quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.8% → 84.8%

Time: 5.9s

Alternatives: 12

Speedup: 1.7×

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: 51.8% 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: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.1 \cdot 10^{+107}:\\ \;\;\;\;\left(-b\_2\right) \cdot \left(\frac{2}{a} + \frac{\frac{c}{b\_2}}{b\_2} \cdot -0.5\right)\\ \mathbf{elif}\;b\_2 \leq 1.45 \cdot 10^{-71}:\\ \;\;\;\;\frac{-b\_2}{a} + \frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.1e+107)
   (* (- b_2) (+ (/ 2.0 a) (* (/ (/ c b_2) b_2) -0.5)))
   (if (<= b_2 1.45e-71)
     (+ (/ (- b_2) a) (/ (sqrt (- (* b_2 b_2) (* c a))) a))
     (/ (+ (* (* a (ratio-of-squares c b_2)) -0.125) (* -0.5 c)) b_2))))

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.1000000000000002e107

   1. Initial program 52.3%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 2.6

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 2.6%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)}{b\_2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{a \cdot c}{{b\_2}^{2}} \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{a \cdot c}{{b\_2}^{2}} \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      10. lift-*.f64 2.6

          \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right) \cdot -0.125 - 0.5\right) \cdot c}{b\_2} \]

   8. Applied rewrites 2.6%

      \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right) \cdot -0.125 - 0.5\right) \cdot c}{b\_2} \]

   9. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

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

       3. lift-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\frac{2}{a} + \frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}}\right) \]

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

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

       13. associate-/r* N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 94.7

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

   11. Applied rewrites 94.7%

       \[\leadsto \color{blue}{\left(-b\_2\right) \cdot \left(\frac{2}{a} + \frac{\frac{c}{b\_2}}{b\_2} \cdot -0.5\right)} \]

   if -5.1000000000000002e107 < b_2 < 1.4499999999999999e-71

   1. Initial program 85.0%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-add N/A

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

      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. lower-/.f64 N/A

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

      15. lift-neg.f64 N/A

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

      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 85.0%

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

   if 1.4499999999999999e-71 < b_2

   1. Initial program 15.0%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 89.2

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.1 \cdot 10^{+107}:\\ \;\;\;\;\left(-b\_2\right) \cdot \left(\frac{2}{a} + \frac{\frac{c}{b\_2}}{b\_2} \cdot -0.5\right)\\ \mathbf{elif}\;b\_2 \leq 1.45 \cdot 10^{-71}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.1e+107)
   (* (- b_2) (+ (/ 2.0 a) (* (/ (/ c b_2) b_2) -0.5)))
   (if (<= b_2 1.45e-71)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (+ (* (* a (ratio-of-squares c b_2)) -0.125) (* -0.5 c)) b_2))))

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.1000000000000002e107

   1. Initial program 52.3%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 2.6

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 2.6%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)}{b\_2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{a \cdot c}{{b\_2}^{2}} \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{a \cdot c}{{b\_2}^{2}} \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} - \frac{1}{2}\right) \cdot c}{b\_2} \]

      10. lift-*.f64 2.6

          \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right) \cdot -0.125 - 0.5\right) \cdot c}{b\_2} \]

   8. Applied rewrites 2.6%

      \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right) \cdot -0.125 - 0.5\right) \cdot c}{b\_2} \]

   9. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

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

       3. lift-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\frac{2}{a} + \frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}}\right) \]

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

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

       13. associate-/r* N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 94.7

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

   11. Applied rewrites 94.7%

       \[\leadsto \color{blue}{\left(-b\_2\right) \cdot \left(\frac{2}{a} + \frac{\frac{c}{b\_2}}{b\_2} \cdot -0.5\right)} \]

   if -5.1000000000000002e107 < b_2 < 1.4499999999999999e-71

   1. Initial program 85.0%

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

   if 1.4499999999999999e-71 < b_2

   1. Initial program 15.0%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 89.2

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.1 \cdot 10^{+107}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.45 \cdot 10^{-71}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.1e+107)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.45e-71)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (+ (* (* a (ratio-of-squares c b_2)) -0.125) (* -0.5 c)) b_2))))

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.1000000000000002e107

   1. Initial program 52.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if -5.1000000000000002e107 < b_2 < 1.4499999999999999e-71

   1. Initial program 85.0%

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

   if 1.4499999999999999e-71 < b_2

   1. Initial program 15.0%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 89.2

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.3 \cdot 10^{-103}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.45 \cdot 10^{-71}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.3e-103)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.45e-71)
     (/ (+ (- b_2) (sqrt (* (- a) c))) a)
     (/ (+ (* (* a (ratio-of-squares c b_2)) -0.125) (* -0.5 c)) b_2))))

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.2999999999999999e-103

   1. Initial program 75.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if -3.2999999999999999e-103 < b_2 < 1.4499999999999999e-71

   1. Initial program 78.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   if 1.4499999999999999e-71 < b_2

   1. Initial program 15.0%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 89.2

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.75 \cdot 10^{-124}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.45 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.75e-124)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.45e-71)
     (/ (sqrt (* (- a) c)) a)
     (/ (+ (* (* a (ratio-of-squares c b_2)) -0.125) (* -0.5 c)) b_2))))

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.7499999999999999e-124

   1. Initial program 76.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -1.7499999999999999e-124 < b_2 < 1.4499999999999999e-71

   1. Initial program 77.0%

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

   3. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if 1.4499999999999999e-71 < b_2

   1. Initial program 15.0%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 89.2

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{-c}{a}}\\ \mathbf{if}\;b\_2 \leq -7.8 \cdot 10^{-135}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.5 \cdot 10^{-203}:\\ \;\;\;\;-t\_0\\ \mathbf{elif}\;b\_2 \leq 2.7 \cdot 10^{-138}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- c) a))))
   (if (<= b_2 -7.8e-135)
     (* -2.0 (/ b_2 a))
     (if (<= b_2 2.5e-203)
       (- t_0)
       (if (<= b_2 2.7e-138) t_0 (/ (* -0.5 c) b_2))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt((-c / a));
	double tmp;
	if (b_2 <= -7.8e-135) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 2.5e-203) {
		tmp = -t_0;
	} else if (b_2 <= 2.7e-138) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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 <= (-7.8d-135)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 2.5d-203) then
        tmp = -t_0
    else if (b_2 <= 2.7d-138) then
        tmp = t_0
    else
        tmp = ((-0.5d0) * c) / b_2
    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 <= -7.8e-135) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 2.5e-203) {
		tmp = -t_0;
	} else if (b_2 <= 2.7e-138) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt((-c / a))
	tmp = 0
	if b_2 <= -7.8e-135:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 2.5e-203:
		tmp = -t_0
	elif b_2 <= 2.7e-138:
		tmp = t_0
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(-c) / a))
	tmp = 0.0
	if (b_2 <= -7.8e-135)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 2.5e-203)
		tmp = Float64(-t_0);
	elseif (b_2 <= 2.7e-138)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt((-c / a));
	tmp = 0.0;
	if (b_2 <= -7.8e-135)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 2.5e-203)
		tmp = -t_0;
	elseif (b_2 <= 2.7e-138)
		tmp = t_0;
	else
		tmp = (-0.5 * c) / b_2;
	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, -7.8e-135], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.5e-203], (-t$95$0), If[LessEqual[b$95$2, 2.7e-138], t$95$0, N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -7.80000000000000043e-135

   1. Initial program 76.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -7.80000000000000043e-135 < b_2 < 2.5000000000000001e-203

   1. Initial program 72.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 17.1

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

   5. Applied rewrites 17.1%

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

   6. Taylor expanded in c around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right)} \]
   7. 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-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 36.3

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

   8. Applied rewrites 36.3%

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

   if 2.5000000000000001e-203 < b_2 < 2.70000000000000029e-138

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   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 58.2

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

   5. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lift-neg.f64 58.2

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

   7. Applied rewrites 58.2%

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

   if 2.70000000000000029e-138 < b_2

   1. Initial program 19.1%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 84.9

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]

   6. Taylor expanded in a around 0

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

      1. lift-*.f64 85.0

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

   8. Applied rewrites 85.0%

      \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.75 \cdot 10^{-124}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.75e-124)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.35e-71) (/ (sqrt (* (- a) c)) a) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.75e-124) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.35e-71) {
		tmp = sqrt((-a * c)) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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 <= (-1.75d-124)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 1.35d-71) then
        tmp = sqrt((-a * c)) / a
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.75e-124) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.35e-71) {
		tmp = Math.sqrt((-a * c)) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.75e-124:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 1.35e-71:
		tmp = math.sqrt((-a * c)) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.75e-124)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 1.35e-71)
		tmp = Float64(sqrt(Float64(Float64(-a) * c)) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.75e-124)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 1.35e-71)
		tmp = sqrt((-a * c)) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.75e-124], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.35e-71], N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.7499999999999999e-124

   1. Initial program 76.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -1.7499999999999999e-124 < b_2 < 1.3500000000000001e-71

   1. Initial program 77.0%

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

   3. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if 1.3500000000000001e-71 < b_2

   1. Initial program 15.0%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 89.2

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]

   6. Taylor expanded in a around 0

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

      1. lift-*.f64 88.9

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

   8. Applied rewrites 88.9%

      \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.3 \cdot 10^{-237}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.7 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.3e-237)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 2.7e-138) (sqrt (/ (- c) a)) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.3e-237) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 2.7e-138) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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 <= (-3.3d-237)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 2.7d-138) then
        tmp = sqrt((-c / a))
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.3e-237) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 2.7e-138) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.3e-237:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 2.7e-138:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.3e-237)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 2.7e-138)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.3e-237)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 2.7e-138)
		tmp = sqrt((-c / a));
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.3e-237], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.7e-138], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.3000000000000001e-237

   1. Initial program 77.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

   if -3.3000000000000001e-237 < b_2 < 2.70000000000000029e-138

   1. Initial program 76.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 30.5

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

   5. Applied rewrites 30.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lift-neg.f64 30.5

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

   7. Applied rewrites 30.5%

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

   if 2.70000000000000029e-138 < b_2

   1. Initial program 19.1%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 84.9

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]

   6. Taylor expanded in a around 0

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

      1. lift-*.f64 85.0

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

   8. Applied rewrites 85.0%

      \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* -2.0 (/ b_2 a)) (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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 <= (-5d-310)) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 76.2%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 33.5%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{\color{blue}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(a \cdot \frac{{c}^{2}}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(a \cdot \frac{c \cdot c}{b\_2 \cdot b\_2}\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot \frac{-1}{8} + \frac{-1}{2} \cdot c}{b\_2} \]

      11. lower-*.f64 66.1

          \[\leadsto \frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2} \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \mathsf{ratio\_of\_squares}\left(c, b\_2\right)\right) \cdot -0.125 + -0.5 \cdot c}{b\_2}} \]

   6. Taylor expanded in a around 0

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

      1. lift-*.f64 66.9

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

   8. Applied rewrites 66.9%

      \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* -2.0 (/ b_2 a)) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-5d-310)) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 76.2%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 33.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.9

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{b\_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -2.0 (/ 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[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b_2 around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 36.2

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

5. Applied rewrites 36.2%

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

Alternative 12: 15.5% accurate, 2.9× 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 54.2%

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

3. Taylor expanded in c around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 15.7

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

5. Applied rewrites 15.7%

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

6. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 14.6

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

8. Applied rewrites 14.6%

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

Reproduce

herbie shell --seed 2025058 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

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