quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.2% → 84.8%

Time: 4.5s

Alternatives: 11

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b\_2}{a} + \frac{\sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e+146)
   (fma -2.0 (/ b_2 a) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.55e-7)
     (+ (/ (- b_2) a) (/ (sqrt (fma (- a) c (* b_2 b_2))) a))
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e+146) {
		tmp = fma(-2.0, (b_2 / a), (0.5 * (c / b_2)));
	} else if (b_2 <= 1.55e-7) {
		tmp = (-b_2 / a) + (sqrt(fma(-a, c, (b_2 * b_2))) / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e+146)
		tmp = fma(-2.0, Float64(b_2 / a), Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.55e-7)
		tmp = Float64(Float64(Float64(-b_2) / a) + Float64(sqrt(fma(Float64(-a), c, Float64(b_2 * b_2))) / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e+146], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.55e-7], N[(N[((-b$95$2) / a), $MachinePrecision] + N[(N[Sqrt[N[((-a) * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.99999999999999973e146

   1. Initial program 35.2%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in b_2 around -inf

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

      1. lift-neg.f64 N/A

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

      2. pow2 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. div-add-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-/.f64 95.5

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

   11. Applied rewrites 95.5%

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

   if -3.99999999999999973e146 < b_2 < 1.55e-7

   1. Initial program 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-add N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. lower-+.f64 N/A

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

      12. associate-*r/ N/A

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

      13. mul-1-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 77.5%

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

   if 1.55e-7 < b_2

   1. Initial program 19.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b\_2}{a} + \frac{\sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e+146)
   (fma -2.0 (/ b_2 a) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.55e-7)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e+146) {
		tmp = fma(-2.0, (b_2 / a), (0.5 * (c / b_2)));
	} else if (b_2 <= 1.55e-7) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e+146)
		tmp = fma(-2.0, Float64(b_2 / a), Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.55e-7)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e+146], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.55e-7], N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.99999999999999973e146

   1. Initial program 35.2%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in b_2 around -inf

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

      1. lift-neg.f64 N/A

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

      2. pow2 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. div-add-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-/.f64 95.5

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

   11. Applied rewrites 95.5%

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

   if -3.99999999999999973e146 < b_2 < 1.55e-7

   1. Initial program 77.5%

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

   if 1.55e-7 < b_2

   1. Initial program 19.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.7e-17)
   (fma -2.0 (/ b_2 a) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.55e-7)
     (/ (+ (- b_2) (sqrt (* (- a) c))) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.7e-17) {
		tmp = fma(-2.0, (b_2 / a), (0.5 * (c / b_2)));
	} else if (b_2 <= 1.55e-7) {
		tmp = (-b_2 + sqrt((-a * c))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.7e-17)
		tmp = fma(-2.0, Float64(b_2 / a), Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.55e-7)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.7e-17], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.55e-7], N[(N[((-b$95$2) + N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.6999999999999999e-17

   1. Initial program 61.1%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in b_2 around -inf

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

      1. lift-neg.f64 N/A

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

      2. pow2 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. div-add-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-/.f64 81.4

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

   11. Applied rewrites 81.4%

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

   if -1.6999999999999999e-17 < b_2 < 1.55e-7

   1. Initial program 71.1%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 63.0

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

   5. Applied rewrites 63.0%

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

   if 1.55e-7 < b_2

   1. Initial program 19.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{-c}{a}}\\ \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-81}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -1.7 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-115}:\\ \;\;\;\;-t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- c) a))))
   (if (<= b_2 -4e-81)
     (/ (* -2.0 b_2) a)
     (if (<= b_2 -1.7e-166)
       t_0
       (if (<= b_2 1.3e-115) (- t_0) (* (/ c b_2) -0.5))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt((-c / a));
	double tmp;
	if (b_2 <= -4e-81) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= -1.7e-166) {
		tmp = t_0;
	} else if (b_2 <= 1.3e-115) {
		tmp = -t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((-c / a))
    if (b_2 <= (-4d-81)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= (-1.7d-166)) then
        tmp = t_0
    else if (b_2 <= 1.3d-115) then
        tmp = -t_0
    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 t_0 = Math.sqrt((-c / a));
	double tmp;
	if (b_2 <= -4e-81) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= -1.7e-166) {
		tmp = t_0;
	} else if (b_2 <= 1.3e-115) {
		tmp = -t_0;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt((-c / a))
	tmp = 0
	if b_2 <= -4e-81:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= -1.7e-166:
		tmp = t_0
	elif b_2 <= 1.3e-115:
		tmp = -t_0
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(-c) / a))
	tmp = 0.0
	if (b_2 <= -4e-81)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= -1.7e-166)
		tmp = t_0;
	elseif (b_2 <= 1.3e-115)
		tmp = Float64(-t_0);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt((-c / a));
	tmp = 0.0;
	if (b_2 <= -4e-81)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= -1.7e-166)
		tmp = t_0;
	elseif (b_2 <= 1.3e-115)
		tmp = -t_0;
	else
		tmp = (c / b_2) * -0.5;
	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, -4e-81], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -1.7e-166], t$95$0, If[LessEqual[b$95$2, 1.3e-115], (-t$95$0), N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.9999999999999998e-81

   1. Initial program 62.9%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -3.9999999999999998e-81 < b_2 < -1.6999999999999999e-166

   1. Initial program 85.2%

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

   3. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.8

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

   5. Applied rewrites 48.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 48.8

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

   7. Applied rewrites 48.8%

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

   if -1.6999999999999999e-166 < b_2 < 1.30000000000000002e-115

   1. Initial program 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-add N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. lower-+.f64 N/A

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

      12. associate-*r/ N/A

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

      13. mul-1-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 69.2%

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

   5. Taylor expanded in a around -inf

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

      1. div-add-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. pow2 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. sqrt-prod N/A

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

      13. lower-sqrt.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

   7. Applied rewrites 45.4%

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

   if 1.30000000000000002e-115 < b_2

   1. Initial program 27.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-81}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -1.7 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-115}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.7e-17)
   (fma -2.0 (/ b_2 a) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.55e-7) (/ (sqrt (* (- a) c)) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.7e-17) {
		tmp = fma(-2.0, (b_2 / a), (0.5 * (c / b_2)));
	} else if (b_2 <= 1.55e-7) {
		tmp = sqrt((-a * c)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.7e-17)
		tmp = fma(-2.0, Float64(b_2 / a), Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.55e-7)
		tmp = Float64(sqrt(Float64(Float64(-a) * c)) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.7e-17], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.55e-7], N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.6999999999999999e-17

   1. Initial program 61.1%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in b_2 around -inf

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

      1. lift-neg.f64 N/A

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

      2. pow2 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. div-add-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-/.f64 81.4

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

   11. Applied rewrites 81.4%

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

   if -1.6999999999999999e-17 < b_2 < 1.55e-7

   1. Initial program 71.1%

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

   3. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if 1.55e-7 < b_2

   1. Initial program 19.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.7e-17)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.55e-7) (/ (sqrt (* (- a) c)) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.7e-17) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.55e-7) {
		tmp = sqrt((-a * 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 <= (-1.7d-17)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.55d-7) then
        tmp = sqrt((-a * 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 <= -1.7e-17) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.55e-7) {
		tmp = Math.sqrt((-a * c)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.7e-17:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.55e-7:
		tmp = math.sqrt((-a * c)) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.7e-17)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.55e-7)
		tmp = Float64(sqrt(Float64(Float64(-a) * 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 <= -1.7e-17)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.55e-7)
		tmp = sqrt((-a * 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, -1.7e-17], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.55e-7], N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.6999999999999999e-17

   1. Initial program 61.1%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -1.6999999999999999e-17 < b_2 < 1.55e-7

   1. Initial program 71.1%

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

   3. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if 1.55e-7 < b_2

   1. Initial program 19.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-81}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.4 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-81)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 2.4e-156) (sqrt (/ (- c) a)) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-81) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2.4e-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 <= (-4d-81)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 2.4d-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 <= -4e-81) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2.4e-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 <= -4e-81:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 2.4e-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 <= -4e-81)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 2.4e-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 <= -4e-81)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 2.4e-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, -4e-81], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.4e-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 < -3.9999999999999998e-81

   1. Initial program 62.9%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -3.9999999999999998e-81 < b_2 < 2.4e-156

   1. Initial program 77.6%

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

   3. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 38.1

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

   5. Applied rewrites 38.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 38.1

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

   7. Applied rewrites 38.1%

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

   if 2.4e-156 < b_2

   1. Initial program 28.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-81}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.4 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.1 \cdot 10^{-269}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5.1e-269) (/ (* -2.0 b_2) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5.1e-269) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 5.1d-269) then
        tmp = ((-2.0d0) * b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5.1e-269) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 5.1e-269:
		tmp = (-2.0 * b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 5.1e-269)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 5.1e-269)
		tmp = (-2.0 * b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 5.1e-269], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 5.1000000000000003e-269

   1. Initial program 66.9%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

   if 5.1000000000000003e-269 < b_2

   1. Initial program 35.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.4

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

   5. Applied rewrites 66.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.1 \cdot 10^{-269}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.1 \cdot 10^{-269}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5.1e-269) (/ (- b_2) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5.1e-269) {
		tmp = -b_2 / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 5.1d-269) then
        tmp = -b_2 / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5.1e-269) {
		tmp = -b_2 / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 5.1e-269:
		tmp = -b_2 / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 5.1e-269)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 5.1e-269)
		tmp = -b_2 / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 5.1e-269], N[((-b$95$2) / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 5.1000000000000003e-269

   1. Initial program 66.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 24.6

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

   8. Applied rewrites 24.6%

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

   if 5.1000000000000003e-269 < b_2

   1. Initial program 35.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.4

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

   5. Applied rewrites 66.4%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.1 \cdot 10^{-269}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 23.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 7.5e+29) (/ (- b_2) a) (* 0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.5e+29) {
		tmp = -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 <= 7.5d+29) then
        tmp = -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 <= 7.5e+29) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 7.5e+29:
		tmp = -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 <= 7.5e+29)
		tmp = Float64(Float64(-b_2) / 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 <= 7.5e+29)
		tmp = -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, 7.5e+29], N[((-b$95$2) / a), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 7.49999999999999945e29

   1. Initial program 65.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 25.0

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

   5. Applied rewrites 25.0%

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

   6. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 18.9

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

   8. Applied rewrites 18.9%

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

   if 7.49999999999999945e29 < b_2

   1. Initial program 18.0%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 18.1

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in b_2 around -inf

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

      1. lift-neg.f64 N/A

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

      2. pow2 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. div-add-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 28.7

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

   11. Applied rewrites 28.7%

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

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 15.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 19.2

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

5. Applied rewrites 19.2%

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

6. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 14.4

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

8. Applied rewrites 14.4%

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

9. Final simplification 14.4%

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

Developer Target 1: 99.7% accurate, 0.2× 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 2025084 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

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

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