Float64(0.3333333333333333 * Float64(c / a))
	tmp = 0.0
	if (abs(b) <= 1.25e-76)
		tmp = Float64(Float64(t_2 * t_2) * t_2);
	else
		tmp = Float64(Float64(Float64(t_0 * Float64(0.0013717421124828531 / Float64(t_1 * t_1))) * Float64(Float64(Float64(c * a) * 3.0) - Float64(abs(b) * abs(b)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c * a) * -3.0;
	t_1 = (a * a) * a;
	t_2 = 0.3333333333333333 * (c / a);
	tmp = 0.0;
	if (abs(b) <= 1.25e-76)
		tmp = (t_2 * t_2) * t_2;
	else
		tmp = ((t_0 * (0.0013717421124828531 / (t_1 * t_1))) * (((c * a) * 3.0) - (abs(b) * abs(b)))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(c / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.25e-76], N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(t$95$0 * N[(0.0013717421124828531 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2499999999999999e-76

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

   2. Taylor expanded in a around inf

      \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

      2. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      3. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      6. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

   4. Applied rewrites 94.0%

      \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

      2. +-commutative N/A

         \[\leadsto {\left(\frac{\frac{1}{3} \cdot c + \frac{-1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

      3. lift-*.f64 N/A

         \[\leadsto {\left(\frac{\frac{1}{3} \cdot c + \frac{-1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

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

         \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

      5. lower--.f64 N/A

         \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

      7. lower-*.f64 94.0%

         \[\leadsto {\left(\frac{0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

      8. lift-/.f64 N/A

         \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

      9. lift-pow.f64 N/A

         \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

      10. pow2 N/A

          \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{b \cdot b}{a}}{a}\right)}^{3} \]

      11. associate-/l* N/A

          \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]

      12. lower-*.f64 N/A

          \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]

      13. lower-/.f64 99.3%

          \[\leadsto {\left(\frac{0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]

   6. Applied rewrites 99.3%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3}} \]

      2. unpow3 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a} \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right) \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a} \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right) \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}} \]

   8. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}} \]

   9. Taylor expanded in a around inf

      \[\leadsto \left(\left(\frac{1}{3} \cdot \color{blue}{\frac{c}{a}}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{c}{\color{blue}{a}}\right) \cdot \frac{\frac{-1}{9} \cdot \left(\frac{b}{a} \cdot b\right) - \frac{-1}{3} \cdot c}{a}\right) \cdot \frac{\frac{-1}{9} \cdot \left(\frac{b}{a} \cdot b\right) - \frac{-1}{3} \cdot c}{a} \]

       2. lower-/.f64 66.9%

          \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]

   11. Applied rewrites 66.9%

       \[\leadsto \left(\left(0.3333333333333333 \cdot \color{blue}{\frac{c}{a}}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]

   12. Taylor expanded in a around inf

       \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\frac{c}{a}}\right)\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{c}{a}\right) \cdot \left(\frac{1}{3} \cdot \frac{c}{\color{blue}{a}}\right)\right) \cdot \frac{\frac{-1}{9} \cdot \left(\frac{b}{a} \cdot b\right) - \frac{-1}{3} \cdot c}{a} \]

       2. lower-/.f64 75.1%

          \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right)\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]

   14. Applied rewrites 75.1%

       \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{c}{a}}\right)\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]

   15. Taylor expanded in a around inf

       \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right)\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\frac{c}{a}}\right) \]
   16. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{c}{a}\right) \cdot \left(\frac{1}{3} \cdot \frac{c}{a}\right)\right) \cdot \left(\frac{1}{3} \cdot \frac{c}{\color{blue}{a}}\right) \]

       2. lower-/.f64 58.9%

          \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right)\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right) \]

   17. Applied rewrites 58.9%

       \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right)\right) \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{c}{a}}\right) \]

   if 1.2499999999999999e-76 < b

   1. Initial program 81.1%

      \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}}^{3} \]

      3. mult-flip N/A

         \[\leadsto {\color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \frac{1}{9 \cdot {a}^{2}}\right)}}^{3} \]

      4. cube-prod N/A

         \[\leadsto \color{blue}{{\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)}^{3} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}} \]

      5. unpow3 N/A

         \[\leadsto \color{blue}{\left(\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right)} \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot \left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right) \cdot \left(\left(3 \cdot \left(a \cdot c\right) - {b}^{2}\right) \cdot {\left(\frac{1}{9 \cdot {a}^{2}}\right)}^{3}\right)} \]

   3. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\left(\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)} \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-3 \cdot \color{blue}{\left(a \cdot c\right)}\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      2. lower-*.f64 18.9%

         \[\leadsto \left(\left(-3 \cdot \left(a \cdot \color{blue}{c}\right)\right) \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   6. Applied rewrites 18.9%

      \[\leadsto \left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)} \cdot \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   7. Taylor expanded in a around inf

      \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \color{blue}{\left(a \cdot c\right)}\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

      2. lower-*.f64 17.4%

         \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot \color{blue}{c}\right)\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

   9. Applied rewrites 17.4%

      \[\leadsto \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right) \cdot \frac{\frac{1}{729}}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right)}\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)} \]

   11. Applied rewrites 35.8%

       \[\leadsto \color{blue}{\left(\left(\left(\left(c \cdot a\right) \cdot -3\right) \cdot \frac{0.0013717421124828531}{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}\right) \cdot \left(\left(c \cdot a\right) \cdot 3 - b \cdot b\right)\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 58.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} t_0 := 0.3333333333333333 \cdot \frac{c}{a}\\ \left(t\_0 \cdot t\_0\right) \cdot t\_0 \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* 0.3333333333333333 (/ c a)))) (* (* t_0 t_0) t_0)))

C

double code(double a, double b, double c) {
	double t_0 = 0.3333333333333333 * (c / a);
	return (t_0 * t_0) * t_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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = 0.3333333333333333d0 * (c / a)
    code = (t_0 * t_0) * t_0
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 0.3333333333333333 * (c / a);
	return (t_0 * t_0) * t_0;
}

Python

def code(a, b, c):
	t_0 = 0.3333333333333333 * (c / a)
	return (t_0 * t_0) * t_0

Julia

function code(a, b, c)
	t_0 = Float64(0.3333333333333333 * Float64(c / a))
	return Float64(Float64(t_0 * t_0) * t_0)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = 0.3333333333333333 * (c / a);
	tmp = (t_0 * t_0) * t_0;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(c / a), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 81.1%

   \[{\left(\frac{3 \cdot \left(a \cdot c\right) - {b}^{2}}{9 \cdot {a}^{2}}\right)}^{3} \]

2. Taylor expanded in a around inf

   \[\leadsto {\color{blue}{\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}}^{3} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{\color{blue}{a}}\right)}^{3} \]

   2. lower-+.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   3. lower-*.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   4. lower-/.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   5. lower-pow.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   6. lower-*.f64 94.0%

      \[\leadsto {\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}^{3} \]

4. Applied rewrites 94.0%

   \[\leadsto {\color{blue}{\left(\frac{-0.1111111111111111 \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot c}{a}\right)}}^{3} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto {\left(\frac{\frac{-1}{9} \cdot \frac{{b}^{2}}{a} + \frac{1}{3} \cdot c}{a}\right)}^{3} \]

   2. +-commutative N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c + \frac{-1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   3. lift-*.f64 N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c + \frac{-1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

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

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   5. lower--.f64 N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   7. lower-*.f64 94.0%

      \[\leadsto {\left(\frac{0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   8. lift-/.f64 N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   9. lift-pow.f64 N/A

      \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{{b}^{2}}{a}}{a}\right)}^{3} \]

   10. pow2 N/A

       \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \frac{b \cdot b}{a}}{a}\right)}^{3} \]

   11. associate-/l* N/A

       \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]

   12. lower-*.f64 N/A

       \[\leadsto {\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]

   13. lower-/.f64 99.3%

       \[\leadsto {\left(\frac{0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]

6. Applied rewrites 99.3%

   \[\leadsto {\left(\frac{0.3333333333333333 \cdot c - 0.1111111111111111 \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right)}^{3}} \]

   2. unpow3 N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a} \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right) \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a} \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}\right) \cdot \frac{\frac{1}{3} \cdot c - \frac{1}{9} \cdot \left(b \cdot \frac{b}{a}\right)}{a}} \]

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\left(\frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}} \]

9. Taylor expanded in a around inf

   \[\leadsto \left(\left(\frac{1}{3} \cdot \color{blue}{\frac{c}{a}}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{c}{\color{blue}{a}}\right) \cdot \frac{\frac{-1}{9} \cdot \left(\frac{b}{a} \cdot b\right) - \frac{-1}{3} \cdot c}{a}\right) \cdot \frac{\frac{-1}{9} \cdot \left(\frac{b}{a} \cdot b\right) - \frac{-1}{3} \cdot c}{a} \]

    2. lower-/.f64 66.9%

       \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]

11. Applied rewrites 66.9%

    \[\leadsto \left(\left(0.3333333333333333 \cdot \color{blue}{\frac{c}{a}}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a}\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]

12. Taylor expanded in a around inf

    \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\frac{c}{a}}\right)\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]
13. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{c}{a}\right) \cdot \left(\frac{1}{3} \cdot \frac{c}{\color{blue}{a}}\right)\right) \cdot \frac{\frac{-1}{9} \cdot \left(\frac{b}{a} \cdot b\right) - \frac{-1}{3} \cdot c}{a} \]

    2. lower-/.f64 75.1%

       \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right)\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]

14. Applied rewrites 75.1%

    \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{c}{a}}\right)\right) \cdot \frac{-0.1111111111111111 \cdot \left(\frac{b}{a} \cdot b\right) - -0.3333333333333333 \cdot c}{a} \]

15. Taylor expanded in a around inf

    \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right)\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\frac{c}{a}}\right) \]
16. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{c}{a}\right) \cdot \left(\frac{1}{3} \cdot \frac{c}{a}\right)\right) \cdot \left(\frac{1}{3} \cdot \frac{c}{\color{blue}{a}}\right) \]

    2. lower-/.f64 58.9%

       \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right)\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right) \]

17. Applied rewrites 58.9%

    \[\leadsto \left(\left(0.3333333333333333 \cdot \frac{c}{a}\right) \cdot \left(0.3333333333333333 \cdot \frac{c}{a}\right)\right) \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{c}{a}}\right) \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (a b c)
  :name "Q^3 (Cubic Equation Discriminant Part)"
  :precision binary64
  :pre (and (and (and (<= -1000000000.0 a) (<= a 1000000000.0)) (and (<= -1000000000.0 b) (<= b 1000000000.0))) (and (<= -1000000000.0 c) (<= c 1000000000.0)))
  (pow (/ (- (* 3.0 (* a c)) (pow b 2.0)) (* 9.0 (pow a 2.0))) 3.0))