Quadratic roots, medium range

Percentage Accurate: 30.9% → 99.3%

Time: 3.9s

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 30.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{c \cdot c} - 16 \cdot \left(a \cdot a\right)\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (- (- (* b b) (* b b)) (* (* a c) -4.0))
  (*
   (-
    (- b)
    (sqrt
     (/
      (* (* c c) (- (/ (pow b 4.0) (* c c)) (* 16.0 (* a a))))
      (- (* b b) (* -4.0 (* c a))))))
   (* 2.0 a))))

C

double code(double a, double b, double c) {
	return (((b * b) - (b * b)) - ((a * c) * -4.0)) / ((-b - sqrt((((c * c) * ((pow(b, 4.0) / (c * c)) - (16.0 * (a * a)))) / ((b * b) - (-4.0 * (c * a)))))) * (2.0 * 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 * b) - (b * b)) - ((a * c) * (-4.0d0))) / ((-b - sqrt((((c * c) * (((b ** 4.0d0) / (c * c)) - (16.0d0 * (a * a)))) / ((b * b) - ((-4.0d0) * (c * a)))))) * (2.0d0 * a))
end function

Java

public static double code(double a, double b, double c) {
	return (((b * b) - (b * b)) - ((a * c) * -4.0)) / ((-b - Math.sqrt((((c * c) * ((Math.pow(b, 4.0) / (c * c)) - (16.0 * (a * a)))) / ((b * b) - (-4.0 * (c * a)))))) * (2.0 * a));
}

Python

def code(a, b, c):
	return (((b * b) - (b * b)) - ((a * c) * -4.0)) / ((-b - math.sqrt((((c * c) * ((math.pow(b, 4.0) / (c * c)) - (16.0 * (a * a)))) / ((b * b) - (-4.0 * (c * a)))))) * (2.0 * a))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(b * b) - Float64(b * b)) - Float64(Float64(a * c) * -4.0)) / Float64(Float64(Float64(-b) - sqrt(Float64(Float64(Float64(c * c) * Float64(Float64((b ^ 4.0) / Float64(c * c)) - Float64(16.0 * Float64(a * a)))) / Float64(Float64(b * b) - Float64(-4.0 * Float64(c * a)))))) * Float64(2.0 * a)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((b * b) - (b * b)) - ((a * c) * -4.0)) / ((-b - sqrt((((c * c) * (((b ^ 4.0) / (c * c)) - (16.0 * (a * a)))) / ((b * b) - (-4.0 * (c * a)))))) * (2.0 * a));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(N[((-b) - N[Sqrt[N[(N[(N[(c * c), $MachinePrecision] * N[(N[(N[Power[b, 4.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision] - N[(16.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

3. Applied rewrites 31.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right) + b \cdot b}}\right) \cdot \left(2 \cdot a\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)} + b \cdot b}\right) \cdot \left(2 \cdot a\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\color{blue}{b \cdot b + -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   4. flip-+ N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}}\right) \cdot \left(2 \cdot a\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}}\right) \cdot \left(2 \cdot a\right)} \]

7. Applied rewrites 99.3%

   \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}}\right) \cdot \left(2 \cdot a\right)} \]

8. Taylor expanded in c around inf

   \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\color{blue}{{c}^{2} \cdot \left(\frac{{b}^{4}}{{c}^{2}} - 16 \cdot {a}^{2}\right)}}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{{c}^{2} \cdot \color{blue}{\left(\frac{{b}^{4}}{{c}^{2}} - 16 \cdot {a}^{2}\right)}}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   2. unpow2 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\color{blue}{\frac{{b}^{4}}{{c}^{2}}} - 16 \cdot {a}^{2}\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\color{blue}{\frac{{b}^{4}}{{c}^{2}}} - 16 \cdot {a}^{2}\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{{c}^{2}} - \color{blue}{16 \cdot {a}^{2}}\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{{c}^{2}} - \color{blue}{16} \cdot {a}^{2}\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{{c}^{2}} - 16 \cdot {a}^{2}\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{c \cdot c} - 16 \cdot {a}^{2}\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{c \cdot c} - 16 \cdot {a}^{2}\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{c \cdot c} - 16 \cdot \color{blue}{{a}^{2}}\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{c \cdot c} - 16 \cdot \left(a \cdot \color{blue}{a}\right)\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   11. lower-*.f64 99.3

       \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{c \cdot c} - 16 \cdot \left(a \cdot \color{blue}{a}\right)\right)}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]

10. Applied rewrites 99.3%

    \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\frac{\color{blue}{\left(c \cdot c\right) \cdot \left(\frac{{b}^{4}}{c \cdot c} - 16 \cdot \left(a \cdot a\right)\right)}}{b \cdot b - -4 \cdot \left(c \cdot a\right)}}\right) \cdot \left(2 \cdot a\right)} \]
11. Add Preprocessing

Alternative 2: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -3e-6)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (+ a a))
   (* -1.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -3e-6) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (a + a);
	} else {
		tmp = -1.0 * (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(4.0 * a) * c)))) / Float64(2.0 * a)) <= -3e-6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(a + a));
	else
		tmp = Float64(-1.0 * 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[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -3e-6], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * 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 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -3.0000000000000001e-6

   1. Initial program 66.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-negate N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.8

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

   3. Applied rewrites 66.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if -3.0000000000000001e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 17.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      3. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

   3. Applied rewrites 18.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 99.4

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}{2 \cdot a}} \]

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 91.2

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

   10. Applied rewrites 91.2%

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{0 - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right)}\right) \cdot \left(2 \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (- 0.0 (* (* a c) -4.0))
  (* (- (- b) (sqrt (* c (fma -4.0 a (/ (* b b) c))))) (* 2.0 a))))

C

double code(double a, double b, double c) {
	return (0.0 - ((a * c) * -4.0)) / ((-b - sqrt((c * fma(-4.0, a, ((b * b) / c))))) * (2.0 * a));
}

Julia

function code(a, b, c)
	return Float64(Float64(0.0 - Float64(Float64(a * c) * -4.0)) / Float64(Float64(Float64(-b) - sqrt(Float64(c * fma(-4.0, a, Float64(Float64(b * b) / c))))) * Float64(2.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(0.0 - N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(N[((-b) - N[Sqrt[N[(c * N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

3. Applied rewrites 31.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. +-inverses 99.4

      \[\leadsto \frac{\color{blue}{0} - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{0} - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

8. Taylor expanded in c around inf

   \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\color{blue}{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)}}\right) \cdot \left(2 \cdot a\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{c \cdot \color{blue}{\left(-4 \cdot a + \frac{{b}^{2}}{c}\right)}}\right) \cdot \left(2 \cdot a\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(-4, \color{blue}{a}, \frac{{b}^{2}}{c}\right)}\right) \cdot \left(2 \cdot a\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(-4, a, \frac{{b}^{2}}{c}\right)}\right) \cdot \left(2 \cdot a\right)} \]

   4. pow2 N/A

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right)}\right) \cdot \left(2 \cdot a\right)} \]

   5. lift-*.f64 99.4

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right)}\right) \cdot \left(2 \cdot a\right)} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\color{blue}{c \cdot \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right)}}\right) \cdot \left(2 \cdot a\right)} \]
11. Add Preprocessing

Alternative 4: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(a \cdot c\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* 4.0 (* a c)) (* (- (- b) (sqrt (fma -4.0 (* c a) (* b b)))) (* 2.0 a))))

C

double code(double a, double b, double c) {
	return (4.0 * (a * c)) / ((-b - sqrt(fma(-4.0, (c * a), (b * b)))) * (2.0 * a));
}

Julia

function code(a, b, c)
	return Float64(Float64(4.0 * Float64(a * c)) / Float64(Float64(Float64(-b) - sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) * Float64(2.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[((-b) - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

3. Applied rewrites 31.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. +-inverses 99.4

      \[\leadsto \frac{\color{blue}{0} - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{0} - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lift-*.f64 99.4

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

10. Applied rewrites 99.4%

    \[\leadsto \frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]
11. Add Preprocessing

Alternative 5: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{0 - \left(a \cdot c\right) \cdot -4}{a \cdot \mathsf{fma}\left(-4, b, 4 \cdot \frac{a \cdot c}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- 0.0 (* (* a c) -4.0)) (* a (fma -4.0 b (* 4.0 (/ (* a c) b))))))

C

double code(double a, double b, double c) {
	return (0.0 - ((a * c) * -4.0)) / (a * fma(-4.0, b, (4.0 * ((a * c) / b))));
}

Julia

function code(a, b, c)
	return Float64(Float64(0.0 - Float64(Float64(a * c) * -4.0)) / Float64(a * fma(-4.0, b, Float64(4.0 * Float64(Float64(a * c) / b)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(0.0 - N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(a * N[(-4.0 * b + N[(4.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

3. Applied rewrites 31.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. +-inverses 99.4

      \[\leadsto \frac{\color{blue}{0} - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{0} - \left(a \cdot c\right) \cdot -4}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{\color{blue}{a \cdot \left(-4 \cdot b + 4 \cdot \frac{a \cdot c}{b}\right)}} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{a \cdot \color{blue}{\left(-4 \cdot b + 4 \cdot \frac{a \cdot c}{b}\right)}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{a \cdot \mathsf{fma}\left(-4, \color{blue}{b}, 4 \cdot \frac{a \cdot c}{b}\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{a \cdot \mathsf{fma}\left(-4, b, 4 \cdot \frac{a \cdot c}{b}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{a \cdot \mathsf{fma}\left(-4, b, 4 \cdot \frac{a \cdot c}{b}\right)} \]

   5. lift-*.f64 91.0

      \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{a \cdot \mathsf{fma}\left(-4, b, 4 \cdot \frac{a \cdot c}{b}\right)} \]

10. Applied rewrites 91.0%

    \[\leadsto \frac{0 - \left(a \cdot c\right) \cdot -4}{\color{blue}{a \cdot \mathsf{fma}\left(-4, b, 4 \cdot \frac{a \cdot c}{b}\right)}} \]
11. Add Preprocessing

Alternative 6: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-1, c, -1 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma -1.0 c (* -1.0 (/ (* a (* c c)) (* b b)))) b))

C

double code(double a, double b, double c) {
	return fma(-1.0, c, (-1.0 * ((a * (c * c)) / (b * b)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-1.0, c, Float64(-1.0 * Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-1.0 * c + N[(-1.0 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 30.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

3. Applied rewrites 31.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}{2 \cdot a}} \]

8. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 91.0

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

10. Applied rewrites 91.0%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, c, -1 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}} \]
11. Add Preprocessing

Alternative 7: 81.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -1.0 (/ c b)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}} \]

3. Applied rewrites 31.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}{2 \cdot a}} \]

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 81.7

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

10. Applied rewrites 81.7%

    \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))