Cubic critical, wide range

Percentage Accurate: 17.1% → 99.4%

Time: 7.0s

Alternatives: 5

Speedup: N/A×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

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 = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Initial Program: 17.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

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 = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Alternative 1: 99.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(3 \cdot a\right) \cdot c}{b + {\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}^{0.5}}}{-3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (* (* 3.0 a) c) (+ b (pow (* a (fma -3.0 c (/ (* b b) a))) 0.5)))
  (- (* 3.0 a))))

C

double code(double a, double b, double c) {
	return (((3.0 * a) * c) / (b + pow((a * fma(-3.0, c, ((b * b) / a))), 0.5))) / -(3.0 * a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(3.0 * a) * c) / Float64(b + (Float64(a * fma(-3.0, c, Float64(Float64(b * b) / a))) ^ 0.5))) / Float64(-Float64(3.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision] / N[(b + N[Power[N[(a * N[(-3.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[(3.0 * a), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. flip-+ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 17.9%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}}}{3 \cdot a} \]

5. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.2

      \[\leadsto \frac{\frac{3 \cdot \left(a \cdot \color{blue}{c}\right)}{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}}{3 \cdot a} \]

7. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}}{3 \cdot a} \]

8. Taylor expanded in a around inf

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

   1. unpow1 N/A

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

   2. unpow1 N/A

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

   3. sqr-abs-rev N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 99.2

      \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right)}{-1 \cdot b - {\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}^{0.5}}}{3 \cdot a} \]

10. Applied rewrites 99.2%

    \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right)}{-1 \cdot b - {\color{blue}{\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}}^{0.5}}}{3 \cdot a} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*r* N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 99.4

       \[\leadsto \frac{\frac{\left(3 \cdot a\right) \cdot c}{-1 \cdot b - {\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}^{0.5}}}{3 \cdot a} \]

12. Applied rewrites 99.4%

    \[\leadsto \frac{\frac{\left(3 \cdot a\right) \cdot \color{blue}{c}}{-1 \cdot b - {\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}^{0.5}}}{3 \cdot a} \]

13. Final simplification 99.4%

    \[\leadsto \frac{\frac{\left(3 \cdot a\right) \cdot c}{b + {\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}^{0.5}}}{-3 \cdot a} \]
14. Add Preprocessing

Alternative 2: 99.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{3 \cdot \left(a \cdot c\right)}{b + {\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}^{0.5}}}{-3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (* 3.0 (* a c)) (+ b (pow (* a (fma -3.0 c (/ (* b b) a))) 0.5)))
  (- (* 3.0 a))))

C

double code(double a, double b, double c) {
	return ((3.0 * (a * c)) / (b + pow((a * fma(-3.0, c, ((b * b) / a))), 0.5))) / -(3.0 * a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(3.0 * Float64(a * c)) / Float64(b + (Float64(a * fma(-3.0, c, Float64(Float64(b * b) / a))) ^ 0.5))) / Float64(-Float64(3.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(b + N[Power[N[(a * N[(-3.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[(3.0 * a), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. flip-+ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 17.9%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}}}{3 \cdot a} \]

5. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.2

      \[\leadsto \frac{\frac{3 \cdot \left(a \cdot \color{blue}{c}\right)}{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}}{3 \cdot a} \]

7. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}}{3 \cdot a} \]

8. Taylor expanded in a around inf

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

   1. unpow1 N/A

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

   2. unpow1 N/A

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

   3. sqr-abs-rev N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 99.2

      \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right)}{-1 \cdot b - {\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}^{0.5}}}{3 \cdot a} \]

10. Applied rewrites 99.2%

    \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right)}{-1 \cdot b - {\color{blue}{\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}}^{0.5}}}{3 \cdot a} \]

11. Final simplification 99.2%

    \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right)}{b + {\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}^{0.5}}}{-3 \cdot a} \]
12. Add Preprocessing

Alternative 3: 99.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)\right)}^{0.25}\\ \frac{\frac{3 \cdot \left(a \cdot c\right)}{\left({b}^{-1} \cdot \left(t\_0 \cdot t\_0\right) + 1\right) \cdot \left(-1 \cdot b\right)}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fma b b (* (* a c) -3.0)) 0.25)))
   (/
    (/ (* 3.0 (* a c)) (* (+ (* (pow b -1.0) (* t_0 t_0)) 1.0) (* -1.0 b)))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fma(b, b, ((a * c) * -3.0)), 0.25);
	return ((3.0 * (a * c)) / (((pow(b, -1.0) * (t_0 * t_0)) + 1.0) * (-1.0 * b))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(a * c) * -3.0)) ^ 0.25
	return Float64(Float64(Float64(3.0 * Float64(a * c)) / Float64(Float64(Float64((b ^ -1.0) * Float64(t_0 * t_0)) + 1.0) * Float64(-1.0 * b))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]}, N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[b, -1.0], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-1.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. flip-+ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 17.9%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}}}{3 \cdot a} \]

5. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.2

      \[\leadsto \frac{\frac{3 \cdot \left(a \cdot \color{blue}{c}\right)}{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}}{3 \cdot a} \]

7. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}}{3 \cdot a} \]

8. Taylor expanded in a around inf

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

   1. unpow1 N/A

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

   2. unpow1 N/A

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

   3. sqr-abs-rev N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 99.2

      \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right)}{-1 \cdot b - {\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}^{0.5}}}{3 \cdot a} \]

10. Applied rewrites 99.2%

    \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right)}{-1 \cdot b - {\color{blue}{\left(a \cdot \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right)\right)}}^{0.5}}}{3 \cdot a} \]

11. Taylor expanded in b around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

13. Applied rewrites 99.1%

    \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right)}{\color{blue}{\left(\left(-1 \cdot {b}^{-1}\right) \cdot \left({\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)\right)}^{0.25}\right) - 1\right) \cdot b}}}{3 \cdot a} \]

14. Final simplification 99.1%

    \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right)}{\left({b}^{-1} \cdot \left({\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)\right)}^{0.25}\right) + 1\right) \cdot \left(-1 \cdot b\right)}}{3 \cdot a} \]
15. Add Preprocessing

Alternative 4: 97.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, -0.16666666666666666, \frac{-0.5625 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{5}}\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (fma
   (fma
    (* a (/ (* (/ (pow c 4.0) (pow b 6.0)) 6.328125) b))
    -0.16666666666666666
    (/ (* -0.5625 (* (* c c) c)) (pow b 5.0)))
   a
   (/ (* -0.375 (* c c)) (pow b 3.0)))
  a
  (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	return fma(fma(fma((a * (((pow(c, 4.0) / pow(b, 6.0)) * 6.328125) / b)), -0.16666666666666666, ((-0.5625 * ((c * c) * c)) / pow(b, 5.0))), a, ((-0.375 * (c * c)) / pow(b, 3.0))), a, ((c / b) * -0.5));
}

Julia

function code(a, b, c)
	return fma(fma(fma(Float64(a * Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125) / b)), -0.16666666666666666, Float64(Float64(-0.5625 * Float64(Float64(c * c) * c)) / (b ^ 5.0))), a, Float64(Float64(-0.375 * Float64(c * c)) / (b ^ 3.0))), a, Float64(Float64(c / b) * -0.5))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(a * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(-0.5625 * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, -0.16666666666666666, \frac{-0.5625 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{5}}\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]
6. Add Preprocessing

Alternative 5: 97.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, a \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, -0.375 \cdot \frac{c \cdot c}{b \cdot b}\right)\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (/ (* (* a a) (* (* c c) c)) (pow b 4.0))
   -0.5625
   (fma
    -0.5
    c
    (*
     a
     (fma
      -1.0546875
      (/ (* (* a a) (pow c 4.0)) (pow b 6.0))
      (* -0.375 (/ (* c c) (* b b)))))))
  b))

C

double code(double a, double b, double c) {
	return fma((((a * a) * ((c * c) * c)) / pow(b, 4.0)), -0.5625, fma(-0.5, c, (a * fma(-1.0546875, (((a * a) * pow(c, 4.0)) / pow(b, 6.0)), (-0.375 * ((c * c) / (b * b))))))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / (b ^ 4.0)), -0.5625, fma(-0.5, c, Float64(a * fma(-1.0546875, Float64(Float64(Float64(a * a) * (c ^ 4.0)) / (b ^ 6.0)), Float64(-0.375 * Float64(Float64(c * c) / Float64(b * b))))))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(a * N[(-1.0546875 * N[(N[(N[(a * a), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right) + \frac{-0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 6.328125\right)}{{b}^{6} \cdot a}\right)}{b}} \]

6. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \frac{-1}{2} \cdot c + a \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{b} \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   6. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}{b} \]

   12. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{c \cdot c}{{b}^{2}}\right)\right)\right)}{b} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{c \cdot c}{{b}^{2}}\right)\right)\right)}{b} \]

   14. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{-3}{8} \cdot \frac{c \cdot c}{b \cdot b}\right)\right)\right)}{b} \]

   15. lift-*.f64 97.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, a \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, -0.375 \cdot \frac{c \cdot c}{b \cdot b}\right)\right)\right)}{b} \]

8. Applied rewrites 97.4%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, a \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, -0.375 \cdot \frac{c \cdot c}{b \cdot b}\right)\right)\right)}{b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))