quadm (p42, negative)

Percentage Accurate: 51.8% → 87.3%

Time: 4.7s

Alternatives: 11

Speedup: 2.5×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.8% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{-b}\\ \mathbf{elif}\;b \leq -2.45 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{a + a}\\ \mathbf{elif}\;b \leq 10^{+123}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e+49)
   (/ (* c (fma a (/ c (* b b)) 1.0)) (- b))
   (if (<= b -2.45e-116)
     (/
      (/ (* 4.0 (* a c)) (fma -1.0 b (sqrt (fma (* -4.0 a) c (* b b)))))
      (+ a a))
     (if (<= b 1e+123)
       (/ (- (- b) (sqrt (fma (* c a) -4.0 (* b b)))) (+ a a))
       (+ (/ (- b) a) (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e+49) {
		tmp = (c * fma(a, (c / (b * b)), 1.0)) / -b;
	} else if (b <= -2.45e-116) {
		tmp = ((4.0 * (a * c)) / fma(-1.0, b, sqrt(fma((-4.0 * a), c, (b * b))))) / (a + a);
	} else if (b <= 1e+123) {
		tmp = (-b - sqrt(fma((c * a), -4.0, (b * b)))) / (a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e+49)
		tmp = Float64(Float64(c * fma(a, Float64(c / Float64(b * b)), 1.0)) / Float64(-b));
	elseif (b <= -2.45e-116)
		tmp = Float64(Float64(Float64(4.0 * Float64(a * c)) / fma(-1.0, b, sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))))) / Float64(a + a));
	elseif (b <= 1e+123)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e+49], N[(N[(c * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision], If[LessEqual[b, -2.45e-116], N[(N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * b + N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+123], N[(N[((-b) - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.1e49

   1. Initial program 17.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in c around 0

      \[\leadsto -\frac{c \cdot \left(1 + \frac{a \cdot c}{{b}^{2}}\right)}{b} \]
   7. 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 92.6

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

   8. Applied rewrites 92.6%

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

   if -1.1e49 < b < -2.44999999999999989e-116

   1. Initial program 45.7%

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

      1. lift-neg.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 45.2%

      \[\leadsto \frac{\color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}}{2 \cdot a} \]

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{4 \cdot \color{blue}{\left(a \cdot c\right)}}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a} \]

      2. lower-*.f64 78.3

         \[\leadsto \frac{\frac{4 \cdot \left(a \cdot \color{blue}{c}\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a} \]

   7. Applied rewrites 78.3%

      \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{\color{blue}{2 \cdot a}} \]

      2. count-2-rev N/A

         \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{\color{blue}{a + a}} \]

      3. lift-+.f64 78.3

         \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{\color{blue}{a + a}} \]

   9. Applied rewrites 78.3%

      \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{\color{blue}{a + a}} \]

   if -2.44999999999999989e-116 < b < 9.99999999999999978e122

   1. Initial program 83.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 83.5

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

   4. Applied rewrites 83.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 83.5

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

   6. Applied rewrites 83.5%

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

   if 9.99999999999999978e122 < b

   1. Initial program 46.9%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.8

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

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{-b}\\ \mathbf{elif}\;b \leq -2.45 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{a + a}\\ \mathbf{elif}\;b \leq 10^{+123}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-113}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{elif}\;b \leq 10^{+123}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.8e+139)
   (/ c (- b))
   (if (<= b -5e-113)
     (/
      (* 4.0 (* a c))
      (* (fma -1.0 b (sqrt (fma (* -4.0 a) c (* b b)))) (* 2.0 a)))
     (if (<= b 1e+123)
       (/ (- (- b) (sqrt (fma (* c a) -4.0 (* b b)))) (+ a a))
       (+ (/ (- b) a) (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.8e+139) {
		tmp = c / -b;
	} else if (b <= -5e-113) {
		tmp = (4.0 * (a * c)) / (fma(-1.0, b, sqrt(fma((-4.0 * a), c, (b * b)))) * (2.0 * a));
	} else if (b <= 1e+123) {
		tmp = (-b - sqrt(fma((c * a), -4.0, (b * b)))) / (a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.8e+139)
		tmp = Float64(c / Float64(-b));
	elseif (b <= -5e-113)
		tmp = Float64(Float64(4.0 * Float64(a * c)) / Float64(fma(-1.0, b, sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))) * Float64(2.0 * a)));
	elseif (b <= 1e+123)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.8e+139], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, -5e-113], N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 * b + N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+123], N[(N[((-b) - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.7999999999999998e139

   1. Initial program 1.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -8.7999999999999998e139 < b < -4.9999999999999997e-113

   1. Initial program 42.9%

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

      1. lift-neg.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 42.5%

      \[\leadsto \frac{\color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}}{2 \cdot a} \]

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{4 \cdot \color{blue}{\left(a \cdot c\right)}}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a} \]

      2. lower-*.f64 81.2

         \[\leadsto \frac{\frac{4 \cdot \left(a \cdot \color{blue}{c}\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a} \]

   7. Applied rewrites 81.2%

      \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{\color{blue}{2 \cdot a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a}} \]

   9. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}} \]

   if -4.9999999999999997e-113 < b < 9.99999999999999978e122

   1. Initial program 83.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 83.5

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

   4. Applied rewrites 83.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 83.5

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

   6. Applied rewrites 83.5%

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

   if 9.99999999999999978e122 < b

   1. Initial program 46.9%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.8

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

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-113}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{elif}\;b \leq 10^{+123}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{-b}\\ \mathbf{elif}\;b \leq 10^{+123}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-31)
   (/ (* c (fma a (/ c (* b b)) 1.0)) (- b))
   (if (<= b 1e+123)
     (/ (- (- b) (sqrt (fma (* c a) -4.0 (* b b)))) (+ a a))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-31) {
		tmp = (c * fma(a, (c / (b * b)), 1.0)) / -b;
	} else if (b <= 1e+123) {
		tmp = (-b - sqrt(fma((c * a), -4.0, (b * b)))) / (a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-31)
		tmp = Float64(Float64(c * fma(a, Float64(c / Float64(b * b)), 1.0)) / Float64(-b));
	elseif (b <= 1e+123)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-31], N[(N[(c * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision], If[LessEqual[b, 1e+123], N[(N[((-b) - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.8e-31

   1. Initial program 21.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in c around 0

      \[\leadsto -\frac{c \cdot \left(1 + \frac{a \cdot c}{{b}^{2}}\right)}{b} \]
   7. 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 84.6

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

   8. Applied rewrites 84.6%

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

   if -3.8e-31 < b < 9.99999999999999978e122

   1. Initial program 79.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 79.4

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

   6. Applied rewrites 79.4%

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

   if 9.99999999999999978e122 < b

   1. Initial program 46.9%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.8

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

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{-b}\\ \mathbf{elif}\;b \leq 10^{+123}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{-b}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-77)
   (/ (* c (fma a (/ c (* b b)) 1.0)) (- b))
   (if (<= b 1.9e-70)
     (/ (+ b (sqrt (* -4.0 (* a c)))) (- (+ a a)))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-77) {
		tmp = (c * fma(a, (c / (b * b)), 1.0)) / -b;
	} else if (b <= 1.9e-70) {
		tmp = (b + sqrt((-4.0 * (a * c)))) / -(a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-77)
		tmp = Float64(Float64(c * fma(a, Float64(c / Float64(b * b)), 1.0)) / Float64(-b));
	elseif (b <= 1.9e-70)
		tmp = Float64(Float64(b + sqrt(Float64(-4.0 * Float64(a * c)))) / Float64(-Float64(a + a)));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-77], N[(N[(c * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision], If[LessEqual[b, 1.9e-70], N[(N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(a + a), $MachinePrecision])), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8999999999999999e-77

   1. Initial program 24.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in c around 0

      \[\leadsto -\frac{c \cdot \left(1 + \frac{a \cdot c}{{b}^{2}}\right)}{b} \]
   7. 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 81.1

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

   8. Applied rewrites 81.1%

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

   if -2.8999999999999999e-77 < b < 1.8999999999999999e-70

   1. Initial program 76.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 76.9

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 75.7

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

   9. Applied rewrites 75.7%

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

   if 1.8999999999999999e-70 < b

   1. Initial program 63.8%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 88.4

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

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{-b}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-77)
   (/ c (- b))
   (if (<= b 1.9e-70)
     (/ (+ b (sqrt (* -4.0 (* a c)))) (- (+ a a)))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-77) {
		tmp = c / -b;
	} else if (b <= 1.9e-70) {
		tmp = (b + sqrt((-4.0 * (a * c)))) / -(a + a);
	} else {
		tmp = (-b / a) + (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 <= (-2.9d-77)) then
        tmp = c / -b
    else if (b <= 1.9d-70) then
        tmp = (b + sqrt(((-4.0d0) * (a * c)))) / -(a + a)
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-77) {
		tmp = c / -b;
	} else if (b <= 1.9e-70) {
		tmp = (b + Math.sqrt((-4.0 * (a * c)))) / -(a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-77:
		tmp = c / -b
	elif b <= 1.9e-70:
		tmp = (b + math.sqrt((-4.0 * (a * c)))) / -(a + a)
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-77)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.9e-70)
		tmp = Float64(Float64(b + sqrt(Float64(-4.0 * Float64(a * c)))) / Float64(-Float64(a + a)));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e-77)
		tmp = c / -b;
	elseif (b <= 1.9e-70)
		tmp = (b + sqrt((-4.0 * (a * c)))) / -(a + a);
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-77], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.9e-70], N[(N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(a + a), $MachinePrecision])), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8999999999999999e-77

   1. Initial program 24.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -2.8999999999999999e-77 < b < 1.8999999999999999e-70

   1. Initial program 76.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 76.9

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 75.7

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

   9. Applied rewrites 75.7%

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

   if 1.8999999999999999e-70 < b

   1. Initial program 63.8%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 88.4

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

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.05 \cdot 10^{-71}:\\ \;\;\;\;\frac{-\sqrt{-4 \cdot \left(c \cdot a\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-77)
   (/ c (- b))
   (if (<= b 3.05e-71)
     (/ (- (sqrt (* -4.0 (* c a)))) (+ a a))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-77) {
		tmp = c / -b;
	} else if (b <= 3.05e-71) {
		tmp = -sqrt((-4.0 * (c * a))) / (a + a);
	} else {
		tmp = (-b / a) + (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 <= (-2.9d-77)) then
        tmp = c / -b
    else if (b <= 3.05d-71) then
        tmp = -sqrt(((-4.0d0) * (c * a))) / (a + a)
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-77) {
		tmp = c / -b;
	} else if (b <= 3.05e-71) {
		tmp = -Math.sqrt((-4.0 * (c * a))) / (a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-77:
		tmp = c / -b
	elif b <= 3.05e-71:
		tmp = -math.sqrt((-4.0 * (c * a))) / (a + a)
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-77)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 3.05e-71)
		tmp = Float64(Float64(-sqrt(Float64(-4.0 * Float64(c * a)))) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e-77)
		tmp = c / -b;
	elseif (b <= 3.05e-71)
		tmp = -sqrt((-4.0 * (c * a))) / (a + a);
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-77], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 3.05e-71], N[((-N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8999999999999999e-77

   1. Initial program 24.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -2.8999999999999999e-77 < b < 3.0499999999999999e-71

   1. Initial program 76.9%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto \frac{-\sqrt{a \cdot c} \cdot \sqrt{-4}}{2 \cdot a} \]

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 75.5

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

   7. Applied rewrites 75.5%

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

   if 3.0499999999999999e-71 < b

   1. Initial program 63.8%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 88.4

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

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.05 \cdot 10^{-71}:\\ \;\;\;\;\frac{-\sqrt{-4 \cdot \left(c \cdot a\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-101}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.28e-101)
   (/ c (- b))
   (if (<= b 5.5e-78) (sqrt (* (/ c a) -1.0)) (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.28e-101) {
		tmp = c / -b;
	} else if (b <= 5.5e-78) {
		tmp = sqrt(((c / a) * -1.0));
	} else {
		tmp = (-b / a) + (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 <= (-1.28d-101)) then
        tmp = c / -b
    else if (b <= 5.5d-78) then
        tmp = sqrt(((c / a) * (-1.0d0)))
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.28e-101) {
		tmp = c / -b;
	} else if (b <= 5.5e-78) {
		tmp = Math.sqrt(((c / a) * -1.0));
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.28e-101:
		tmp = c / -b
	elif b <= 5.5e-78:
		tmp = math.sqrt(((c / a) * -1.0))
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.28e-101)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5.5e-78)
		tmp = sqrt(Float64(Float64(c / a) * -1.0));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.28e-101)
		tmp = c / -b;
	elseif (b <= 5.5e-78)
		tmp = sqrt(((c / a) * -1.0));
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.28e-101], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5.5e-78], N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.27999999999999995e-101

   1. Initial program 26.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

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

   if -1.27999999999999995e-101 < b < 5.50000000000000017e-78

   1. Initial program 77.0%

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

   3. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      4. lower-/.f64 38.7

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

   5. Applied rewrites 38.7%

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

   if 5.50000000000000017e-78 < b

   1. Initial program 64.5%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 86.6

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

   7. Applied rewrites 86.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-101}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-101}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.28e-101)
   (/ c (- b))
   (if (<= b 4.6e-74) (sqrt (* (/ c a) -1.0)) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.28e-101) {
		tmp = c / -b;
	} else if (b <= 4.6e-74) {
		tmp = sqrt(((c / a) * -1.0));
	} else {
		tmp = -b / a;
	}
	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 <= (-1.28d-101)) then
        tmp = c / -b
    else if (b <= 4.6d-74) then
        tmp = sqrt(((c / a) * (-1.0d0)))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.28e-101) {
		tmp = c / -b;
	} else if (b <= 4.6e-74) {
		tmp = Math.sqrt(((c / a) * -1.0));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.28e-101:
		tmp = c / -b
	elif b <= 4.6e-74:
		tmp = math.sqrt(((c / a) * -1.0))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.28e-101)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 4.6e-74)
		tmp = sqrt(Float64(Float64(c / a) * -1.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.28e-101)
		tmp = c / -b;
	elseif (b <= 4.6e-74)
		tmp = sqrt(((c / a) * -1.0));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.28e-101], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 4.6e-74], N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.27999999999999995e-101

   1. Initial program 26.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

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

   if -1.27999999999999995e-101 < b < 4.59999999999999961e-74

   1. Initial program 77.4%

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

   3. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      4. lower-/.f64 38.2

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

   5. Applied rewrites 38.2%

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

   if 4.59999999999999961e-74 < b

   1. Initial program 64.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-neg.f64 87.5

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

   5. Applied rewrites 87.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-101}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	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 <= (-5d-311)) then
        tmp = c / -b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 38.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -5.00000000000023e-311 < b

   1. Initial program 69.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-neg.f64 68.0

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

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.9% 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 52.6%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 34.6

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

5. Applied rewrites 34.6%

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

6. Final simplification 34.6%

   \[\leadsto \frac{c}{-b} \]
7. Add Preprocessing

Alternative 11: 10.4% accurate, 4.2× 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 / 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 52.6%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 33.1

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

5. Applied rewrites 33.1%

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

6. Taylor expanded in a around inf

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

   1. lift-/.f64 8.8

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

8. Applied rewrites 8.8%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2025073 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))