Cubic critical, medium range

Percentage Accurate: 31.7% → 99.4%

Time: 4.6s

Alternatives: 7

Speedup: 2.9×

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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 32.6%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.2

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0} - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. sub0-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. metadata-eval N/A

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

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

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

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

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

   21. metadata-eval N/A

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

   22. lift-*.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 2: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.6e-8)
   (/ (+ (- b) (sqrt (fma b b (* (* c -3.0) a)))) (* 3.0 a))
   (* -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)) <= -1.6e-8) {
		tmp = (-b + sqrt(fma(b, b, ((c * -3.0) * a)))) / (3.0 * a);
	} else {
		tmp = -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)) <= -1.6e-8)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(c * -3.0) * a)))) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(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], -1.6e-8], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(c * -3.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $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)) < -1.6000000000000001e-8

   1. Initial program 66.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-negate1 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.1

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

   3. Applied rewrites 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if -1.6000000000000001e-8 < (/.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 15.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-negate1 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 15.3

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

   3. Applied rewrites 15.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 15.3

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

   5. Applied rewrites 15.3%

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 92.6

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

   10. Applied rewrites 92.6%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.6e-8)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (* -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)) <= -1.6e-8) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = -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)) <= -1.6e-8)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(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], -1.6e-8], 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]), $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)) < -1.6000000000000001e-8

   1. Initial program 66.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-negate1 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.1

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

   3. Applied rewrites 66.1%

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

   if -1.6000000000000001e-8 < (/.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 15.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-negate1 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 15.3

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

   3. Applied rewrites 15.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 15.3

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

   5. Applied rewrites 15.3%

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 92.6

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

   10. Applied rewrites 92.6%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 32.6%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.2

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0} - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. sub0-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

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

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

   16. metadata-eval N/A

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

   17. lower-*.f64 99.2

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

7. Applied rewrites 99.2%

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

Alternative 5: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 32.6%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.2

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

Alternative 6: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $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. 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. sub-negate1 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. 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%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 31.7

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

5. Applied rewrites 31.7%

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
9. 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. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 90.8

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

10. Applied rewrites 90.8%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}} \]
11. Add Preprocessing

Alternative 7: 81.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -0.5 * (c / 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 = (-0.5d0) * (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. sub-negate1 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. 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%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 31.7

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

5. Applied rewrites 31.7%

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 81.1

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

10. Applied rewrites 81.1%

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

Reproduce

herbie shell --seed 2025107 
(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)))