quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.4% → 86.2%

Time: 4.4s

Alternatives: 14

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.4% 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: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.55 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(-b\_2, \frac{1}{a}, \frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.55e+99)
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))
   (if (<= b_2 3.5e-92)
     (fma (- b_2) (/ 1.0 a) (/ (sqrt (- (* b_2 b_2) (* c a))) a))
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.55e+99) {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	} else if (b_2 <= 3.5e-92) {
		tmp = fma(-b_2, (1.0 / a), (sqrt(((b_2 * b_2) - (c * a))) / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.55e+99)
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 3.5e-92)
		tmp = fma(Float64(-b_2), Float64(1.0 / a), Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.55e+99], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-92], N[((-b$95$2) * N[(1.0 / a), $MachinePrecision] + N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.55e99

   1. Initial program 55.2%

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

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   3. 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

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

   if -1.55e99 < b_2 < 3.5e-92

   1. Initial program 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. division-flip N/A

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

      12. lift-neg.f64 N/A

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

      13. div-add N/A

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

      14. mult-flip N/A

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

   5. Applied rewrites 82.5%

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

   if 3.5e-92 < b_2

   1. Initial program 18.6%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 84.3

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

   4. Applied rewrites 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 84.3

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

   6. Applied rewrites 84.3%

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

Alternative 2: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.55 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.02 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.55e+99)
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))
   (if (<= b_2 1.02e-111)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.55e+99) {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	} else if (b_2 <= 1.02e-111) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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 <= -1.55e+99)
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 1.02e-111)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.55e+99], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.02e-111], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.55e99

   1. Initial program 55.2%

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

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   3. 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

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

   if -1.55e99 < b_2 < 1.02000000000000003e-111

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. division-flip N/A

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

   5. Applied rewrites 83.2%

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

   if 1.02000000000000003e-111 < b_2

   1. Initial program 20.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 82.8

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

   4. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 82.8

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

   6. Applied rewrites 82.8%

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

Alternative 3: 81.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 4.4 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{-\frac{\sqrt{\left(-a\right) \cdot c} + b\_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-61)
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))
   (if (<= b_2 4.4e-39)
     (/ 1.0 (- (/ (+ (sqrt (* (- a) c)) b_2) c)))
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-61) {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	} else if (b_2 <= 4.4e-39) {
		tmp = 1.0 / -((sqrt((-a * c)) + b_2) / c);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-61)
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 4.4e-39)
		tmp = Float64(1.0 / Float64(-Float64(Float64(sqrt(Float64(Float64(-a) * c)) + b_2) / c)));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-61], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.4e-39], N[(1.0 / (-N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] + b$95$2), $MachinePrecision] / c), $MachinePrecision])), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.0000000000000006e-61

   1. Initial program 69.7%

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

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   3. 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

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.5

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

   7. Applied rewrites 87.5%

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

   if -7.0000000000000006e-61 < b_2 < 4.40000000000000002e-39

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 73.5%

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

   4. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. *-commutative N/A

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

      8. lower-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 66.3

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

   6. Applied rewrites 66.3%

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

   if 4.40000000000000002e-39 < b_2

   1. Initial program 15.4%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 88.6

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

   6. Applied rewrites 88.6%

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

Alternative 4: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.02 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.12e-60)
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))
   (if (<= b_2 1.02e-111)
     (/ (- (sqrt (* (- a) c)) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.12e-60) {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	} else if (b_2 <= 1.02e-111) {
		tmp = (sqrt((-a * c)) - 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 <= -1.12e-60)
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 1.02e-111)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.12e-60], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.02e-111], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.12e-60

   1. Initial program 69.7%

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

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   3. 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

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -1.12e-60 < b_2 < 1.02000000000000003e-111

   1. Initial program 78.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 78.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. division-flip N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. *-commutative N/A

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

      4. lower-sqrt.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 69.7

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

   8. Applied rewrites 69.7%

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

   if 1.02000000000000003e-111 < b_2

   1. Initial program 20.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 82.8

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

   4. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 82.8

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

   6. Applied rewrites 82.8%

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

Alternative 5: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.02 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.12e-60)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.02e-111)
     (/ (- (sqrt (* (- a) c)) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.12e-60) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.02e-111) {
		tmp = (sqrt((-a * c)) - 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 <= (-1.12d-60)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.02d-111) then
        tmp = (sqrt((-a * c)) - 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 <= -1.12e-60) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.02e-111) {
		tmp = (Math.sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.12e-60:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.02e-111:
		tmp = (math.sqrt((-a * c)) - 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 <= -1.12e-60)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.02e-111)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - 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 <= -1.12e-60)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.02e-111)
		tmp = (sqrt((-a * c)) - 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, -1.12e-60], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.02e-111], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.12e-60

   1. Initial program 69.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 87.1

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

   4. Applied rewrites 87.1%

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

   if -1.12e-60 < b_2 < 1.02000000000000003e-111

   1. Initial program 78.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 78.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. division-flip N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. *-commutative N/A

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

      4. lower-sqrt.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 69.7

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

   8. Applied rewrites 69.7%

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

   if 1.02000000000000003e-111 < b_2

   1. Initial program 20.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 82.8

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

   4. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 82.8

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

   6. Applied rewrites 82.8%

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

Alternative 6: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-61}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.02 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-61)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.02e-111) (/ (sqrt (* (- a) c)) a) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-61) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.02e-111) {
		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 <= (-7d-61)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.02d-111) 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 <= -7e-61) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.02e-111) {
		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 <= -7e-61:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.02e-111:
		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 <= -7e-61)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.02e-111)
		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 <= -7e-61)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.02e-111)
		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, -7e-61], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.02e-111], 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 < -7.0000000000000006e-61

   1. Initial program 69.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 87.1

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

   4. Applied rewrites 87.1%

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

   if -7.0000000000000006e-61 < b_2 < 1.02000000000000003e-111

   1. Initial program 78.2%

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

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\sqrt{a \cdot c} \cdot \sqrt{-1}}}{a} \]
   3. 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 67.9

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

   4. Applied rewrites 67.9%

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

   if 1.02000000000000003e-111 < b_2

   1. Initial program 20.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 82.8

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

   4. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 82.8

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

   6. Applied rewrites 82.8%

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

Alternative 7: 72.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -4.8 \cdot 10^{-260}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{elif}\;b\_2 \leq 4.1 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{-c}}{\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.5e-115)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 -4.8e-260)
     (- (sqrt (/ (- c) a)))
     (if (<= b_2 4.1e-137) (/ (sqrt (- c)) (sqrt a)) (/ (* -0.5 c) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.5e-115) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= -4.8e-260) {
		tmp = -sqrt((-c / a));
	} else if (b_2 <= 4.1e-137) {
		tmp = sqrt(-c) / sqrt(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 <= (-6.5d-115)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= (-4.8d-260)) then
        tmp = -sqrt((-c / a))
    else if (b_2 <= 4.1d-137) then
        tmp = sqrt(-c) / sqrt(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 <= -6.5e-115) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= -4.8e-260) {
		tmp = -Math.sqrt((-c / a));
	} else if (b_2 <= 4.1e-137) {
		tmp = Math.sqrt(-c) / Math.sqrt(a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.5e-115:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= -4.8e-260:
		tmp = -math.sqrt((-c / a))
	elif b_2 <= 4.1e-137:
		tmp = math.sqrt(-c) / math.sqrt(a)
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.5e-115)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= -4.8e-260)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	elseif (b_2 <= 4.1e-137)
		tmp = Float64(sqrt(Float64(-c)) / sqrt(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 <= -6.5e-115)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= -4.8e-260)
		tmp = -sqrt((-c / a));
	elseif (b_2 <= 4.1e-137)
		tmp = sqrt(-c) / sqrt(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, -6.5e-115], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -4.8e-260], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), If[LessEqual[b$95$2, 4.1e-137], N[(N[Sqrt[(-c)], $MachinePrecision] / N[Sqrt[a], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -6.50000000000000033e-115

   1. Initial program 71.8%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 82.8

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

   4. Applied rewrites 82.8%

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

   if -6.50000000000000033e-115 < b_2 < -4.8000000000000001e-260

   1. Initial program 80.6%

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

   2. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 30.8

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

   4. Applied rewrites 30.8%

      \[\leadsto \color{blue}{-\sqrt{\frac{c}{a} \cdot -1}} \]
   5. 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. associate-*l/ N/A

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

      4. *-commutative 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. lower-neg.f64 30.8

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

   6. Applied rewrites 30.8%

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

   if -4.8000000000000001e-260 < b_2 < 4.0999999999999999e-137

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 73.8%

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

   4. Taylor expanded in a around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. division-flip N/A

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

      6. sqrt-prod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 33.8

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

   6. Applied rewrites 33.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 43.5

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

   8. Applied rewrites 43.5%

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

   if 4.0999999999999999e-137 < b_2

   1. Initial program 21.6%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 8: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{-c}{a}}\\ \mathbf{if}\;b\_2 \leq -6.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -1.35 \cdot 10^{-265}:\\ \;\;\;\;-t\_0\\ \mathbf{elif}\;b\_2 \leq 1.42 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- c) a))))
   (if (<= b_2 -6.5e-115)
     (/ (* -2.0 b_2) a)
     (if (<= b_2 -1.35e-265)
       (- t_0)
       (if (<= b_2 1.42e-156) 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 <= -6.5e-115) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= -1.35e-265) {
		tmp = -t_0;
	} else if (b_2 <= 1.42e-156) {
		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 <= (-6.5d-115)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= (-1.35d-265)) then
        tmp = -t_0
    else if (b_2 <= 1.42d-156) 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 <= -6.5e-115) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= -1.35e-265) {
		tmp = -t_0;
	} else if (b_2 <= 1.42e-156) {
		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 <= -6.5e-115:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= -1.35e-265:
		tmp = -t_0
	elif b_2 <= 1.42e-156:
		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 <= -6.5e-115)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= -1.35e-265)
		tmp = Float64(-t_0);
	elseif (b_2 <= 1.42e-156)
		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 <= -6.5e-115)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= -1.35e-265)
		tmp = -t_0;
	elseif (b_2 <= 1.42e-156)
		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, -6.5e-115], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -1.35e-265], (-t$95$0), If[LessEqual[b$95$2, 1.42e-156], t$95$0, N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -6.50000000000000033e-115

   1. Initial program 71.8%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 82.8

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

   4. Applied rewrites 82.8%

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

   if -6.50000000000000033e-115 < b_2 < -1.3500000000000001e-265

   1. Initial program 80.6%

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

   2. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 31.0

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

   4. Applied rewrites 31.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{c}{a} \cdot -1}} \]
   5. 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. associate-*l/ N/A

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

      4. *-commutative 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. lower-neg.f64 31.0

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

   6. Applied rewrites 31.0%

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

   if -1.3500000000000001e-265 < b_2 < 1.42000000000000004e-156

   1. Initial program 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in a around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. division-flip N/A

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

      6. sqrt-prod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 34.2

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

   6. Applied rewrites 34.2%

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

   if 1.42000000000000004e-156 < b_2

   1. Initial program 22.8%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 79.5

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

   6. Applied rewrites 79.5%

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

Alternative 9: 71.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.25 \cdot 10^{-214}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.42 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.25e-214)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.42e-156) (sqrt (/ (- c) a)) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.25e-214) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.42e-156) {
		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 <= (-1.25d-214)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.42d-156) 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 <= -1.25e-214) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.42e-156) {
		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 <= -1.25e-214:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.42e-156:
		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 <= -1.25e-214)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.42e-156)
		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 <= -1.25e-214)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.42e-156)
		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, -1.25e-214], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.42e-156], 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 < -1.2499999999999999e-214

   1. Initial program 73.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   if -1.2499999999999999e-214 < b_2 < 1.42000000000000004e-156

   1. Initial program 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in a around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. division-flip N/A

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

      6. sqrt-prod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 34.8

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

   6. Applied rewrites 34.8%

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

   if 1.42000000000000004e-156 < b_2

   1. Initial program 22.8%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 79.5

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

   6. Applied rewrites 79.5%

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

Alternative 10: 53.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.42 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-18)
   (/ (- b_2) a)
   (if (<= b_2 1.42e-156) (sqrt (/ (- c) a)) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-18) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.42e-156) {
		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 <= (-2.7d-18)) then
        tmp = -b_2 / a
    else if (b_2 <= 1.42d-156) 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 <= -2.7e-18) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.42e-156) {
		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 <= -2.7e-18:
		tmp = -b_2 / a
	elif b_2 <= 1.42e-156:
		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 <= -2.7e-18)
		tmp = Float64(Float64(-b_2) / a);
	elseif (b_2 <= 1.42e-156)
		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 <= -2.7e-18)
		tmp = -b_2 / a;
	elseif (b_2 <= 1.42e-156)
		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, -2.7e-18], N[((-b$95$2) / a), $MachinePrecision], If[LessEqual[b$95$2, 1.42e-156], 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 < -2.69999999999999989e-18

   1. Initial program 67.7%

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

   2. Taylor expanded in a around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 21.6

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

   4. Applied rewrites 21.6%

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

   5. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 39.0

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

   7. Applied rewrites 39.0%

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

   if -2.69999999999999989e-18 < b_2 < 1.42000000000000004e-156

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in a around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. division-flip N/A

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

      6. sqrt-prod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 29.5

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

   6. Applied rewrites 29.5%

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

   if 1.42000000000000004e-156 < b_2

   1. Initial program 22.8%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 79.5

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

   6. Applied rewrites 79.5%

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

Alternative 11: 53.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.42 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-18)
   (/ (- b_2) a)
   (if (<= b_2 1.42e-156) (sqrt (/ (- c) a)) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-18) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.42e-156) {
		tmp = sqrt((-c / 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 <= (-2.7d-18)) then
        tmp = -b_2 / a
    else if (b_2 <= 1.42d-156) then
        tmp = sqrt((-c / 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 <= -2.7e-18) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.42e-156) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.7e-18:
		tmp = -b_2 / a
	elif b_2 <= 1.42e-156:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.7e-18)
		tmp = Float64(Float64(-b_2) / a);
	elseif (b_2 <= 1.42e-156)
		tmp = sqrt(Float64(Float64(-c) / 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 <= -2.7e-18)
		tmp = -b_2 / a;
	elseif (b_2 <= 1.42e-156)
		tmp = sqrt((-c / 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, -2.7e-18], N[((-b$95$2) / a), $MachinePrecision], If[LessEqual[b$95$2, 1.42e-156], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.69999999999999989e-18

   1. Initial program 67.7%

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

   2. Taylor expanded in a around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 21.6

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

   4. Applied rewrites 21.6%

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

   5. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 39.0

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

   7. Applied rewrites 39.0%

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

   if -2.69999999999999989e-18 < b_2 < 1.42000000000000004e-156

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in a around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. division-flip N/A

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

      6. sqrt-prod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 29.5

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

   6. Applied rewrites 29.5%

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

   if 1.42000000000000004e-156 < b_2

   1. Initial program 22.8%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 79.5

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

   4. Applied rewrites 79.5%

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

Alternative 12: 31.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.35 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-18)
   (/ (- b_2) a)
   (if (<= b_2 3.35e-34) (sqrt (/ (- c) a)) (* 0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-18) {
		tmp = -b_2 / a;
	} else if (b_2 <= 3.35e-34) {
		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 <= (-2.7d-18)) then
        tmp = -b_2 / a
    else if (b_2 <= 3.35d-34) 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 <= -2.7e-18) {
		tmp = -b_2 / a;
	} else if (b_2 <= 3.35e-34) {
		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 <= -2.7e-18:
		tmp = -b_2 / a
	elif b_2 <= 3.35e-34:
		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 <= -2.7e-18)
		tmp = Float64(Float64(-b_2) / a);
	elseif (b_2 <= 3.35e-34)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.7e-18)
		tmp = -b_2 / a;
	elseif (b_2 <= 3.35e-34)
		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, -2.7e-18], N[((-b$95$2) / a), $MachinePrecision], If[LessEqual[b$95$2, 3.35e-34], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.69999999999999989e-18

   1. Initial program 67.7%

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

   2. Taylor expanded in a around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 21.6

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

   4. Applied rewrites 21.6%

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

   5. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 39.0

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

   7. Applied rewrites 39.0%

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

   if -2.69999999999999989e-18 < b_2 < 3.3500000000000002e-34

   1. Initial program 74.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in a around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. division-flip N/A

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

      6. sqrt-prod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 28.1

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

   6. Applied rewrites 28.1%

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

   if 3.3500000000000002e-34 < b_2

   1. Initial program 15.2%

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

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   3. 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

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 2.5

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

   4. Applied rewrites 2.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 26.8

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

   7. Applied rewrites 26.8%

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

Alternative 13: 26.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-18) (/ (- b_2) a) (sqrt (/ (- c) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-18) {
		tmp = -b_2 / a;
	} else {
		tmp = sqrt((-c / a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.7d-18)) then
        tmp = -b_2 / a
    else
        tmp = sqrt((-c / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-18) {
		tmp = -b_2 / a;
	} else {
		tmp = Math.sqrt((-c / a));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.7e-18:
		tmp = -b_2 / a
	else:
		tmp = math.sqrt((-c / a))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.7e-18)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = sqrt(Float64(Float64(-c) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.7e-18)
		tmp = -b_2 / a;
	else
		tmp = sqrt((-c / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.7e-18], N[((-b$95$2) / a), $MachinePrecision], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.69999999999999989e-18

   1. Initial program 67.7%

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

   2. Taylor expanded in a around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 21.6

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

   4. Applied rewrites 21.6%

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

   5. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 39.0

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

   7. Applied rewrites 39.0%

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

   if -2.69999999999999989e-18 < b_2

   1. Initial program 45.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      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

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

      7. lift--.f64 N/A

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

      8. division-flip N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 45.3%

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

   4. Taylor expanded in a around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. division-flip N/A

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

      6. sqrt-prod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 20.9

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

   6. Applied rewrites 20.9%

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

Alternative 14: 15.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

2. Taylor expanded in a around -inf

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

   1. lower-+.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. mul-1-neg N/A

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

   10. lift-neg.f64 N/A

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

   11. lower-/.f64 19.3

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

4. Applied rewrites 19.3%

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

5. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 15.2

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

7. Applied rewrites 15.2%

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

Developer Target 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025126 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

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

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