quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.1% → 85.8%

Time: 4.9s

Alternatives: 11

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{+139}:\\ \;\;\;\;\frac{-b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 7e+139)
     (- (/ (- b_2) a) (/ (sqrt (- (* b_2 b_2) (* c a))) a))
     (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e+139) {
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a);
	} else {
		tmp = (-2.0 * b_2) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.26d-123)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 7d+139) then
        tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a)
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e+139) {
		tmp = (-b_2 / a) - (Math.sqrt(((b_2 * b_2) - (c * a))) / a);
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.26e-123:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 7e+139:
		tmp = (-b_2 / a) - (math.sqrt(((b_2 * b_2) - (c * a))) / a)
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 7e+139)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) / a));
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.26e-123)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 7e+139)
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a);
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7e+139], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 84.9

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 84.9%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 6.99999999999999957e139

   1. Initial program 83.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      7. div-sub N/A

         \[\leadsto \color{blue}{\frac{-b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\_2\right)}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      12. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{-1 \cdot b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\_2\right)}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      14. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{-b\_2}{a} - \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   4. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\frac{-b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}} \]

   if 6.99999999999999957e139 < b_2

   1. Initial program 44.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 98.1

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   5. Applied rewrites 98.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 7e+139)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e+139) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-2.0 * b_2) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.26d-123)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 7d+139) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e+139) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.26e-123:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 7e+139:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 7e+139)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.26e-123)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 7e+139)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7e+139], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 84.9

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 84.9%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 6.99999999999999957e139

   1. Initial program 83.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 6.99999999999999957e139 < b_2

   1. Initial program 44.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 98.1

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   5. Applied rewrites 98.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.2e-116)
     (/ (- (- b_2) (sqrt (* (- a) c))) a)
     (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.2e-116) {
		tmp = (-b_2 - sqrt((-a * c))) / a;
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.2e-116)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.2e-116], N[(N[((-b$95$2) - N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 84.9

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 84.9%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 1.19999999999999996e-116

   1. Initial program 73.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(-1 \cdot a\right) \cdot \color{blue}{c}}}{a} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot c}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{c}}}{a} \]

      4. lower-neg.f64 72.7

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a} \]

   5. Applied rewrites 72.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   if 1.19999999999999996e-116 < b_2

   1. Initial program 71.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{1}{2} + \color{blue}{-2} \cdot \frac{b\_2}{a} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \color{blue}{\frac{1}{2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, -2 \cdot \frac{b\_2}{a}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      7. lower-/.f64 86.5

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right) \]

   5. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.46 \cdot 10^{-116}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.46e-116)
     (/ (- (- b_2) (sqrt (* (- a) c))) a)
     (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.46e-116) {
		tmp = (-b_2 - sqrt((-a * c))) / a;
	} else {
		tmp = (-2.0 * b_2) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.26d-123)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.46d-116) then
        tmp = (-b_2 - sqrt((-a * c))) / a
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.46e-116) {
		tmp = (-b_2 - Math.sqrt((-a * c))) / a;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.26e-123:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.46e-116:
		tmp = (-b_2 - math.sqrt((-a * c))) / a
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.46e-116)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.26e-123)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.46e-116)
		tmp = (-b_2 - sqrt((-a * c))) / a;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.46e-116], N[(N[((-b$95$2) - N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 84.9

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 84.9%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 2.46e-116

   1. Initial program 73.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(-1 \cdot a\right) \cdot \color{blue}{c}}}{a} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot c}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{c}}}{a} \]

      4. lower-neg.f64 72.7

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a} \]

   5. Applied rewrites 72.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   if 2.46e-116 < b_2

   1. Initial program 71.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 86.2

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   5. Applied rewrites 86.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 8.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 8.2e-117) (/ (- (sqrt (* (- a) c))) a) (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 8.2e-117) {
		tmp = -sqrt((-a * c)) / a;
	} else {
		tmp = (-2.0 * b_2) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.26d-123)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 8.2d-117) then
        tmp = -sqrt((-a * c)) / a
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 8.2e-117) {
		tmp = -Math.sqrt((-a * c)) / a;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.26e-123:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 8.2e-117:
		tmp = -math.sqrt((-a * c)) / a
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 8.2e-117)
		tmp = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.26e-123)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 8.2e-117)
		tmp = -sqrt((-a * c)) / a;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 8.2e-117], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 84.9

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 84.9%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 8.20000000000000063e-117

   1. Initial program 73.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{a \cdot c} \cdot \sqrt{-1}\right)}{a} \]

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if 8.20000000000000063e-117 < b_2

   1. Initial program 71.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 86.2

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   5. Applied rewrites 86.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{c}{-a}}\\ \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -5.6 \cdot 10^{-268}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq 1.8 \cdot 10^{-197}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-117}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (/ c (- a)))))
   (if (<= b_2 -1.2e-128)
     (/ (* -0.5 c) b_2)
     (if (<= b_2 -5.6e-268)
       t_0
       (if (<= b_2 1.8e-197)
         (- (sqrt (* (/ c a) -1.0)))
         (if (<= b_2 1.3e-117) t_0 (/ (* -2.0 b_2) a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt((c / -a));
	double tmp;
	if (b_2 <= -1.2e-128) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -5.6e-268) {
		tmp = t_0;
	} else if (b_2 <= 1.8e-197) {
		tmp = -sqrt(((c / a) * -1.0));
	} else if (b_2 <= 1.3e-117) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * b_2) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((c / -a))
    if (b_2 <= (-1.2d-128)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= (-5.6d-268)) then
        tmp = t_0
    else if (b_2 <= 1.8d-197) then
        tmp = -sqrt(((c / a) * (-1.0d0)))
    else if (b_2 <= 1.3d-117) then
        tmp = t_0
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt((c / -a));
	double tmp;
	if (b_2 <= -1.2e-128) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -5.6e-268) {
		tmp = t_0;
	} else if (b_2 <= 1.8e-197) {
		tmp = -Math.sqrt(((c / a) * -1.0));
	} else if (b_2 <= 1.3e-117) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt((c / -a))
	tmp = 0
	if b_2 <= -1.2e-128:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= -5.6e-268:
		tmp = t_0
	elif b_2 <= 1.8e-197:
		tmp = -math.sqrt(((c / a) * -1.0))
	elif b_2 <= 1.3e-117:
		tmp = t_0
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(c / Float64(-a)))
	tmp = 0.0
	if (b_2 <= -1.2e-128)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -5.6e-268)
		tmp = t_0;
	elseif (b_2 <= 1.8e-197)
		tmp = Float64(-sqrt(Float64(Float64(c / a) * -1.0)));
	elseif (b_2 <= 1.3e-117)
		tmp = t_0;
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt((c / -a));
	tmp = 0.0;
	if (b_2 <= -1.2e-128)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= -5.6e-268)
		tmp = t_0;
	elseif (b_2 <= 1.8e-197)
		tmp = -sqrt(((c / a) * -1.0));
	elseif (b_2 <= 1.3e-117)
		tmp = t_0;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.2e-128], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -5.6e-268], t$95$0, If[LessEqual[b$95$2, 1.8e-197], (-N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[b$95$2, 1.3e-117], t$95$0, N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.1999999999999999e-128

   1. Initial program 16.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.1

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 84.2

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 84.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -1.1999999999999999e-128 < b_2 < -5.6000000000000003e-268 or 1.7999999999999999e-197 < b_2 < 1.29999999999999992e-117

   1. Initial program 81.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 50.3

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

   8. Applied rewrites 50.3%

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

   if -5.6000000000000003e-268 < b_2 < 1.7999999999999999e-197

   1. Initial program 50.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right) \]

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if 1.29999999999999992e-117 < b_2

   1. Initial program 71.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 86.2

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   5. Applied rewrites 86.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -5.6 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{elif}\;b\_2 \leq 1.8 \cdot 10^{-197}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e-128)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.3e-117) (sqrt (/ c (- a))) (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-128) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.3e-117) {
		tmp = sqrt((c / -a));
	} else {
		tmp = (-2.0 * b_2) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.2d-128)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.3d-117) then
        tmp = sqrt((c / -a))
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-128) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.3e-117) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.2e-128:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.3e-117:
		tmp = math.sqrt((c / -a))
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.2e-128)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.3e-117)
		tmp = sqrt(Float64(c / Float64(-a)));
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.2e-128)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.3e-117)
		tmp = sqrt((c / -a));
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.2e-128], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.3e-117], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.1999999999999999e-128

   1. Initial program 16.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.1

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 84.2

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 84.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -1.1999999999999999e-128 < b_2 < 1.29999999999999992e-117

   1. Initial program 73.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 38.8

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

   8. Applied rewrites 38.8%

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

   if 1.29999999999999992e-117 < b_2

   1. Initial program 71.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 86.2

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   5. Applied rewrites 86.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e-128)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3.7e+45) (sqrt (/ c (- a))) (/ (- b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-128) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.7e+45) {
		tmp = sqrt((c / -a));
	} else {
		tmp = -b_2 / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.2d-128)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 3.7d+45) then
        tmp = sqrt((c / -a))
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-128) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.7e+45) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.2e-128:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 3.7e+45:
		tmp = math.sqrt((c / -a))
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.2e-128)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3.7e+45)
		tmp = sqrt(Float64(c / Float64(-a)));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.2e-128)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 3.7e+45)
		tmp = sqrt((c / -a));
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.2e-128], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.7e+45], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.1999999999999999e-128

   1. Initial program 16.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.1

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 84.2

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 84.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -1.1999999999999999e-128 < b_2 < 3.69999999999999977e45

   1. Initial program 79.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 33.2

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

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 33.2

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

   8. Applied rewrites 33.2%

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

   if 3.69999999999999977e45 < b_2

   1. Initial program 64.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} + \sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}}} \cdot \sqrt{-1} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} + \sqrt{\color{blue}{\frac{c}{a}}} \cdot \sqrt{-1} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\color{blue}{\frac{c}{a}}} \cdot \sqrt{-1} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}}} \cdot \sqrt{-1} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

      9. lower-/.f64 20.9

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

   5. Applied rewrites 20.9%

      \[\leadsto \color{blue}{\frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1}} \]

   6. Taylor expanded in b_2 around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b\_2}{a}\right) \]

      2. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{-b\_2}{a} \]

      4. lift-/.f64 40.4

         \[\leadsto \frac{-b\_2}{a} \]

   8. Applied rewrites 40.4%

      \[\leadsto \frac{-b\_2}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-128}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e-128)
   (* -0.5 (/ c b_2))
   (if (<= b_2 3.7e+45) (sqrt (/ c (- a))) (/ (- b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-128) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3.7e+45) {
		tmp = sqrt((c / -a));
	} else {
		tmp = -b_2 / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.2d-128)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 3.7d+45) then
        tmp = sqrt((c / -a))
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-128) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3.7e+45) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.2e-128:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 3.7e+45:
		tmp = math.sqrt((c / -a))
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.2e-128)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 3.7e+45)
		tmp = sqrt(Float64(c / Float64(-a)));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.2e-128)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 3.7e+45)
		tmp = sqrt((c / -a));
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.2e-128], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.7e+45], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.1999999999999999e-128

   1. Initial program 16.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.1

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -1.1999999999999999e-128 < b_2 < 3.69999999999999977e45

   1. Initial program 79.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 33.2

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

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 33.2

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

   8. Applied rewrites 33.2%

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

   if 3.69999999999999977e45 < b_2

   1. Initial program 64.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} + \sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}}} \cdot \sqrt{-1} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} + \sqrt{\color{blue}{\frac{c}{a}}} \cdot \sqrt{-1} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\color{blue}{\frac{c}{a}}} \cdot \sqrt{-1} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}}} \cdot \sqrt{-1} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

      9. lower-/.f64 20.9

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

   5. Applied rewrites 20.9%

      \[\leadsto \color{blue}{\frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1}} \]

   6. Taylor expanded in b_2 around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b\_2}{a}\right) \]

      2. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{-b\_2}{a} \]

      4. lift-/.f64 40.4

         \[\leadsto \frac{-b\_2}{a} \]

   8. Applied rewrites 40.4%

      \[\leadsto \frac{-b\_2}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-128}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 23.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= a -2.3e-210) (sqrt (/ c (- a))) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (a <= -2.3e-210) {
		tmp = sqrt((c / -a));
	} else {
		tmp = -b_2 / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (a <= (-2.3d-210)) then
        tmp = sqrt((c / -a))
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (a <= -2.3e-210) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if a <= -2.3e-210:
		tmp = math.sqrt((c / -a))
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (a <= -2.3e-210)
		tmp = sqrt(Float64(c / Float64(-a)));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (a <= -2.3e-210)
		tmp = sqrt((c / -a));
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[a, -2.3e-210], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3e-210

   1. Initial program 52.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 34.4

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

   5. Applied rewrites 34.4%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 34.4

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

   8. Applied rewrites 34.4%

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

   if -2.3e-210 < a

   1. Initial program 47.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} + \sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}}} \cdot \sqrt{-1} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} + \sqrt{\color{blue}{\frac{c}{a}}} \cdot \sqrt{-1} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\color{blue}{\frac{c}{a}}} \cdot \sqrt{-1} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}}} \cdot \sqrt{-1} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

      9. lower-/.f64 6.5

         \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

   5. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1}} \]

   6. Taylor expanded in b_2 around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b\_2}{a}\right) \]

      2. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{-b\_2}{a} \]

      4. lift-/.f64 19.5

         \[\leadsto \frac{-b\_2}{a} \]

   8. Applied rewrites 19.5%

      \[\leadsto \frac{-b\_2}{\color{blue}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 15.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{-b\_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (- b_2) a))

C

double code(double a, double b_2, double c) {
	return -b_2 / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = -b_2 / a
end function

Java

public static double code(double a, double b_2, double c) {
	return -b_2 / a;
}

Python

def code(a, b_2, c):
	return -b_2 / a

Julia

function code(a, b_2, c)
	return Float64(Float64(-b_2) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -b_2 / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[((-b$95$2) / a), $MachinePrecision]

Derivation

1. Initial program 49.4%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in a around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} + \sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
4. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}}} \cdot \sqrt{-1} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} + \sqrt{\color{blue}{\frac{c}{a}}} \cdot \sqrt{-1} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{-b\_2}{a} + \sqrt{\color{blue}{\frac{c}{a}}} \cdot \sqrt{-1} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{-b\_2}{a} + \color{blue}{\sqrt{\frac{c}{a}}} \cdot \sqrt{-1} \]

   6. sqrt-unprod N/A

      \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

   9. lower-/.f64 18.8

      \[\leadsto \frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1} \]

5. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\frac{-b\_2}{a} + \sqrt{\frac{c}{a} \cdot -1}} \]

6. Taylor expanded in b_2 around inf

   \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{b\_2}{a}\right) \]

   2. distribute-frac-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{-b\_2}{a} \]

   4. lift-/.f64 15.3

      \[\leadsto \frac{-b\_2}{a} \]

8. Applied rewrites 15.3%

   \[\leadsto \frac{-b\_2}{\color{blue}{a}} \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

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

Reproduce

herbie shell --seed 2025064 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, 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_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))