quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.5% → 91.1%

Time: 5.4s

Alternatives: 8

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: 52.5% 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: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(0.5, \frac{c}{b\_2} \cdot a, -2 \cdot b\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.5e+83)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 -2e-302)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (if (<= b_2 5e+140)
       (/ c (- (- b_2) (sqrt (fma (- a) c (* b_2 b_2)))))
       (/ c (fma 0.5 (* (/ c b_2) a) (* -2.0 b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.5e+83) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= -2e-302) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else if (b_2 <= 5e+140) {
		tmp = c / (-b_2 - sqrt(fma(-a, c, (b_2 * b_2))));
	} else {
		tmp = c / fma(0.5, ((c / b_2) * a), (-2.0 * b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9.5e+83)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= -2e-302)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	elseif (b_2 <= 5e+140)
		tmp = Float64(c / Float64(Float64(-b_2) - sqrt(fma(Float64(-a), c, Float64(b_2 * b_2)))));
	else
		tmp = Float64(c / fma(0.5, Float64(Float64(c / b_2) * a), Float64(-2.0 * b_2)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9.5e+83], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -2e-302], 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], If[LessEqual[b$95$2, 5e+140], N[(c / N[((-b$95$2) - N[Sqrt[N[((-a) * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.5 * N[(N[(c / b$95$2), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -9.5000000000000002e83

   1. Initial program 51.7%

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

   5. Applied rewrites 90.5%

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

   if -9.5000000000000002e83 < b_2 < -1.9999999999999999e-302

   1. Initial program 85.1%

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

   if -1.9999999999999999e-302 < b_2 < 5.00000000000000008e140

   1. Initial program 55.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 54.9%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 89.2%

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

   if 5.00000000000000008e140 < b_2

   1. Initial program 4.2%

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

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 2.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 2.6%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.3%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 93.3%

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

Alternative 2: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \frac{c}{b\_2} \cdot a, -2 \cdot b\_2\right)\\ \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{t\_0}{a}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (fma 0.5 (* (/ c b_2) a) (* -2.0 b_2))))
   (if (<= b_2 -4.2e-102)
     (/ t_0 a)
     (if (<= b_2 5e+140)
       (/ c (- (- b_2) (sqrt (fma (- a) c (* b_2 b_2)))))
       (/ c t_0)))))

C

double code(double a, double b_2, double c) {
	double t_0 = fma(0.5, ((c / b_2) * a), (-2.0 * b_2));
	double tmp;
	if (b_2 <= -4.2e-102) {
		tmp = t_0 / a;
	} else if (b_2 <= 5e+140) {
		tmp = c / (-b_2 - sqrt(fma(-a, c, (b_2 * b_2))));
	} else {
		tmp = c / t_0;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = fma(0.5, Float64(Float64(c / b_2) * a), Float64(-2.0 * b_2))
	tmp = 0.0
	if (b_2 <= -4.2e-102)
		tmp = Float64(t_0 / a);
	elseif (b_2 <= 5e+140)
		tmp = Float64(c / Float64(Float64(-b_2) - sqrt(fma(Float64(-a), c, Float64(b_2 * b_2)))));
	else
		tmp = Float64(c / t_0);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(0.5 * N[(N[(c / b$95$2), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -4.2e-102], N[(t$95$0 / a), $MachinePrecision], If[LessEqual[b$95$2, 5e+140], N[(c / N[((-b$95$2) - N[Sqrt[N[((-a) * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.2e-102

   1. Initial program 66.3%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 86.3%

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

   if -4.2e-102 < b_2 < 5.00000000000000008e140

   1. Initial program 59.2%

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

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 83.1%

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

   if 5.00000000000000008e140 < b_2

   1. Initial program 4.2%

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

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 2.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 2.6%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.3%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 93.3%

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

Alternative 3: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \frac{c}{b\_2} \cdot a, -2 \cdot b\_2\right)\\ \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{t\_0}{a}\\ \mathbf{elif}\;b\_2 \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-c}{b\_2 + \sqrt{\left(-c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (fma 0.5 (* (/ c b_2) a) (* -2.0 b_2))))
   (if (<= b_2 -3.8e-102)
     (/ t_0 a)
     (if (<= b_2 3.2e-89) (/ (- c) (+ b_2 (sqrt (* (- c) a)))) (/ c t_0)))))

C

double code(double a, double b_2, double c) {
	double t_0 = fma(0.5, ((c / b_2) * a), (-2.0 * b_2));
	double tmp;
	if (b_2 <= -3.8e-102) {
		tmp = t_0 / a;
	} else if (b_2 <= 3.2e-89) {
		tmp = -c / (b_2 + sqrt((-c * a)));
	} else {
		tmp = c / t_0;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = fma(0.5, Float64(Float64(c / b_2) * a), Float64(-2.0 * b_2))
	tmp = 0.0
	if (b_2 <= -3.8e-102)
		tmp = Float64(t_0 / a);
	elseif (b_2 <= 3.2e-89)
		tmp = Float64(Float64(-c) / Float64(b_2 + sqrt(Float64(Float64(-c) * a))));
	else
		tmp = Float64(c / t_0);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(0.5 * N[(N[(c / b$95$2), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -3.8e-102], N[(t$95$0 / a), $MachinePrecision], If[LessEqual[b$95$2, 3.2e-89], N[((-c) / N[(b$95$2 + N[Sqrt[N[((-c) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.80000000000000026e-102

   1. Initial program 66.3%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 86.3%

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

   if -3.80000000000000026e-102 < b_2 < 3.19999999999999998e-89

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 69.9%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 74.2%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 71.1%

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

   if 3.19999999999999998e-89 < b_2

   1. Initial program 24.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 23.9%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 69.1%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 82.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b\_2} \cdot a, -2 \cdot b\_2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-c}{b\_2 + \sqrt{\left(-c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(0.5, \frac{c}{b\_2} \cdot a, -2 \cdot b\_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b\_2} \cdot a, -2 \cdot b\_2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-c}{b\_2 + \sqrt{\left(-c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e-102)
   (/ (fma 0.5 (* (/ c b_2) a) (* -2.0 b_2)) a)
   (if (<= b_2 3.2e-89)
     (/ (- c) (+ b_2 (sqrt (* (- c) a))))
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-102) {
		tmp = fma(0.5, ((c / b_2) * a), (-2.0 * b_2)) / a;
	} else if (b_2 <= 3.2e-89) {
		tmp = -c / (b_2 + 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 <= -3.8e-102)
		tmp = Float64(fma(0.5, Float64(Float64(c / b_2) * a), Float64(-2.0 * b_2)) / a);
	elseif (b_2 <= 3.2e-89)
		tmp = Float64(Float64(-c) / Float64(b_2 + sqrt(Float64(Float64(-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, -3.8e-102], N[(N[(0.5 * N[(N[(c / b$95$2), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.2e-89], N[((-c) / N[(b$95$2 + N[Sqrt[N[((-c) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.80000000000000026e-102

   1. Initial program 66.3%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 86.3%

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

   if -3.80000000000000026e-102 < b_2 < 3.19999999999999998e-89

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 69.9%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 74.2%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 71.1%

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

   if 3.19999999999999998e-89 < b_2

   1. Initial program 24.6%

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

   5. Applied rewrites 81.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b\_2} \cdot a, -2 \cdot b\_2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-c}{b\_2 + \sqrt{\left(-c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-c}{b\_2 + \sqrt{\left(-c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e-102)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 3.2e-89)
     (/ (- c) (+ b_2 (sqrt (* (- c) a))))
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-102) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3.2e-89) {
		tmp = -c / (b_2 + 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 <= (-3.8d-102)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 3.2d-89) then
        tmp = -c / (b_2 + 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 <= -3.8e-102) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3.2e-89) {
		tmp = -c / (b_2 + 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 <= -3.8e-102:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 3.2e-89:
		tmp = -c / (b_2 + 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 <= -3.8e-102)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 3.2e-89)
		tmp = Float64(Float64(-c) / Float64(b_2 + 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 <= -3.8e-102)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 3.2e-89)
		tmp = -c / (b_2 + 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, -3.8e-102], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.2e-89], N[((-c) / N[(b$95$2 + N[Sqrt[N[((-c) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.80000000000000026e-102

   1. Initial program 66.3%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 85.8%

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

   if -3.80000000000000026e-102 < b_2 < 3.19999999999999998e-89

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 69.9%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 74.2%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 71.1%

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

   if 3.19999999999999998e-89 < b_2

   1. Initial program 24.6%

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

   5. Applied rewrites 81.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-c}{b\_2 + \sqrt{\left(-c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.35 \cdot 10^{-307}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.35e-307) (/ (* -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 <= 1.35e-307) {
		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 <= 1.35d-307) 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 <= 1.35e-307) {
		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 <= 1.35e-307:
		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 <= 1.35e-307)
		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 <= 1.35e-307)
		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, 1.35e-307], 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 < 1.34999999999999993e-307

   1. Initial program 67.7%

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

   5. Applied rewrites 70.3%

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

   if 1.34999999999999993e-307 < b_2

   1. Initial program 38.3%

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

   5. Applied rewrites 65.1%

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

Alternative 7: 35.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b\_2} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* (/ c b_2) -0.5))

C

double code(double a, double b_2, double c) {
	return (c / b_2) * -0.5;
}

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 = (c / b_2) * (-0.5d0)
end function

Java

public static double code(double a, double b_2, double c) {
	return (c / b_2) * -0.5;
}

Python

def code(a, b_2, c):
	return (c / b_2) * -0.5

Julia

function code(a, b_2, c)
	return Float64(Float64(c / b_2) * -0.5)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (c / b_2) * -0.5;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 32.7%

   \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
6. Add Preprocessing

Alternative 8: 35.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* c (/ -0.5 b_2)))

C

double code(double a, double b_2, double c) {
	return c * (-0.5 / b_2);
}

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 = c * ((-0.5d0) / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return c * (-0.5 / b_2);
}

Python

def code(a, b_2, c):
	return c * (-0.5 / b_2)

Julia

function code(a, b_2, c)
	return Float64(c * Float64(-0.5 / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = c * (-0.5 / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 32.7%

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

7. Applied rewrites 32.6%

   \[\leadsto c \cdot \color{blue}{\frac{-0.5}{b\_2}} \]
8. 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 2025021 
(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))