Quadratic roots, full range

Percentage Accurate: 52.2% → 90.8%

Time: 7.5s

Alternatives: 8

Speedup: 2.5×

Specification

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: 52.2% 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: 90.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b} \cdot -2, \frac{c}{b}, 2\right) \cdot \left(-b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-221}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -2e+153)
     (/ (* (fma (* (/ a b) -2.0) (/ c b) 2.0) (- b)) (* 2.0 a))
     (if (<= b 2.3e-221)
       (/ (- t_0 b) (* 2.0 a))
       (if (<= b 2e+80) (/ (* 2.0 c) (- (- b) t_0)) (/ 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 <= -2e+153) {
		tmp = (fma(((a / b) * -2.0), (c / b), 2.0) * -b) / (2.0 * a);
	} else if (b <= 2.3e-221) {
		tmp = (t_0 - b) / (2.0 * a);
	} else if (b <= 2e+80) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = 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 (b <= -2e+153)
		tmp = Float64(Float64(fma(Float64(Float64(a / b) * -2.0), Float64(c / b), 2.0) * Float64(-b)) / Float64(2.0 * a));
	elseif (b <= 2.3e-221)
		tmp = Float64(Float64(t_0 - b) / Float64(2.0 * a));
	elseif (b <= 2e+80)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(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[b, -2e+153], N[(N[(N[(N[(N[(a / b), $MachinePrecision] * -2.0), $MachinePrecision] * N[(c / b), $MachinePrecision] + 2.0), $MachinePrecision] * (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.3e-221], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+80], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e153

   1. Initial program 24.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 24.5

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 24.6

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

   4. Applied rewrites 24.6%

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

   5. Taylor expanded in b around -inf

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

   7. Applied rewrites 95.0%

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

   if -2e153 < b < 2.3e-221

   1. Initial program 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 86.1

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 86.2

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

   4. Applied rewrites 86.2%

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

   if 2.3e-221 < b < 2e80

   1. Initial program 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 78.1

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

   9. Applied rewrites 78.1%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 92.9%

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

   if 2e80 < b

   1. Initial program 7.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 93.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b} \cdot -2, \frac{c}{b}, 2\right) \cdot \left(-b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+136}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-221}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -4.5e+136)
     (- (fma (/ (- c) (* b b)) b (/ b a)))
     (if (<= b 2.3e-221)
       (/ (- t_0 b) (* 2.0 a))
       (if (<= b 2e+80) (/ (* 2.0 c) (- (- b) t_0)) (/ 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 <= -4.5e+136) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else if (b <= 2.3e-221) {
		tmp = (t_0 - b) / (2.0 * a);
	} else if (b <= 2e+80) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = 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 (b <= -4.5e+136)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	elseif (b <= 2.3e-221)
		tmp = Float64(Float64(t_0 - b) / Float64(2.0 * a));
	elseif (b <= 2e+80)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(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[b, -4.5e+136], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 2.3e-221], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+80], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.4999999999999999e136

   1. Initial program 29.3%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 93.1%

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

   if -4.4999999999999999e136 < b < 2.3e-221

   1. Initial program 86.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 86.6

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 86.6

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

   4. Applied rewrites 86.6%

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

   if 2.3e-221 < b < 2e80

   1. Initial program 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 78.1

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

   9. Applied rewrites 78.1%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 92.9%

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

   if 2e80 < b

   1. Initial program 7.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 93.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+136}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+136}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+136)
   (- (fma (/ (- c) (* b b)) b (/ b a)))
   (if (<= b 2.3e-221)
     (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
     (if (<= b 2e+80)
       (/ (* 2.0 c) (- (- b) (sqrt (fma (* -4.0 a) c (* b b)))))
       (/ c (- b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+136) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else if (b <= 2.3e-221) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else if (b <= 2e+80) {
		tmp = (2.0 * c) / (-b - sqrt(fma((-4.0 * a), c, (b * b))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+136)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	elseif (b <= 2.3e-221)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	elseif (b <= 2e+80)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+136], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 2.3e-221], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+80], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.4999999999999999e136

   1. Initial program 29.3%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 93.1%

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

   if -4.4999999999999999e136 < b < 2.3e-221

   1. Initial program 86.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 86.6

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

   4. Applied rewrites 86.6%

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

   if 2.3e-221 < b < 2e80

   1. Initial program 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 78.1

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

   9. Applied rewrites 78.1%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 92.9%

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

   if 2e80 < b

   1. Initial program 7.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 93.1%

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

Alternative 4: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-17}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-17)
   (/ (- b) a)
   (if (<= b 2e+80)
     (/ (* 2.0 c) (- (- b) (sqrt (fma (* -4.0 a) c (* b b)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-17) {
		tmp = -b / a;
	} else if (b <= 2e+80) {
		tmp = (2.0 * c) / (-b - sqrt(fma((-4.0 * a), c, (b * b))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-17)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2e+80)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-17], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2e+80], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.99999999999999957e-17

   1. Initial program 56.2%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 85.1%

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

   if -8.99999999999999957e-17 < b < 2e80

   1. Initial program 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 73.8

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

   9. Applied rewrites 73.8%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 80.9%

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

   if 2e80 < b

   1. Initial program 7.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 93.1%

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

Alternative 5: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e-7)
   (/ (- b) a)
   (if (<= b 2.7e-95) (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-7) {
		tmp = -b / a;
	} else if (b <= 2.7e-95) {
		tmp = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b <= (-1.95d-7)) then
        tmp = -b / a
    else if (b <= 2.7d-95) then
        tmp = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-7) {
		tmp = -b / a;
	} else if (b <= 2.7e-95) {
		tmp = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e-7:
		tmp = -b / a
	elif b <= 2.7e-95:
		tmp = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e-7)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.7e-95)
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.95e-7)
		tmp = -b / a;
	elseif (b <= 2.7e-95)
		tmp = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e-7], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.7e-95], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.95000000000000012e-7

   1. Initial program 53.7%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 86.8%

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

   if -1.95000000000000012e-7 < b < 2.7e-95

   1. Initial program 83.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 83.7

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 83.7

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

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 72.3

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

   9. Applied rewrites 72.3%

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

   if 2.7e-95 < b

   1. Initial program 18.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (- b) a) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -b / a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = -b / a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -b / a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -b / a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = -b / a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[((-b) / a), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 67.5%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 62.1%

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

   if -9.999999999999969e-311 < b

   1. Initial program 33.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 67.2%

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

Alternative 7: 35.7% 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(c / Float64(-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 49.7%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 36.0%

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

Alternative 8: 11.6% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 49.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 49.7

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 49.7

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

4. Applied rewrites 49.7%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lower-+.f64 N/A

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

6. Applied rewrites 48.8%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 7.9%

   \[\leadsto \color{blue}{\frac{b}{a} \cdot 0} \]

10. Taylor expanded in a around 0

    \[\leadsto 0 \]
11. Step-by-step derivation

12. Applied rewrites 11.7%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025056 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))