quadm (p42, negative)

Percentage Accurate: 52.2% → 84.1%

Time: 5.6s

Alternatives: 14

Speedup: 1.2×

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: 52.2% 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: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \mathbf{elif}\;b \leq -3.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{\left(\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.32e+154)
   (/ (* -1.0 (/ (* a c) b)) a)
   (if (<= b -3.1e-120)
     (/
      (* (- (- (* b b) (* b b)) (* (* a c) -4.0)) (/ 0.5 a))
      (+ (sqrt (fma -4.0 (* c a) (* b b))) (- b)))
     (if (<= b 5.4e+85)
       (/ (- (- b) (sqrt (fma (* -4.0 c) a (* b b)))) (+ a a))
       (* -1.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -3.1e-120) {
		tmp = ((((b * b) - (b * b)) - ((a * c) * -4.0)) * (0.5 / a)) / (sqrt(fma(-4.0, (c * a), (b * b))) + -b);
	} else if (b <= 5.4e+85) {
		tmp = (-b - sqrt(fma((-4.0 * c), a, (b * b)))) / (a + a);
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.32e+154)
		tmp = Float64(Float64(-1.0 * Float64(Float64(a * c) / b)) / a);
	elseif (b <= -3.1e-120)
		tmp = Float64(Float64(Float64(Float64(Float64(b * b) - Float64(b * b)) - Float64(Float64(a * c) * -4.0)) * Float64(0.5 / a)) / Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + Float64(-b)));
	elseif (b <= 5.4e+85)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.32e+154], N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -3.1e-120], N[(N[(N[(N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.4e+85], N[(N[((-b) - N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.31999999999999998e154

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -1.31999999999999998e154 < b < -3.10000000000000019e-120

   1. Initial program 52.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + \left(-b\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 38.9

         \[\leadsto \frac{\left(\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -4}\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + \left(-b\right)} \]

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 38.9

          \[\leadsto \frac{\left(\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(a \cdot c\right)} \cdot -4\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + \left(-b\right)} \]

   5. Applied rewrites 38.9%

      \[\leadsto \frac{\color{blue}{\left(\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -4\right)} \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + \left(-b\right)} \]

   if -3.10000000000000019e-120 < b < 5.39999999999999966e85

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if 5.39999999999999966e85 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+115}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-6}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + \left(-b\right)\right) \cdot \left(a + a\right)}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.9e+115)
   (/ (* -1.0 (/ (* a c) b)) a)
   (if (<= b -1.26e-6)
     (/
      (* 4.0 (* a c))
      (* (+ (sqrt (fma -4.0 (* c a) (* b b))) (- b)) (+ a a)))
     (if (<= b 5.4e+85)
       (/ (- (- b) (sqrt (fma (* -4.0 c) a (* b b)))) (+ a a))
       (* -1.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.9e+115) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -1.26e-6) {
		tmp = (4.0 * (a * c)) / ((sqrt(fma(-4.0, (c * a), (b * b))) + -b) * (a + a));
	} else if (b <= 5.4e+85) {
		tmp = (-b - sqrt(fma((-4.0 * c), a, (b * b)))) / (a + a);
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.9e+115)
		tmp = Float64(Float64(-1.0 * Float64(Float64(a * c) / b)) / a);
	elseif (b <= -1.26e-6)
		tmp = Float64(Float64(4.0 * Float64(a * c)) / Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + Float64(-b)) * Float64(a + a)));
	elseif (b <= 5.4e+85)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.9e+115], N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -1.26e-6], N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.4e+85], N[(N[((-b) - N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.90000000000000006e115

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -3.90000000000000006e115 < b < -1.26000000000000001e-6

   1. Initial program 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 23.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 41.2

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

   8. Applied rewrites 41.2%

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

   if -1.26000000000000001e-6 < b < 5.39999999999999966e85

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if 5.39999999999999966e85 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \mathbf{elif}\;b \leq -0.000195:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b} - b}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.32e+154)
   (/ (* -1.0 (/ (* a c) b)) a)
   (if (<= b -0.000195)
     (* 2.0 (/ c (- (sqrt (* b b)) b)))
     (if (<= b 5.4e+85)
       (/ (- (- b) (sqrt (fma (* -4.0 c) a (* b b)))) (+ a a))
       (* -1.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -0.000195) {
		tmp = 2.0 * (c / (sqrt((b * b)) - b));
	} else if (b <= 5.4e+85) {
		tmp = (-b - sqrt(fma((-4.0 * c), a, (b * b)))) / (a + a);
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.32e+154)
		tmp = Float64(Float64(-1.0 * Float64(Float64(a * c) / b)) / a);
	elseif (b <= -0.000195)
		tmp = Float64(2.0 * Float64(c / Float64(sqrt(Float64(b * b)) - b)));
	elseif (b <= 5.4e+85)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.32e+154], N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -0.000195], N[(2.0 * N[(c / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.4e+85], N[(N[((-b) - N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.31999999999999998e154

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -1.31999999999999998e154 < b < -1.94999999999999996e-4

   1. Initial program 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 23.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{c}{\sqrt{{b}^{2}} - b}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\color{blue}{\sqrt{{b}^{2}} - b}} \]

      3. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\sqrt{{b}^{2}} - \color{blue}{b}} \]

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 25.9

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

   8. Applied rewrites 25.9%

      \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{b \cdot b} - b}} \]

   if -1.94999999999999996e-4 < b < 5.39999999999999966e85

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if 5.39999999999999966e85 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \mathbf{elif}\;b \leq -0.000195:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b} - b}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.32e+154)
   (/ (* -1.0 (/ (* a c) b)) a)
   (if (<= b -0.000195)
     (* 2.0 (/ c (- (sqrt (* b b)) b)))
     (if (<= b 5.4e+85)
       (/ (* (+ (sqrt (fma (* a c) -4.0 (* b b))) b) -0.5) a)
       (* -1.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -0.000195) {
		tmp = 2.0 * (c / (sqrt((b * b)) - b));
	} else if (b <= 5.4e+85) {
		tmp = ((sqrt(fma((a * c), -4.0, (b * b))) + b) * -0.5) / a;
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.32e+154)
		tmp = Float64(Float64(-1.0 * Float64(Float64(a * c) / b)) / a);
	elseif (b <= -0.000195)
		tmp = Float64(2.0 * Float64(c / Float64(sqrt(Float64(b * b)) - b)));
	elseif (b <= 5.4e+85)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) + b) * -0.5) / a);
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.32e+154], N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -0.000195], N[(2.0 * N[(c / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.4e+85], N[(N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.31999999999999998e154

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -1.31999999999999998e154 < b < -1.94999999999999996e-4

   1. Initial program 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 23.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{c}{\sqrt{{b}^{2}} - b}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\color{blue}{\sqrt{{b}^{2}} - b}} \]

      3. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\sqrt{{b}^{2}} - \color{blue}{b}} \]

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 25.9

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

   8. Applied rewrites 25.9%

      \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{b \cdot b} - b}} \]

   if -1.94999999999999996e-4 < b < 5.39999999999999966e85

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   if 5.39999999999999966e85 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \mathbf{elif}\;b \leq -0.000195:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b} - b}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+85}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.32e+154)
   (/ (* -1.0 (/ (* a c) b)) a)
   (if (<= b -0.000195)
     (* 2.0 (/ c (- (sqrt (* b b)) b)))
     (if (<= b 5.4e+85)
       (* (+ (sqrt (fma (* a c) -4.0 (* b b))) b) (/ -0.5 a))
       (* -1.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -0.000195) {
		tmp = 2.0 * (c / (sqrt((b * b)) - b));
	} else if (b <= 5.4e+85) {
		tmp = (sqrt(fma((a * c), -4.0, (b * b))) + b) * (-0.5 / a);
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.32e+154)
		tmp = Float64(Float64(-1.0 * Float64(Float64(a * c) / b)) / a);
	elseif (b <= -0.000195)
		tmp = Float64(2.0 * Float64(c / Float64(sqrt(Float64(b * b)) - b)));
	elseif (b <= 5.4e+85)
		tmp = Float64(Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) + b) * Float64(-0.5 / a));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.32e+154], N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -0.000195], N[(2.0 * N[(c / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.4e+85], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.31999999999999998e154

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -1.31999999999999998e154 < b < -1.94999999999999996e-4

   1. Initial program 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 23.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{c}{\sqrt{{b}^{2}} - b}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\color{blue}{\sqrt{{b}^{2}} - b}} \]

      3. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\sqrt{{b}^{2}} - \color{blue}{b}} \]

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 25.9

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

   8. Applied rewrites 25.9%

      \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{b \cdot b} - b}} \]

   if -1.94999999999999996e-4 < b < 5.39999999999999966e85

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot \frac{-0.5}{a}} \]

   if 5.39999999999999966e85 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b} - b}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.32e+154)
   (/ (* -1.0 (/ (* a c) b)) a)
   (if (<= b -7e-148)
     (* 2.0 (/ c (- (sqrt (* b b)) b)))
     (if (<= b 2.4e-81)
       (/ (* (+ b (sqrt (* -4.0 (* a c)))) -0.5) a)
       (* -1.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -7e-148) {
		tmp = 2.0 * (c / (sqrt((b * b)) - b));
	} else if (b <= 2.4e-81) {
		tmp = ((b + sqrt((-4.0 * (a * c)))) * -0.5) / a;
	} else {
		tmp = -1.0 * (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 <= (-1.32d+154)) then
        tmp = ((-1.0d0) * ((a * c) / b)) / a
    else if (b <= (-7d-148)) then
        tmp = 2.0d0 * (c / (sqrt((b * b)) - b))
    else if (b <= 2.4d-81) then
        tmp = ((b + sqrt(((-4.0d0) * (a * c)))) * (-0.5d0)) / a
    else
        tmp = (-1.0d0) * (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -7e-148) {
		tmp = 2.0 * (c / (Math.sqrt((b * b)) - b));
	} else if (b <= 2.4e-81) {
		tmp = ((b + Math.sqrt((-4.0 * (a * c)))) * -0.5) / a;
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.32e+154:
		tmp = (-1.0 * ((a * c) / b)) / a
	elif b <= -7e-148:
		tmp = 2.0 * (c / (math.sqrt((b * b)) - b))
	elif b <= 2.4e-81:
		tmp = ((b + math.sqrt((-4.0 * (a * c)))) * -0.5) / a
	else:
		tmp = -1.0 * (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.32e+154)
		tmp = Float64(Float64(-1.0 * Float64(Float64(a * c) / b)) / a);
	elseif (b <= -7e-148)
		tmp = Float64(2.0 * Float64(c / Float64(sqrt(Float64(b * b)) - b)));
	elseif (b <= 2.4e-81)
		tmp = Float64(Float64(Float64(b + sqrt(Float64(-4.0 * Float64(a * c)))) * -0.5) / a);
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.32e+154)
		tmp = (-1.0 * ((a * c) / b)) / a;
	elseif (b <= -7e-148)
		tmp = 2.0 * (c / (sqrt((b * b)) - b));
	elseif (b <= 2.4e-81)
		tmp = ((b + sqrt((-4.0 * (a * c)))) * -0.5) / a;
	else
		tmp = -1.0 * (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.32e+154], N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -7e-148], N[(2.0 * N[(c / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-81], N[(N[(N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.31999999999999998e154

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -1.31999999999999998e154 < b < -7.0000000000000001e-148

   1. Initial program 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 23.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{c}{\sqrt{{b}^{2}} - b}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\color{blue}{\sqrt{{b}^{2}} - b}} \]

      3. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\sqrt{{b}^{2}} - \color{blue}{b}} \]

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 25.9

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

   8. Applied rewrites 25.9%

      \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{b \cdot b} - b}} \]

   if -7.0000000000000001e-148 < b < 2.3999999999999999e-81

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2}}{a} \]

      4. lift-*.f64 33.1

         \[\leadsto \frac{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot -0.5}{a} \]

   8. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right)}\right)} \cdot -0.5}{a} \]

   if 2.3999999999999999e-81 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b} - b}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-81}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.32e+154)
   (/ (* -1.0 (/ (* a c) b)) a)
   (if (<= b -7e-148)
     (* 2.0 (/ c (- (sqrt (* b b)) b)))
     (if (<= b 2.4e-81)
       (* 2.0 (/ c (sqrt (* -4.0 (* a c)))))
       (* -1.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -7e-148) {
		tmp = 2.0 * (c / (sqrt((b * b)) - b));
	} else if (b <= 2.4e-81) {
		tmp = 2.0 * (c / sqrt((-4.0 * (a * c))));
	} else {
		tmp = -1.0 * (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 <= (-1.32d+154)) then
        tmp = ((-1.0d0) * ((a * c) / b)) / a
    else if (b <= (-7d-148)) then
        tmp = 2.0d0 * (c / (sqrt((b * b)) - b))
    else if (b <= 2.4d-81) then
        tmp = 2.0d0 * (c / sqrt(((-4.0d0) * (a * c))))
    else
        tmp = (-1.0d0) * (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -7e-148) {
		tmp = 2.0 * (c / (Math.sqrt((b * b)) - b));
	} else if (b <= 2.4e-81) {
		tmp = 2.0 * (c / Math.sqrt((-4.0 * (a * c))));
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.32e+154:
		tmp = (-1.0 * ((a * c) / b)) / a
	elif b <= -7e-148:
		tmp = 2.0 * (c / (math.sqrt((b * b)) - b))
	elif b <= 2.4e-81:
		tmp = 2.0 * (c / math.sqrt((-4.0 * (a * c))))
	else:
		tmp = -1.0 * (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.32e+154)
		tmp = Float64(Float64(-1.0 * Float64(Float64(a * c) / b)) / a);
	elseif (b <= -7e-148)
		tmp = Float64(2.0 * Float64(c / Float64(sqrt(Float64(b * b)) - b)));
	elseif (b <= 2.4e-81)
		tmp = Float64(2.0 * Float64(c / sqrt(Float64(-4.0 * Float64(a * c)))));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.32e+154)
		tmp = (-1.0 * ((a * c) / b)) / a;
	elseif (b <= -7e-148)
		tmp = 2.0 * (c / (sqrt((b * b)) - b));
	elseif (b <= 2.4e-81)
		tmp = 2.0 * (c / sqrt((-4.0 * (a * c))));
	else
		tmp = -1.0 * (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.32e+154], N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -7e-148], N[(2.0 * N[(c / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-81], N[(2.0 * N[(c / N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.31999999999999998e154

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -1.31999999999999998e154 < b < -7.0000000000000001e-148

   1. Initial program 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 23.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{c}{\sqrt{{b}^{2}} - b}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\color{blue}{\sqrt{{b}^{2}} - b}} \]

      3. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\sqrt{{b}^{2}} - \color{blue}{b}} \]

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 25.9

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

   8. Applied rewrites 25.9%

      \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{b \cdot b} - b}} \]

   if -7.0000000000000001e-148 < b < 2.3999999999999999e-81

   1. Initial program 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 23.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 28.8

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

   8. Applied rewrites 28.8%

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

   if 2.3999999999999999e-81 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-150}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b} - b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-193}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.32e+154)
   (/ (* -1.0 (/ (* a c) b)) a)
   (if (<= b -1.8e-150)
     (* 2.0 (/ c (- (sqrt (* b b)) b)))
     (if (<= b 6.8e-193) (* -0.5 (sqrt (* -4.0 (/ c a)))) (* -1.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -1.8e-150) {
		tmp = 2.0 * (c / (sqrt((b * b)) - b));
	} else if (b <= 6.8e-193) {
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (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 <= (-1.32d+154)) then
        tmp = ((-1.0d0) * ((a * c) / b)) / a
    else if (b <= (-1.8d-150)) then
        tmp = 2.0d0 * (c / (sqrt((b * b)) - b))
    else if (b <= 6.8d-193) then
        tmp = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = (-1.0d0) * (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e+154) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= -1.8e-150) {
		tmp = 2.0 * (c / (Math.sqrt((b * b)) - b));
	} else if (b <= 6.8e-193) {
		tmp = -0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.32e+154:
		tmp = (-1.0 * ((a * c) / b)) / a
	elif b <= -1.8e-150:
		tmp = 2.0 * (c / (math.sqrt((b * b)) - b))
	elif b <= 6.8e-193:
		tmp = -0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = -1.0 * (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.32e+154)
		tmp = Float64(Float64(-1.0 * Float64(Float64(a * c) / b)) / a);
	elseif (b <= -1.8e-150)
		tmp = Float64(2.0 * Float64(c / Float64(sqrt(Float64(b * b)) - b)));
	elseif (b <= 6.8e-193)
		tmp = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.32e+154)
		tmp = (-1.0 * ((a * c) / b)) / a;
	elseif (b <= -1.8e-150)
		tmp = 2.0 * (c / (sqrt((b * b)) - b));
	elseif (b <= 6.8e-193)
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = -1.0 * (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.32e+154], N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -1.8e-150], N[(2.0 * N[(c / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-193], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.31999999999999998e154

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -1.31999999999999998e154 < b < -1.8000000000000001e-150

   1. Initial program 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 23.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{c}{\sqrt{{b}^{2}} - b}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\color{blue}{\sqrt{{b}^{2}} - b}} \]

      3. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{c}{\sqrt{{b}^{2}} - \color{blue}{b}} \]

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 25.9

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

   8. Applied rewrites 25.9%

      \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{b \cdot b} - b}} \]

   if -1.8000000000000001e-150 < b < 6.8000000000000004e-193

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      4. lower-/.f64 17.5

         \[\leadsto -0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

   8. Applied rewrites 17.5%

      \[\leadsto \color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]

   if 6.8000000000000004e-193 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-193}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e-60)
   (/ (* -1.0 (/ (* a c) b)) a)
   (if (<= b 6.8e-193) (* -0.5 (sqrt (* -4.0 (/ c a)))) (* -1.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-60) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= 6.8e-193) {
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (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 <= (-7.5d-60)) then
        tmp = ((-1.0d0) * ((a * c) / b)) / a
    else if (b <= 6.8d-193) then
        tmp = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = (-1.0d0) * (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-60) {
		tmp = (-1.0 * ((a * c) / b)) / a;
	} else if (b <= 6.8e-193) {
		tmp = -0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e-60:
		tmp = (-1.0 * ((a * c) / b)) / a
	elif b <= 6.8e-193:
		tmp = -0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = -1.0 * (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e-60)
		tmp = Float64(Float64(-1.0 * Float64(Float64(a * c) / b)) / a);
	elseif (b <= 6.8e-193)
		tmp = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e-60)
		tmp = (-1.0 * ((a * c) / b)) / a;
	elseif (b <= 6.8e-193)
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = -1.0 * (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e-60], N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6.8e-193], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5000000000000002e-60

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -7.5000000000000002e-60 < b < 6.8000000000000004e-193

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      4. lower-/.f64 17.5

         \[\leadsto -0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

   8. Applied rewrites 17.5%

      \[\leadsto \color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]

   if 6.8000000000000004e-193 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 50.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{+169}:\\ \;\;\;\;\frac{-1 \cdot \frac{c}{b}}{a + a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-193}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.65e+169)
   (/ (* -1.0 (/ c b)) (+ a a))
   (if (<= b 6.8e-193) (* -0.5 (sqrt (* -4.0 (/ c a)))) (* -1.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e+169) {
		tmp = (-1.0 * (c / b)) / (a + a);
	} else if (b <= 6.8e-193) {
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (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 <= (-2.65d+169)) then
        tmp = ((-1.0d0) * (c / b)) / (a + a)
    else if (b <= 6.8d-193) then
        tmp = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = (-1.0d0) * (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e+169) {
		tmp = (-1.0 * (c / b)) / (a + a);
	} else if (b <= 6.8e-193) {
		tmp = -0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.65e+169:
		tmp = (-1.0 * (c / b)) / (a + a)
	elif b <= 6.8e-193:
		tmp = -0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = -1.0 * (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.65e+169)
		tmp = Float64(Float64(-1.0 * Float64(c / b)) / Float64(a + a));
	elseif (b <= 6.8e-193)
		tmp = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.65e+169)
		tmp = (-1.0 * (c / b)) / (a + a);
	elseif (b <= 6.8e-193)
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = -1.0 * (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.65e+169], N[(N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-193], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.64999999999999995e169

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 52.2

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 6.3%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 11.7

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

   10. Applied rewrites 11.7%

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

   if -2.64999999999999995e169 < b < 6.8000000000000004e-193

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      4. lower-/.f64 17.5

         \[\leadsto -0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

   8. Applied rewrites 17.5%

      \[\leadsto \color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]

   if 6.8000000000000004e-193 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 50.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{+169}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-193}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.65e+169)
   (/ c b)
   (if (<= b 6.8e-193) (* -0.5 (sqrt (* -4.0 (/ c a)))) (* -1.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e+169) {
		tmp = c / b;
	} else if (b <= 6.8e-193) {
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (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 <= (-2.65d+169)) then
        tmp = c / b
    else if (b <= 6.8d-193) then
        tmp = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = (-1.0d0) * (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e+169) {
		tmp = c / b;
	} else if (b <= 6.8e-193) {
		tmp = -0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.65e+169:
		tmp = c / b
	elif b <= 6.8e-193:
		tmp = -0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = -1.0 * (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.65e+169)
		tmp = Float64(c / b);
	elseif (b <= 6.8e-193)
		tmp = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.65e+169)
		tmp = c / b;
	elseif (b <= 6.8e-193)
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = -1.0 * (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.65e+169], N[(c / b), $MachinePrecision], If[LessEqual[b, 6.8e-193], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.64999999999999995e169

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 52.2

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 11.0

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

   10. Applied rewrites 11.0%

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

   if -2.64999999999999995e169 < b < 6.8000000000000004e-193

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      4. lower-/.f64 17.5

         \[\leadsto -0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

   8. Applied rewrites 17.5%

      \[\leadsto \color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]

   if 6.8000000000000004e-193 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 44.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+49) (/ c b) (* -1.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+49) {
		tmp = c / b;
	} else {
		tmp = -1.0 * (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 <= (-8.5d+49)) then
        tmp = c / b
    else
        tmp = (-1.0d0) * (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+49) {
		tmp = c / b;
	} else {
		tmp = -1.0 * (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+49:
		tmp = c / b
	else:
		tmp = -1.0 * (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+49)
		tmp = Float64(c / b);
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e+49)
		tmp = c / b;
	else
		tmp = -1.0 * (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+49], N[(c / b), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.4999999999999996e49

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 52.2

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 11.0

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

   10. Applied rewrites 11.0%

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

   if -8.4999999999999996e49 < b

   1. Initial program 52.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. 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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 52.2

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 11.0% accurate, 5.5× 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 52.2%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. 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. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 52.2

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

3. Applied rewrites 52.2%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 52.2

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

5. Applied rewrites 52.2%

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

7. Applied rewrites 32.5%

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

8. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{c}{b}} \]
9. Step-by-step derivation

   1. lower-/.f64 11.0

      \[\leadsto \frac{c}{\color{blue}{b}} \]

10. Applied rewrites 11.0%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
11. Add Preprocessing

Alternative 14: 2.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b a))

C

double code(double a, double b, double c) {
	return b / 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 / a
end function

Java

public static double code(double a, double b, double c) {
	return b / a;
}

Python

def code(a, b, c):
	return b / a

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / a;
end

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. 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. sub-flip N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)\right)}}}{2 \cdot a} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)\right) + b \cdot b}}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

   5. distribute-lft-neg-out N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b}}{2 \cdot a} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(a \cdot c\right)} + b \cdot b}}{2 \cdot a} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(c \cdot a\right)} + b \cdot b}}{2 \cdot a} \]

   8. associate-*r* N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot c\right) \cdot a} + b \cdot b}}{2 \cdot a} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(4\right)\right) \cdot c, a, b \cdot b\right)}}}{2 \cdot a} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot c}, a, b \cdot b\right)}}{2 \cdot a} \]

   11. metadata-eval 52.2

       \[\leadsto \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{-4} \cdot c, a, b \cdot b\right)}}{2 \cdot a} \]

3. Applied rewrites 52.2%

   \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}{2 \cdot a} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{\color{blue}{2 \cdot a}} \]

   2. count-2-rev N/A

      \[\leadsto \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{\color{blue}{a + a}} \]

   3. lower-+.f64 52.2

      \[\leadsto \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{\color{blue}{a + a}} \]

5. Applied rewrites 52.2%

   \[\leadsto \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{\color{blue}{a + a}} \]

7. Applied rewrites 1.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, a, \left(a + a\right) \cdot \frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2}\right)}{\left(a + a\right) \cdot a}} \]

8. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{b}{a}} \]
9. Step-by-step derivation

   1. lower-/.f64 2.5

      \[\leadsto \frac{b}{\color{blue}{a}} \]

10. Applied rewrites 2.5%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.4× 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 2025136 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform c (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)))