quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.3% → 87.6%

Time: 2.7s

Alternatives: 10

Speedup: 1.7×

Specification

Math

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

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: 52.3% accurate, 1.0× speedup

Math

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

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: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{b\_2 \cdot b\_2 - c \cdot a}\\ \mathbf{if}\;b\_2 \leq -6099999999999999841728801650498289360189552030478590321787836116030972686527957979843325463625728:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq \frac{-4678880108828875}{346583711765101857447301773017885462929554634421977071896309947576827663475703202879996800763017447262173901370175446478621769728}:\\ \;\;\;\;\frac{\frac{0 + c \cdot a}{a}}{t\_0 - b\_2}\\ \mathbf{elif}\;b\_2 \leq 240000000000000007829447601155211065538340938625732871707250923667456:\\ \;\;\;\;\frac{-t\_0}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
  (if (<=
       b_2
       -6099999999999999841728801650498289360189552030478590321787836116030972686527957979843325463625728)
    (* -1/2 (/ c b_2))
    (if (<=
         b_2
         -4678880108828875/346583711765101857447301773017885462929554634421977071896309947576827663475703202879996800763017447262173901370175446478621769728)
      (/ (/ (+ 0 (* c a)) a) (- t_0 b_2))
      (if (<=
           b_2
           240000000000000007829447601155211065538340938625732871707250923667456)
        (- (/ (- t_0) a) (/ b_2 a))
        (* -2 (/ b_2 a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -6.1e+96) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -1.35e-113) {
		tmp = ((0.0 + (c * a)) / a) / (t_0 - b_2);
	} else if (b_2 <= 2.4e+68) {
		tmp = (-t_0 / a) - (b_2 / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    if (b_2 <= (-6.1d+96)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= (-1.35d-113)) then
        tmp = ((0.0d0 + (c * a)) / a) / (t_0 - b_2)
    else if (b_2 <= 2.4d+68) then
        tmp = (-t_0 / a) - (b_2 / 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 t_0 = Math.sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -6.1e+96) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -1.35e-113) {
		tmp = ((0.0 + (c * a)) / a) / (t_0 - b_2);
	} else if (b_2 <= 2.4e+68) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(((b_2 * b_2) - (c * a)))
	tmp = 0
	if b_2 <= -6.1e+96:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= -1.35e-113:
		tmp = ((0.0 + (c * a)) / a) / (t_0 - b_2)
	elif b_2 <= 2.4e+68:
		tmp = (-t_0 / a) - (b_2 / a)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -6.1e+96)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -1.35e-113)
		tmp = Float64(Float64(Float64(0.0 + Float64(c * a)) / a) / Float64(t_0 - b_2));
	elseif (b_2 <= 2.4e+68)
		tmp = Float64(Float64(Float64(-t_0) / a) - Float64(b_2 / a));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt(((b_2 * b_2) - (c * a)));
	tmp = 0.0;
	if (b_2 <= -6.1e+96)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= -1.35e-113)
		tmp = ((0.0 + (c * a)) / a) / (t_0 - b_2);
	elseif (b_2 <= 2.4e+68)
		tmp = (-t_0 / a) - (b_2 / a);
	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[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -6099999999999999841728801650498289360189552030478590321787836116030972686527957979843325463625728], N[(-1/2 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -4678880108828875/346583711765101857447301773017885462929554634421977071896309947576827663475703202879996800763017447262173901370175446478621769728], N[(N[(N[(0 + N[(c * a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / N[(t$95$0 - b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 240000000000000007829447601155211065538340938625732871707250923667456], N[(N[((-t$95$0) / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(-2 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -6.0999999999999998e96

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   4. Applied rewrites 35.8%

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

   if -6.0999999999999998e96 < b_2 < -1.35e-113

   1. Initial program 52.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. 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. frac-2neg N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. sub-flip N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg 52.2%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 52.2%

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\frac{\frac{0 + c \cdot a}{a}}{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}} \]

   if -1.35e-113 < b_2 < 2.4000000000000001e68

   1. Initial program 52.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. 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. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. div-sub N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   3. Applied rewrites 51.7%

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

   if 2.4000000000000001e68 < b_2

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

Alternative 2: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{b\_2 \cdot b\_2 - c \cdot a}\\ \mathbf{if}\;b\_2 \leq -54999999999999998043547987204697905647054195046560117650923133451965273229273060805030445056:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq \frac{-7225426368408531}{1852673427797059126777135760139006525652319754650249024631321344126610074238976}:\\ \;\;\;\;c \cdot \frac{a}{\left(t\_0 - b\_2\right) \cdot a}\\ \mathbf{elif}\;b\_2 \leq 240000000000000007829447601155211065538340938625732871707250923667456:\\ \;\;\;\;\frac{-t\_0}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
  (if (<=
       b_2
       -54999999999999998043547987204697905647054195046560117650923133451965273229273060805030445056)
    (* -1/2 (/ c b_2))
    (if (<=
         b_2
         -7225426368408531/1852673427797059126777135760139006525652319754650249024631321344126610074238976)
      (* c (/ a (* (- t_0 b_2) a)))
      (if (<=
           b_2
           240000000000000007829447601155211065538340938625732871707250923667456)
        (- (/ (- t_0) a) (/ b_2 a))
        (* -2 (/ b_2 a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -5.5e+91) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -3.9e-63) {
		tmp = c * (a / ((t_0 - b_2) * a));
	} else if (b_2 <= 2.4e+68) {
		tmp = (-t_0 / a) - (b_2 / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    if (b_2 <= (-5.5d+91)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= (-3.9d-63)) then
        tmp = c * (a / ((t_0 - b_2) * a))
    else if (b_2 <= 2.4d+68) then
        tmp = (-t_0 / a) - (b_2 / 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 t_0 = Math.sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -5.5e+91) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -3.9e-63) {
		tmp = c * (a / ((t_0 - b_2) * a));
	} else if (b_2 <= 2.4e+68) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(((b_2 * b_2) - (c * a)))
	tmp = 0
	if b_2 <= -5.5e+91:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= -3.9e-63:
		tmp = c * (a / ((t_0 - b_2) * a))
	elif b_2 <= 2.4e+68:
		tmp = (-t_0 / a) - (b_2 / a)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -5.5e+91)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -3.9e-63)
		tmp = Float64(c * Float64(a / Float64(Float64(t_0 - b_2) * a)));
	elseif (b_2 <= 2.4e+68)
		tmp = Float64(Float64(Float64(-t_0) / a) - Float64(b_2 / a));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt(((b_2 * b_2) - (c * a)));
	tmp = 0.0;
	if (b_2 <= -5.5e+91)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= -3.9e-63)
		tmp = c * (a / ((t_0 - b_2) * a));
	elseif (b_2 <= 2.4e+68)
		tmp = (-t_0 / a) - (b_2 / a);
	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[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -54999999999999998043547987204697905647054195046560117650923133451965273229273060805030445056], N[(-1/2 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -7225426368408531/1852673427797059126777135760139006525652319754650249024631321344126610074238976], N[(c * N[(a / N[(N[(t$95$0 - b$95$2), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 240000000000000007829447601155211065538340938625732871707250923667456], N[(N[((-t$95$0) / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(-2 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -5.4999999999999998e91

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   4. Applied rewrites 35.8%

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

   if -5.4999999999999998e91 < b_2 < -3.9000000000000002e-63

   1. Initial program 52.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. 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. frac-2neg N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. sub-flip N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg 52.2%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 52.2%

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{0 + c \cdot a}{\left(\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2\right) \cdot a}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 + c \cdot a}{\left(\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2\right) \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{0 + c \cdot a}}{\left(\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2\right) \cdot a} \]

      3. +-lft-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 46.0%

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

   7. Applied rewrites 46.0%

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

   if -3.9000000000000002e-63 < b_2 < 2.4000000000000001e68

   1. Initial program 52.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. 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. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. div-sub N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   3. Applied rewrites 51.7%

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

   if 2.4000000000000001e68 < b_2

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

Alternative 3: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b\_2 \leq -54999999999999998043547987204697905647054195046560117650923133451965273229273060805030445056:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq \frac{-7225426368408531}{1852673427797059126777135760139006525652319754650249024631321344126610074238976}:\\ \;\;\;\;c \cdot \frac{a}{\left(\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2\right) \cdot a}\\ \mathbf{elif}\;b\_2 \leq 240000000000000007829447601155211065538340938625732871707250923667456:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (if (<=
     b_2
     -54999999999999998043547987204697905647054195046560117650923133451965273229273060805030445056)
  (* -1/2 (/ c b_2))
  (if (<=
       b_2
       -7225426368408531/1852673427797059126777135760139006525652319754650249024631321344126610074238976)
    (* c (/ a (* (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)))
    (if (<=
         b_2
         240000000000000007829447601155211065538340938625732871707250923667456)
      (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
      (* -2 (/ b_2 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.5e+91) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -3.9e-63) {
		tmp = c * (a / ((sqrt(((b_2 * b_2) - (c * a))) - b_2) * a));
	} else if (b_2 <= 2.4e+68) {
		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 <= (-5.5d+91)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= (-3.9d-63)) then
        tmp = c * (a / ((sqrt(((b_2 * b_2) - (c * a))) - b_2) * a))
    else if (b_2 <= 2.4d+68) 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 <= -5.5e+91) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -3.9e-63) {
		tmp = c * (a / ((Math.sqrt(((b_2 * b_2) - (c * a))) - b_2) * a));
	} else if (b_2 <= 2.4e+68) {
		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 <= -5.5e+91:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= -3.9e-63:
		tmp = c * (a / ((math.sqrt(((b_2 * b_2) - (c * a))) - b_2) * a))
	elif b_2 <= 2.4e+68:
		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 <= -5.5e+91)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -3.9e-63)
		tmp = Float64(c * Float64(a / Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) * a)));
	elseif (b_2 <= 2.4e+68)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.5e+91)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= -3.9e-63)
		tmp = c * (a / ((sqrt(((b_2 * b_2) - (c * a))) - b_2) * a));
	elseif (b_2 <= 2.4e+68)
		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, -54999999999999998043547987204697905647054195046560117650923133451965273229273060805030445056], N[(-1/2 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -7225426368408531/1852673427797059126777135760139006525652319754650249024631321344126610074238976], N[(c * N[(a / N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 240000000000000007829447601155211065538340938625732871707250923667456], 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[(-2 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -5.4999999999999998e91

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   4. Applied rewrites 35.8%

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

   if -5.4999999999999998e91 < b_2 < -3.9000000000000002e-63

   1. Initial program 52.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. 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. frac-2neg N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. sub-flip N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg 52.2%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 52.2%

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

   3. Applied rewrites 52.2%

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

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{0 + c \cdot a}{\left(\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2\right) \cdot a}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 + c \cdot a}{\left(\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2\right) \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{0 + c \cdot a}}{\left(\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2\right) \cdot a} \]

      3. +-lft-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 46.0%

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

   7. Applied rewrites 46.0%

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

   if -3.9000000000000002e-63 < b_2 < 2.4000000000000001e68

   1. Initial program 52.3%

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

   if 2.4000000000000001e68 < b_2

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

Alternative 4: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b\_2 \leq \frac{-7225426368408531}{1852673427797059126777135760139006525652319754650249024631321344126610074238976}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 240000000000000007829447601155211065538340938625732871707250923667456:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (if (<=
     b_2
     -7225426368408531/1852673427797059126777135760139006525652319754650249024631321344126610074238976)
  (* -1/2 (/ c b_2))
  (if (<=
       b_2
       240000000000000007829447601155211065538340938625732871707250923667456)
    (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
    (* -2 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.9e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.4e+68) {
		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 <= (-3.9d-63)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 2.4d+68) 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 <= -3.9e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.4e+68) {
		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 <= -3.9e-63:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 2.4e+68:
		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 <= -3.9e-63)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 2.4e+68)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.9e-63)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 2.4e+68)
		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, -7225426368408531/1852673427797059126777135760139006525652319754650249024631321344126610074238976], N[(-1/2 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 240000000000000007829447601155211065538340938625732871707250923667456], 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[(-2 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.9000000000000002e-63

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   4. Applied rewrites 35.8%

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

   if -3.9000000000000002e-63 < b_2 < 2.4000000000000001e68

   1. Initial program 52.3%

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

   if 2.4000000000000001e68 < b_2

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

Alternative 5: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b\_2 \leq \frac{-7040159025628825}{1852673427797059126777135760139006525652319754650249024631321344126610074238976}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq \frac{4199521391583383}{139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444736}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (if (<=
     b_2
     -7040159025628825/1852673427797059126777135760139006525652319754650249024631321344126610074238976)
  (* -1/2 (/ c b_2))
  (if (<=
       b_2
       4199521391583383/139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444736)
    (/ (- (- b_2) (sqrt (- (* a c)))) a)
    (* -2 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3e-86) {
		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 <= (-3.8d-63)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 3d-86) 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 <= -3.8e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3e-86) {
		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 <= -3.8e-63:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 3e-86:
		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 <= -3.8e-63)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 3e-86)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(-Float64(a * c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.8e-63)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 3e-86)
		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, -7040159025628825/1852673427797059126777135760139006525652319754650249024631321344126610074238976], N[(-1/2 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4199521391583383/139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444736], N[(N[((-b$95$2) - N[Sqrt[(-N[(a * c), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.8000000000000002e-63

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   4. Applied rewrites 35.8%

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

   if -3.8000000000000002e-63 < b_2 < 3.0000000000000001e-86

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 33.7%

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

   4. Applied rewrites 33.7%

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

   if 3.0000000000000001e-86 < b_2

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b\_2 \leq \frac{-7040159025628825}{1852673427797059126777135760139006525652319754650249024631321344126610074238976}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq \frac{4199521391583383}{139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444736}:\\ \;\;\;\;\frac{c}{\sqrt{-a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (if (<=
     b_2
     -7040159025628825/1852673427797059126777135760139006525652319754650249024631321344126610074238976)
  (* -1/2 (/ c b_2))
  (if (<=
       b_2
       4199521391583383/139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444736)
    (/ c (sqrt (- (* a c))))
    (* -2 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3e-86) {
		tmp = c / sqrt(-(a * c));
	} 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 <= (-3.8d-63)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 3d-86) then
        tmp = c / sqrt(-(a * c))
    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 <= -3.8e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3e-86) {
		tmp = c / Math.sqrt(-(a * c));
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.8e-63:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 3e-86:
		tmp = c / math.sqrt(-(a * c))
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.8e-63)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 3e-86)
		tmp = Float64(c / sqrt(Float64(-Float64(a * c))));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.8e-63)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 3e-86)
		tmp = c / sqrt(-(a * c));
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7040159025628825/1852673427797059126777135760139006525652319754650249024631321344126610074238976], N[(-1/2 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4199521391583383/139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444736], N[(c / N[Sqrt[(-N[(a * c), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(-2 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.8000000000000002e-63

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   4. Applied rewrites 35.8%

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

   if -3.8000000000000002e-63 < b_2 < 3.0000000000000001e-86

   1. Initial program 52.3%

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

   3. Applied rewrites 29.1%

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

   4. Taylor expanded in b_2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 29.5%

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

   6. Applied rewrites 29.5%

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

   if 3.0000000000000001e-86 < b_2

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

Alternative 7: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b\_2 \leq \frac{-8997827589086393}{112472844863579909570263462692149546471742427957547915827518889315295939516787196757976017152597271428748022765838022378080206651387357492225212879521629096378368}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq \frac{183729060881773}{8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296}:\\ \;\;\;\;\sqrt{-1 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (if (<=
     b_2
     -8997827589086393/112472844863579909570263462692149546471742427957547915827518889315295939516787196757976017152597271428748022765838022378080206651387357492225212879521629096378368)
  (* -1/2 (/ c b_2))
  (if (<=
       b_2
       183729060881773/8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296)
    (sqrt (* -1 (/ c a)))
    (* -2 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e-146) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.1e-86) {
		tmp = sqrt((-1.0 * (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 <= (-8d-146)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 2.1d-86) then
        tmp = sqrt(((-1.0d0) * (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 <= -8e-146) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.1e-86) {
		tmp = Math.sqrt((-1.0 * (c / a)));
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8e-146:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 2.1e-86:
		tmp = math.sqrt((-1.0 * (c / a)))
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8e-146)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 2.1e-86)
		tmp = sqrt(Float64(-1.0 * Float64(c / a)));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8e-146)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 2.1e-86)
		tmp = sqrt((-1.0 * (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, -8997827589086393/112472844863579909570263462692149546471742427957547915827518889315295939516787196757976017152597271428748022765838022378080206651387357492225212879521629096378368], N[(-1/2 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 183729060881773/8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296], N[Sqrt[N[(-1 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(-2 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.0000000000000002e-146

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   4. Applied rewrites 35.8%

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

   if -8.0000000000000002e-146 < b_2 < 2.1e-86

   1. Initial program 52.3%

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

   2. Taylor expanded in a around -inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 16.5%

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

   4. Applied rewrites 16.5%

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

   if 2.1e-86 < b_2

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

Alternative 8: 67.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b\_2 \leq \frac{-5464863684898495}{12420144738405671481191835907700020442055088136933572889112416304208407621491015090647027270629171823603901845577048585649372640352918515131554298200329449113635639808166799244402122285052787558602103993549731750007142774830528462848}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (if (<=
     b_2
     -5464863684898495/12420144738405671481191835907700020442055088136933572889112416304208407621491015090647027270629171823603901845577048585649372640352918515131554298200329449113635639808166799244402122285052787558602103993549731750007142774830528462848)
  (* -1/2 (/ c b_2))
  (* -2 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.4e-217) {
		tmp = -0.5 * (c / b_2);
	} 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 <= (-4.4d-217)) then
        tmp = (-0.5d0) * (c / b_2)
    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 <= -4.4e-217) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.4e-217:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.4e-217)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.4e-217)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5464863684898495/12420144738405671481191835907700020442055088136933572889112416304208407621491015090647027270629171823603901845577048585649372640352918515131554298200329449113635639808166799244402122285052787558602103993549731750007142774830528462848], N[(-1/2 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.3999999999999996e-217

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   4. Applied rewrites 35.8%

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

   if -4.3999999999999996e-217 < b_2

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

Alternative 9: 43.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b\_2 \leq \frac{1454766195646295}{6325070415853456823515479584966165845298645305129441198653167438357198111499854590373761990669910140474596183259900372230931523043306046152094168748148078435047419508642698792639590866940413010663742739952273283392562733857021646831815729864036236135650314266011211548510419206725953204130822734645187695728365866909171712}:\\ \;\;\;\;-2 \cdot \frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (if (<=
     b_2
     1454766195646295/6325070415853456823515479584966165845298645305129441198653167438357198111499854590373761990669910140474596183259900372230931523043306046152094168748148078435047419508642698792639590866940413010663742739952273283392562733857021646831815729864036236135650314266011211548510419206725953204130822734645187695728365866909171712)
  (* -2 (/ 0 a))
  (* -2 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.3e-307) {
		tmp = -2.0 * (0.0 / 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 <= 2.3d-307) then
        tmp = (-2.0d0) * (0.0d0 / 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 <= 2.3e-307) {
		tmp = -2.0 * (0.0 / a);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.3e-307:
		tmp = -2.0 * (0.0 / a)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.3e-307)
		tmp = Float64(-2.0 * Float64(0.0 / a));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.3e-307)
		tmp = -2.0 * (0.0 / 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, 1454766195646295/6325070415853456823515479584966165845298645305129441198653167438357198111499854590373761990669910140474596183259900372230931523043306046152094168748148078435047419508642698792639590866940413010663742739952273283392562733857021646831815729864036236135650314266011211548510419206725953204130822734645187695728365866909171712], N[(-2 * N[(0 / a), $MachinePrecision]), $MachinePrecision], N[(-2 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.2999999999999999e-307

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto -2 \cdot \frac{0}{a} \]
   6. Step-by-step derivation

   7. Applied rewrites 11.8%

      \[\leadsto -2 \cdot \frac{0}{a} \]

   if 2.2999999999999999e-307 < b_2

   1. Initial program 52.3%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6%

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

   4. Applied rewrites 34.6%

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

Alternative 10: 34.6% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (a b_2 c)
  :precision binary64
  (* -2 (/ b_2 a)))

C

double code(double a, double b_2, double c) {
	return -2.0 * (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 = (-2.0d0) * (b_2 / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Python

def code(a, b_2, c):
	return -2.0 * (b_2 / a)

Julia

function code(a, b_2, c)
	return Float64(-2.0 * Float64(b_2 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -2.0 * (b_2 / a);
end

Wolfram

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

Derivation

1. Initial program 52.3%

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

2. Taylor expanded in b_2 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 34.6%

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

4. Applied rewrites 34.6%

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025274 -o generate:evaluate
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))