Quadratic roots, narrow range

Percentage Accurate: 55.4% → 91.7%

Time: 6.6s

Alternatives: 7

Speedup: 3.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.4% 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: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -300:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t\_0}^{3}}{\mathsf{fma}\left(b, b, t\_0 \cdot t\_0 + b \cdot t\_0\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -300.0)
     (/
      (/ (+ (pow (- b) 3.0) (pow t_0 3.0)) (fma b b (+ (* t_0 t_0) (* b t_0))))
      (* 2.0 a))
     (/
      (+
       (*
        (* c c)
        (fma
         c
         (* (* a a) (fma 5.0 (/ (* a c) (pow b 6.0)) (* 2.0 (pow b -4.0))))
         (/ a (* b b))))
       c)
      (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -300.0) {
		tmp = ((pow(-b, 3.0) + pow(t_0, 3.0)) / fma(b, b, ((t_0 * t_0) + (b * t_0)))) / (2.0 * a);
	} else {
		tmp = (((c * c) * fma(c, ((a * a) * fma(5.0, ((a * c) / pow(b, 6.0)), (2.0 * pow(b, -4.0)))), (a / (b * b)))) + c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -300.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_0 ^ 3.0)) / fma(b, b, Float64(Float64(t_0 * t_0) + Float64(b * t_0)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * fma(c, Float64(Float64(a * a) * fma(5.0, Float64(Float64(a * c) / (b ^ 6.0)), Float64(2.0 * (b ^ -4.0)))), Float64(a / Float64(b * b)))) + c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -300.0], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[(N[(a * a), $MachinePrecision] * N[(5.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -300

   1. Initial program 96.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 97.1%

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

   if -300 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

   8. Applied rewrites 92.9%

      \[\leadsto -\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{4}}, 2, \mathsf{fma}\left({a}^{3} \cdot \frac{{c}^{4}}{{b}^{6}}, 5, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)\right) + c}{b} \]

   9. Taylor expanded in c around 0

      \[\leadsto -\frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -\frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]

       2. pow2 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]

       3. lift-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]

       4. lower-fma.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, 2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}, \frac{a}{{b}^{2}}\right) + c}{b} \]

   11. Applied rewrites 92.9%

       \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(5, {a}^{3} \cdot \frac{c}{{b}^{6}}, \frac{2 \cdot \left(a \cdot a\right)}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

   12. Taylor expanded in a around 0

       \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, {a}^{2} \cdot \left(5 \cdot \frac{a \cdot c}{{b}^{6}} + 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, {a}^{2} \cdot \left(5 \cdot \frac{a \cdot c}{{b}^{6}} + 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       2. pow2 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \left(5 \cdot \frac{a \cdot c}{{b}^{6}} + 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       3. lift-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \left(5 \cdot \frac{a \cdot c}{{b}^{6}} + 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       4. lower-fma.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       5. lower-/.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       6. lower-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       7. lift-pow.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       8. lower-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       9. pow-flip N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{\left(\mathsf{neg}\left(4\right)\right)}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       10. metadata-eval N/A

           \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       11. lower-pow.f64 92.9

           \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

   14. Applied rewrites 92.9%

       \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -300:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}^{3}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -300:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -300.0)
   (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) (* 2.0 a))
   (/
    (+
     (*
      (* c c)
      (fma
       c
       (* (* a a) (fma 5.0 (/ (* a c) (pow b 6.0)) (* 2.0 (pow b -4.0))))
       (/ a (* b b))))
     c)
    (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -300.0) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = (((c * c) * fma(c, ((a * a) * fma(5.0, ((a * c) / pow(b, 6.0)), (2.0 * pow(b, -4.0)))), (a / (b * b)))) + c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -300.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * fma(c, Float64(Float64(a * a) * fma(5.0, Float64(Float64(a * c) / (b ^ 6.0)), Float64(2.0 * (b ^ -4.0)))), Float64(a / Float64(b * b)))) + c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -300.0], 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], N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[(N[(a * a), $MachinePrecision] * N[(5.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -300

   1. Initial program 96.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 96.5%

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

   if -300 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

   8. Applied rewrites 92.9%

      \[\leadsto -\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{4}}, 2, \mathsf{fma}\left({a}^{3} \cdot \frac{{c}^{4}}{{b}^{6}}, 5, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)\right) + c}{b} \]

   9. Taylor expanded in c around 0

      \[\leadsto -\frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -\frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]

       2. pow2 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]

       3. lift-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]

       4. lower-fma.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, 2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}, \frac{a}{{b}^{2}}\right) + c}{b} \]

   11. Applied rewrites 92.9%

       \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(5, {a}^{3} \cdot \frac{c}{{b}^{6}}, \frac{2 \cdot \left(a \cdot a\right)}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

   12. Taylor expanded in a around 0

       \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, {a}^{2} \cdot \left(5 \cdot \frac{a \cdot c}{{b}^{6}} + 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, {a}^{2} \cdot \left(5 \cdot \frac{a \cdot c}{{b}^{6}} + 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       2. pow2 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \left(5 \cdot \frac{a \cdot c}{{b}^{6}} + 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       3. lift-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \left(5 \cdot \frac{a \cdot c}{{b}^{6}} + 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       4. lower-fma.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       5. lower-/.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       6. lower-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       7. lift-pow.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       8. lower-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot \frac{1}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       9. pow-flip N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{\left(\mathsf{neg}\left(4\right)\right)}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       10. metadata-eval N/A

           \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

       11. lower-pow.f64 92.9

           \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

   14. Applied rewrites 92.9%

       \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -300:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \left(a \cdot a\right) \cdot \mathsf{fma}\left(5, \frac{a \cdot c}{{b}^{6}}, 2 \cdot {b}^{-4}\right), \frac{a}{b \cdot b}\right) + c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.02:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.02)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.02) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.02)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0200000000000000004

   1. Initial program 80.5%

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

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

      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. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 80.6

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

   4. Applied rewrites 80.6%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 89.9

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

   8. Applied rewrites 89.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.02:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \frac{a + \frac{\left(\left(2 \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}}{b \cdot b} + c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.5)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/
    (+ (* (* c c) (/ (+ a (/ (* (* (* 2.0 a) a) c) (* b b))) (* b b))) c)
    (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (((c * c) * ((a + ((((2.0 * a) * a) * c) / (b * b))) / (b * b))) + c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * Float64(Float64(a + Float64(Float64(Float64(Float64(2.0 * a) * a) * c) / Float64(b * b))) / Float64(b * b))) + c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.5], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * N[(N[(a + N[(N[(N[(N[(2.0 * a), $MachinePrecision] * a), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5

   1. Initial program 84.1%

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

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

      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. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.3

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

   4. Applied rewrites 84.3%

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

   if 2.5 < b

   1. Initial program 49.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

   8. Applied rewrites 94.2%

      \[\leadsto -\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{4}}, 2, \mathsf{fma}\left({a}^{3} \cdot \frac{{c}^{4}}{{b}^{6}}, 5, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)\right) + c}{b} \]

   9. Taylor expanded in c around 0

      \[\leadsto -\frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -\frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]

       2. pow2 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]

       3. lift-*.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{b} \]

       4. lower-fma.f64 N/A

          \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, 2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}, \frac{a}{{b}^{2}}\right) + c}{b} \]

   11. Applied rewrites 94.2%

       \[\leadsto -\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(5, {a}^{3} \cdot \frac{c}{{b}^{6}}, \frac{2 \cdot \left(a \cdot a\right)}{{b}^{4}}\right), \frac{a}{b \cdot b}\right) + c}{b} \]

   12. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 92.1

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

   14. Applied rewrites 92.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \frac{a + \frac{\left(\left(2 \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}}{b \cdot b} + c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

5. Applied rewrites 90.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 81.8

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

8. Applied rewrites 81.8%

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

9. Final simplification 81.8%

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

Alternative 6: 81.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

5. Applied rewrites 90.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in b around -inf

   \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}\right) \]

   2. lower-neg.f64 N/A

      \[\leadsto -\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto -\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

8. Applied rewrites 90.5%

   \[\leadsto -\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{4}}, 2, \mathsf{fma}\left({a}^{3} \cdot \frac{{c}^{4}}{{b}^{6}}, 5, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)\right) + c}{b} \]

9. Taylor expanded in c around 0

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

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. associate-/l* N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 81.7

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

11. Applied rewrites 81.7%

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

12. Final simplification 81.7%

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

Alternative 7: 64.3% 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 55.5%

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

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 64.5

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

5. Applied rewrites 64.5%

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

Reproduce

herbie shell --seed 2025075 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))