Quadratic roots, medium range

Percentage Accurate: 31.5% → 99.7%

Time: 10.6s

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: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. sqr-abs-rev N/A

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

   10. fabs-fabs N/A

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

   11. fabs-fabs N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. pow2 N/A

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

   14. sqrt-pow1 N/A

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

   15. metadata-eval N/A

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left|b\right| \cdot {b}^{\color{blue}{1}} + \left(\mathsf{neg}\left(a\right)\right) \cdot \left(4 \cdot c\right)}}{2 \cdot a} \]

   16. unpow1 N/A

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

   17. associate-*l* N/A

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

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

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

   19. *-commutative N/A

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

   20. lift-*.f64 N/A

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

4. Applied rewrites 35.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 36.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. sqr-abs-rev N/A

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

   10. fabs-fabs N/A

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

   11. fabs-fabs N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. pow2 N/A

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

   14. sqrt-pow1 N/A

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

   15. metadata-eval N/A

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left|b\right| \cdot {b}^{\color{blue}{1}} + \left(\mathsf{neg}\left(a\right)\right) \cdot \left(4 \cdot c\right)}}{2 \cdot a} \]

   16. unpow1 N/A

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

   17. associate-*l* N/A

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

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

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

   19. *-commutative N/A

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

   20. lift-*.f64 N/A

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

4. Applied rewrites 35.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 36.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.4

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. sqr-abs-rev N/A

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

   10. fabs-fabs N/A

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

   11. fabs-fabs N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. pow2 N/A

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

   14. sqrt-pow1 N/A

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

   15. metadata-eval N/A

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left|b\right| \cdot {b}^{\color{blue}{1}} + \left(\mathsf{neg}\left(a\right)\right) \cdot \left(4 \cdot c\right)}}{2 \cdot a} \]

   16. unpow1 N/A

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

   17. associate-*l* N/A

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

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

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

   19. *-commutative N/A

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

   20. lift-*.f64 N/A

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

4. Applied rewrites 35.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 36.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. +-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 90.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. sqr-abs-rev N/A

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

   10. fabs-fabs N/A

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

   11. fabs-fabs N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. pow2 N/A

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

   14. sqrt-pow1 N/A

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

   15. metadata-eval N/A

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left|b\right| \cdot {b}^{\color{blue}{1}} + \left(\mathsf{neg}\left(a\right)\right) \cdot \left(4 \cdot c\right)}}{2 \cdot a} \]

   16. unpow1 N/A

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

   17. associate-*l* N/A

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

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

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

   19. *-commutative N/A

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

   20. lift-*.f64 N/A

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

4. Applied rewrites 35.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 36.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 88.8

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

11. Applied rewrites 88.8%

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

Alternative 5: 90.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. sqr-abs-rev N/A

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

   10. fabs-fabs N/A

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

   11. fabs-fabs N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. pow2 N/A

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

   14. sqrt-pow1 N/A

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

   15. metadata-eval N/A

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left|b\right| \cdot {b}^{\color{blue}{1}} + \left(\mathsf{neg}\left(a\right)\right) \cdot \left(4 \cdot c\right)}}{2 \cdot a} \]

   16. unpow1 N/A

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

   17. associate-*l* N/A

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

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

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

   19. *-commutative N/A

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

   20. lift-*.f64 N/A

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

4. Applied rewrites 35.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 36.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. associate-/l* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-/.f64 88.8

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

11. Applied rewrites 88.8%

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

Alternative 6: 90.8% 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 35.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 88.7%

   \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}} \]
6. 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 35.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 77.9

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

5. Applied rewrites 77.9%

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

Reproduce

herbie shell --seed 2025003 
(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)))