Cubic critical, medium range

Percentage Accurate: 31.7% → 99.2%

Time: 5.3s

Alternatives: 9

Speedup: 3.3×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.7% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}\\ \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-2.25, t\_0, 2.25 \cdot t\_0\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* (* a a) (* c c)) (* b b))))
   (/
    (/
     (/
      (fma -3.0 (* a c) (fma -2.25 t_0 (* 2.25 t_0)))
      (- (sqrt (fma (* -3.0 a) c (* b b))) (- b)))
     3.0)
    a)))

C

double code(double a, double b, double c) {
	double t_0 = ((a * a) * (c * c)) / (b * b);
	return ((fma(-3.0, (a * c), fma(-2.25, t_0, (2.25 * t_0))) / (sqrt(fma((-3.0 * a), c, (b * b))) - -b)) / 3.0) / a;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(a * a) * Float64(c * c)) / Float64(b * b))
	return Float64(Float64(Float64(fma(-3.0, Float64(a * c), fma(-2.25, t_0, Float64(2.25 * t_0))) / Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) - Float64(-b))) / 3.0) / a)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(-3.0 * N[(a * c), $MachinePrecision] + N[(-2.25 * t$95$0 + N[(2.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - (-b)), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Initial program 31.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

3. Applied rewrites 31.7%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. flip-+ N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 31.7%

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

6. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\frac{\color{blue}{-3 \cdot \left(a \cdot c\right) + \left(\frac{-9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, \color{blue}{a \cdot c}, \frac{-9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot \color{blue}{c}, \frac{-9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   4. pow2 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{{a}^{2} \cdot {c}^{2}}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot {c}^{2}}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   11. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

8. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-2.25, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 2.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]
9. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot c}{b \cdot b}\\ \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(-2.25, t\_0, 2.25 \cdot t\_0\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c c) (* b b))))
   (/
    (/
     (/
      (* a (fma -3.0 c (* a (fma -2.25 t_0 (* 2.25 t_0)))))
      (- (sqrt (fma (* -3.0 a) c (* b b))) (- b)))
     3.0)
    a)))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) / (b * b);
	return (((a * fma(-3.0, c, (a * fma(-2.25, t_0, (2.25 * t_0))))) / (sqrt(fma((-3.0 * a), c, (b * b))) - -b)) / 3.0) / a;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) / Float64(b * b))
	return Float64(Float64(Float64(Float64(a * fma(-3.0, c, Float64(a * fma(-2.25, t_0, Float64(2.25 * t_0))))) / Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) - Float64(-b))) / 3.0) / a)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(a * N[(-3.0 * c + N[(a * N[(-2.25 * t$95$0 + N[(2.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - (-b)), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Initial program 31.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

3. Applied rewrites 31.7%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. flip-+ N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 31.7%

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

6. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\frac{\color{blue}{-3 \cdot \left(a \cdot c\right) + \left(\frac{-9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, \color{blue}{a \cdot c}, \frac{-9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot \color{blue}{c}, \frac{-9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   4. pow2 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{{a}^{2} \cdot {c}^{2}}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot {c}^{2}}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   11. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

8. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-2.25, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 2.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

9. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{\frac{a \cdot \color{blue}{\left(-3 \cdot c + a \cdot \left(\frac{-9}{4} \cdot \frac{{c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{a \cdot \left(-3 \cdot c + \color{blue}{a \cdot \left(\frac{-9}{4} \cdot \frac{{c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    2. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \left(\frac{-9}{4} \cdot \frac{{c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \left(\frac{-9}{4} \cdot \frac{{c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{{c}^{2}}{{b}^{2}}, \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    5. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{{c}^{2}}{{b}^{2}}, \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    6. pow2 N/A

       \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{c \cdot c}{{b}^{2}}, \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{c \cdot c}{{b}^{2}}, \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    8. pow2 N/A

       \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{c \cdot c}{b \cdot b}, \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    9. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{c \cdot c}{b \cdot b}, \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{c \cdot c}{b \cdot b}, \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    11. lower-/.f64 N/A

        \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{c \cdot c}{b \cdot b}, \frac{9}{4} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    12. pow2 N/A

        \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{c \cdot c}{b \cdot b}, \frac{9}{4} \cdot \frac{c \cdot c}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    13. lift-*.f64 N/A

        \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{c \cdot c}{b \cdot b}, \frac{9}{4} \cdot \frac{c \cdot c}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    14. pow2 N/A

        \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(\frac{-9}{4}, \frac{c \cdot c}{b \cdot b}, \frac{9}{4} \cdot \frac{c \cdot c}{b \cdot b}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

    15. lift-*.f64 99.2

        \[\leadsto \frac{\frac{\frac{a \cdot \mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(-2.25, \frac{c \cdot c}{b \cdot b}, 2.25 \cdot \frac{c \cdot c}{b \cdot b}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

11. Applied rewrites 99.2%

    \[\leadsto \frac{\frac{\frac{a \cdot \color{blue}{\mathsf{fma}\left(-3, c, a \cdot \mathsf{fma}\left(-2.25, \frac{c \cdot c}{b \cdot b}, 2.25 \cdot \frac{c \cdot c}{b \cdot b}\right)\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]
12. Add Preprocessing

Alternative 3: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

3. Applied rewrites 31.7%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. flip-+ N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 31.7%

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

6. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\frac{\color{blue}{-3 \cdot \left(a \cdot c\right) + \left(\frac{-9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, \color{blue}{a \cdot c}, \frac{-9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot \color{blue}{c}, \frac{-9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   4. pow2 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{{a}^{2} \cdot {c}^{2}}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot {c}^{2}}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   11. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-9}{4}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{9}{4} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

8. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-2.25, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 2.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{3}}{a} \]

9. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. lift-*.f64 99.2

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

11. Applied rewrites 99.2%

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

Alternative 4: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \frac{\left(a \cdot 3\right) \cdot c}{b \cdot b}}, \left|b\right|, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
   (/
    (fma (sqrt (- 1.0 (/ (* (* a 3.0) c) (* b b)))) (fabs b) (- b))
    (* 3.0 a))
   (fma -0.5 (/ c b) (* -0.375 (/ (* a (* c c)) (* (* b b) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = fma(sqrt((1.0 - (((a * 3.0) * c) / (b * b)))), fabs(b), -b) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (-0.375 * ((a * (c * c)) / ((b * b) * b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(Float64(a * 3.0) * c) / Float64(b * b)))), abs(b), Float64(-b)) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(Float64(b * b) * b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.1], N[(N[(N[Sqrt[N[(1.0 - N[(N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 31.7%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. sub-to-mult N/A

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

      10. sqrt-prod N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 32.1%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.7%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} + \color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{\color{blue}{b}}, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right) \]

      10. lift-*.f64 90.7

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

   7. Applied rewrites 90.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{c}{b}}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (fma -0.5 (/ c b) (* -0.375 (/ (* a (* c c)) (* (* b b) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (-0.375 * ((a * (c * c)) / ((b * b) * b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(Float64(b * b) * b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.1], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 31.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 31.7

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

   3. Applied rewrites 31.7%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.7%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} + \color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{\color{blue}{b}}, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right) \]

      10. lift-*.f64 90.7

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

   7. Applied rewrites 90.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{c}{b}}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.1], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 31.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 31.7

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

   3. Applied rewrites 31.7%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.7%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{\color{blue}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}{b} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b}^{2}} \cdot \frac{-3}{8} + \frac{-1}{2} \cdot c}{b} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      12. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* c (- (* -0.375 (/ (* a c) (* b b))) 0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = (c * ((-0.375 * ((a * c) / (b * b))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / Float64(b * b))) - 0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.1], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 31.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 31.7

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

   3. Applied rewrites 31.7%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.7%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

      6. pow2 N/A

         \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{b \cdot b} - \frac{1}{2}\right)}{b} \]

      7. lift-*.f64 90.6

         \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5\right)}{b} \]

   7. Applied rewrites 90.6%

      \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (- (* -0.375 (/ (* a c) (* b b))) 0.5)) b))

C

double code(double a, double b, double c) {
	return (c * ((-0.375 * ((a * c) / (b * b))) - 0.5)) / b;
}

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 = (c * (((-0.375d0) * ((a * c) / (b * b))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((-0.375 * ((a * c) / (b * b))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((-0.375 * ((a * c) / (b * b))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / Float64(b * b))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((-0.375 * ((a * c) / (b * b))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.3%

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

5. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

   6. pow2 N/A

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{b \cdot b} - \frac{1}{2}\right)}{b} \]

   7. lift-*.f64 90.6

      \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5\right)}{b} \]

7. Applied rewrites 90.6%

   \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5\right)}{b} \]
8. Add Preprocessing

Alternative 9: 81.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -0.5))

C

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

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 = (c / b) * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -0.5;
end

Wolfram

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

Derivation

1. Initial program 31.7%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

   3. lower-/.f64 81.1

      \[\leadsto \frac{c}{b} \cdot -0.5 \]

4. Applied rewrites 81.1%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))