quadm (p42, negative)

Percentage Accurate: 52.4% → 90.7%

Time: 7.8s

Alternatives: 8

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * 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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * 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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right) \cdot c}{-b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-274}:\\ \;\;\;\;\frac{-2 \cdot c}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{+125}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.45e+154)
   (/ (* (fma (/ a b) (/ c b) 1.0) c) (- b))
   (if (<= b 1.02e-274)
     (/ (* -2.0 c) (- b (sqrt (fma (* a c) -4.0 (* b b)))))
     (if (<= b 1.28e+125)
       (/ (+ (sqrt (fma -4.0 (* c a) (* b b))) b) (* -2.0 a))
       (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e+154) {
		tmp = (fma((a / b), (c / b), 1.0) * c) / -b;
	} else if (b <= 1.02e-274) {
		tmp = (-2.0 * c) / (b - sqrt(fma((a * c), -4.0, (b * b))));
	} else if (b <= 1.28e+125) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e+154)
		tmp = Float64(Float64(fma(Float64(a / b), Float64(c / b), 1.0) * c) / Float64(-b));
	elseif (b <= 1.02e-274)
		tmp = Float64(Float64(-2.0 * c) / Float64(b - sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))));
	elseif (b <= 1.28e+125)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e+154], N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 1.0), $MachinePrecision] * c), $MachinePrecision] / (-b)), $MachinePrecision], If[LessEqual[b, 1.02e-274], N[(N[(-2.0 * c), $MachinePrecision] / N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.28e+125], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.4500000000000001e154

   1. Initial program 1.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 100.0%

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

   if -2.4500000000000001e154 < b < 1.01999999999999997e-274

   1. Initial program 60.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(-2 \cdot a\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

      6. lower-/.f64 59.8

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(c \cdot a\right) \cdot 4 + b \cdot b\right)} - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4 + \left(b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(c \cdot a\right)} + \left(b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(4, c \cdot a, b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{c \cdot a}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      15. +-inverses 80.7

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{0}\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   6. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(4, a \cdot c, 0\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 91.1

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]

   9. Applied rewrites 91.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]

   if 1.01999999999999997e-274 < b < 1.27999999999999997e125

   1. Initial program 94.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}} \]

   if 1.27999999999999997e125 < b

   1. Initial program 54.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{+154}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-274}:\\ \;\;\;\;\frac{-2 \cdot c}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{+125}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.45e+154)
   (/ c (- b))
   (if (<= b 1.02e-274)
     (/ (* -2.0 c) (- b (sqrt (fma (* a c) -4.0 (* b b)))))
     (if (<= b 1.28e+125)
       (/ (+ (sqrt (fma -4.0 (* c a) (* b b))) b) (* -2.0 a))
       (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e+154) {
		tmp = c / -b;
	} else if (b <= 1.02e-274) {
		tmp = (-2.0 * c) / (b - sqrt(fma((a * c), -4.0, (b * b))));
	} else if (b <= 1.28e+125) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e+154)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.02e-274)
		tmp = Float64(Float64(-2.0 * c) / Float64(b - sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))));
	elseif (b <= 1.28e+125)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e+154], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.02e-274], N[(N[(-2.0 * c), $MachinePrecision] / N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.28e+125], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.4500000000000001e154

   1. Initial program 1.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 98.5

         \[\leadsto \frac{c}{\color{blue}{-b}} \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]

   if -2.4500000000000001e154 < b < 1.01999999999999997e-274

   1. Initial program 60.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(-2 \cdot a\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

      6. lower-/.f64 59.8

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(c \cdot a\right) \cdot 4 + b \cdot b\right)} - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4 + \left(b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(c \cdot a\right)} + \left(b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(4, c \cdot a, b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{c \cdot a}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      15. +-inverses 80.7

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{0}\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   6. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(4, a \cdot c, 0\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 91.1

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]

   9. Applied rewrites 91.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]

   if 1.01999999999999997e-274 < b < 1.27999999999999997e125

   1. Initial program 94.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}} \]

   if 1.27999999999999997e125 < b

   1. Initial program 54.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{+154}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{-2 \cdot c}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.45e+154)
   (/ c (- b))
   (if (<= b 6.8e-96)
     (/ (* -2.0 c) (- b (sqrt (fma (* a c) -4.0 (* b b)))))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e+154) {
		tmp = c / -b;
	} else if (b <= 6.8e-96) {
		tmp = (-2.0 * c) / (b - sqrt(fma((a * c), -4.0, (b * b))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e+154)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 6.8e-96)
		tmp = Float64(Float64(-2.0 * c) / Float64(b - sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e+154], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 6.8e-96], N[(N[(-2.0 * c), $MachinePrecision] / N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4500000000000001e154

   1. Initial program 1.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 98.5

         \[\leadsto \frac{c}{\color{blue}{-b}} \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]

   if -2.4500000000000001e154 < b < 6.8000000000000002e-96

   1. Initial program 65.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(-2 \cdot a\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

      6. lower-/.f64 62.7

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(c \cdot a\right) \cdot 4 + b \cdot b\right)} - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4 + \left(b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(c \cdot a\right)} + \left(b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(4, c \cdot a, b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{c \cdot a}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      15. +-inverses 79.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{0}\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   6. Applied rewrites 79.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(4, a \cdot c, 0\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 88.5

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]

   9. Applied rewrites 88.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]

   if 6.8000000000000002e-96 < b

   1. Initial program 76.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.7

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

   5. Applied rewrites 89.7%

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

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-84)
   (/ c (- b))
   (if (<= b 3.8e-51)
     (/ (+ (sqrt (* -4.0 (* a c))) b) (* -2.0 a))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-84) {
		tmp = c / -b;
	} else if (b <= 3.8e-51) {
		tmp = (sqrt((-4.0 * (a * c))) + b) / (-2.0 * a);
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e-84)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 3.8e-51)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e-84], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 3.8e-51], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.59999999999999961e-84

   1. Initial program 24.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.9

         \[\leadsto \frac{c}{\color{blue}{-b}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]

   if -4.59999999999999961e-84 < b < 3.80000000000000003e-51

   1. Initial program 88.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}} \]

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}} + b}{-2 \cdot a} \]

      2. lower-*.f64 81.7

         \[\leadsto \frac{\sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}} + b}{-2 \cdot a} \]

   7. Applied rewrites 81.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}} + b}{-2 \cdot a} \]

   if 3.80000000000000003e-51 < b

   1. Initial program 75.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.1

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

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{-2 \cdot c}{b - \sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.4e-84)
   (/ c (- b))
   (if (<= b 6.8e-96)
     (/ (* -2.0 c) (- b (sqrt (* -4.0 (* c a)))))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-84) {
		tmp = c / -b;
	} else if (b <= 6.8e-96) {
		tmp = (-2.0 * c) / (b - sqrt((-4.0 * (c * a))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.4d-84)) then
        tmp = c / -b
    else if (b <= 6.8d-96) then
        tmp = ((-2.0d0) * c) / (b - sqrt(((-4.0d0) * (c * a))))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-84) {
		tmp = c / -b;
	} else if (b <= 6.8e-96) {
		tmp = (-2.0 * c) / (b - Math.sqrt((-4.0 * (c * a))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.4e-84:
		tmp = c / -b
	elif b <= 6.8e-96:
		tmp = (-2.0 * c) / (b - math.sqrt((-4.0 * (c * a))))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.4e-84)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 6.8e-96)
		tmp = Float64(Float64(-2.0 * c) / Float64(b - sqrt(Float64(-4.0 * Float64(c * a)))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.4e-84)
		tmp = c / -b;
	elseif (b <= 6.8e-96)
		tmp = (-2.0 * c) / (b - sqrt((-4.0 * (c * a))));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.4e-84], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 6.8e-96], N[(N[(-2.0 * c), $MachinePrecision] / N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.3999999999999998e-84

   1. Initial program 24.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.9

         \[\leadsto \frac{c}{\color{blue}{-b}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]

   if -5.3999999999999998e-84 < b < 6.8000000000000002e-96

   1. Initial program 87.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(-2 \cdot a\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

      6. lower-/.f64 82.4

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(c \cdot a\right) \cdot 4 + b \cdot b\right)} - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4 + \left(b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(c \cdot a\right)} + \left(b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(4, c \cdot a, b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{c \cdot a}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

      15. +-inverses 82.4

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{0}\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   6. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(4, a \cdot c, 0\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 84.7

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]

   9. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \]

   10. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{-2 \cdot c}{b - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{-2 \cdot c}{b - \sqrt{-4 \cdot \color{blue}{\left(c \cdot a\right)}}} \]

       3. lower-*.f64 83.4

          \[\leadsto \frac{-2 \cdot c}{b - \sqrt{-4 \cdot \color{blue}{\left(c \cdot a\right)}}} \]

   12. Applied rewrites 83.4%

       \[\leadsto \frac{-2 \cdot c}{b - \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)}}} \]

   if 6.8000000000000002e-96 < b

   1. Initial program 76.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.7

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

   5. Applied rewrites 89.7%

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

Alternative 6: 67.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-307) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-307) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.5d-307)) then
        tmp = c / -b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-307) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-307:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-307)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e-307)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-307], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.5e-307

   1. Initial program 40.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 65.3

         \[\leadsto \frac{c}{\color{blue}{-b}} \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]

   if -9.5e-307 < b

   1. Initial program 80.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 7: 34.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / (-b)), $MachinePrecision]

Derivation

1. Initial program 60.3%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 33.8

      \[\leadsto \frac{c}{\color{blue}{-b}} \]

5. Applied rewrites 33.8%

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

Alternative 8: 10.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

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

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

Java

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

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 60.3%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 27.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(-2 \cdot a\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}} \]

   6. lower-/.f64 32.4

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{-2 \cdot a}}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   7. lift--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(c \cdot a\right) \cdot 4 + b \cdot b\right)} - b \cdot b}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   9. associate--l+ N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4 + \left(b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(c \cdot a\right)} + \left(b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(4, c \cdot a, b \cdot b - b \cdot b\right)}}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{c \cdot a}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \color{blue}{a \cdot c}, b \cdot b - b \cdot b\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   15. +-inverses 48.0

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{0}\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

6. Applied rewrites 48.0%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(4, a \cdot c, 0\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

7. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 38.3

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

9. Applied rewrites 38.3%

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

10. Taylor expanded in a around inf

    \[\leadsto \frac{c}{\color{blue}{b}} \]
11. Step-by-step derivation

12. Applied rewrites 13.3%

    \[\leadsto \frac{c}{\color{blue}{b}} \]
13. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024357 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))