Quadratic roots, medium range

Percentage Accurate: 31.5% → 99.4%

Time: 11.1s

Alternatives: 7

Speedup: 3.6×

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: 31.5% 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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 29.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sqr-neg-rev N/A

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

   11. lift-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 30.0

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

6. Applied rewrites 30.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. lower-*.f64 99.5

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

8. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

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

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

   16. lower-fma.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 99.5

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

10. Applied rewrites 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 29.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sqr-neg-rev N/A

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

   11. lift-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 30.0

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

6. Applied rewrites 30.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. lower-*.f64 99.5

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

8. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 99.5

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

10. Applied rewrites 99.5%

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

11. Final simplification 99.5%

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

Alternative 3: 90.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 29.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sqr-neg-rev N/A

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

   11. lift-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 30.0

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

6. Applied rewrites 30.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. lower-*.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in c around 0

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

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

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

    2. metadata-eval N/A

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

    3. distribute-lft-out-- N/A

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

    4. lower-*.f64 N/A

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

    5. lower--.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 91.9

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

11. Applied rewrites 91.9%

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

Alternative 4: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 29.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sqr-neg-rev N/A

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

   11. lift-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 30.0

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

6. Applied rewrites 30.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. lower-*.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in a around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

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

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

    4. metadata-eval N/A

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

    5. distribute-lft-out-- N/A

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

    6. lower-*.f64 N/A

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

    7. lower--.f64 N/A

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

    8. associate-/l* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-/.f64 91.9

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

11. Applied rewrites 91.9%

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

Alternative 5: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l* N/A

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

   6. div-add N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 91.9%

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

Alternative 6: 90.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. associate-*l/ N/A

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

   6. distribute-neg-frac N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

5. Applied rewrites 91.6%

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

6. Taylor expanded in b around -inf

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

8. Applied rewrites 91.6%

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

Alternative 7: 81.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(c\right)}{b}} \]

   4. lower-neg.f64 82.9

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

5. Applied rewrites 82.9%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(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)))