quadm (p42, negative)

Percentage Accurate: 53.6% → 86.9%

Time: 4.9s

Alternatives: 7

Speedup: N/A×

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: 53.6% 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: 86.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{1}\\ \mathbf{if}\;b \leq -16500000000000:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{elif}\;b \leq -4.6 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{c}{a}}{\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{{\left(\mathsf{fma}\left(t\_0, t\_0, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 1.0)))
   (if (<= b -16500000000000.0)
     (* (/ c b) -1.0)
     (if (<= b -4.6e-142)
       (/
        (/ c a)
        (fma
         (/ b a)
         -0.5
         (/ (pow (fma t_0 t_0 (* (* c a) -4.0)) 0.5) (* a 2.0))))
       (if (<= b 2e+129)
         (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a))
         (fma (/ b a) -1.0 (/ c b)))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fabs(b), 1.0);
	double tmp;
	if (b <= -16500000000000.0) {
		tmp = (c / b) * -1.0;
	} else if (b <= -4.6e-142) {
		tmp = (c / a) / fma((b / a), -0.5, (pow(fma(t_0, t_0, ((c * a) * -4.0)), 0.5) / (a * 2.0)));
	} else if (b <= 2e+129) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = abs(b) ^ 1.0
	tmp = 0.0
	if (b <= -16500000000000.0)
		tmp = Float64(Float64(c / b) * -1.0);
	elseif (b <= -4.6e-142)
		tmp = Float64(Float64(c / a) / fma(Float64(b / a), -0.5, Float64((fma(t_0, t_0, Float64(Float64(c * a) * -4.0)) ^ 0.5) / Float64(a * 2.0))));
	elseif (b <= 2e+129)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 1.0], $MachinePrecision]}, If[LessEqual[b, -16500000000000.0], N[(N[(c / b), $MachinePrecision] * -1.0), $MachinePrecision], If[LessEqual[b, -4.6e-142], N[(N[(c / a), $MachinePrecision] / N[(N[(b / a), $MachinePrecision] * -0.5 + N[(N[Power[N[(t$95$0 * t$95$0 + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+129], 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], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.65e13

   1. Initial program 14.0%

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

   2. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if -1.65e13 < b < -4.60000000000000005e-142

   1. Initial program 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{\color{blue}{2 \cdot a}} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

   3. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{2 \cdot a}} \]

   5. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\frac{\left(\frac{b}{a} \cdot -0.5\right) \cdot \left(\frac{b}{a} \cdot -0.5\right) - \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2} \cdot \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}}{\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}\right)}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\frac{c}{a}}}{\mathsf{fma}\left(\frac{b}{a}, \frac{-1}{2}, \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{\frac{1}{2}}}{a \cdot 2}\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 69.7

         \[\leadsto \frac{\frac{c}{\color{blue}{a}}}{\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}\right)} \]

   8. Applied rewrites 69.7%

      \[\leadsto \frac{\color{blue}{\frac{c}{a}}}{\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}\right)} \]

   if -4.60000000000000005e-142 < b < 2e129

   1. Initial program 84.4%

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

   if 2e129 < b

   1. Initial program 48.1%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{1}\\ \mathbf{if}\;b \leq -16500000000000:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{elif}\;b \leq -4.6 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{c}{a}}{\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{{\left(\mathsf{fma}\left(t\_0, t\_0, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 1.0)))
   (if (<= b -16500000000000.0)
     (* (/ c b) -1.0)
     (if (<= b -4.6e-142)
       (/
        (/ c a)
        (fma
         (/ b a)
         -0.5
         (/ (pow (fma t_0 t_0 (* (* c a) -4.0)) 0.5) (* a 2.0))))
       (if (<= b 2e+129)
         (-
          (/ (* -1.0 b) (* 2.0 a))
          (/
           (pow (fma (pow b 1.0) (pow b 1.0) (* -4.0 (* c a))) 0.5)
           (* 2.0 a)))
         (fma (/ b a) -1.0 (/ c b)))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fabs(b), 1.0);
	double tmp;
	if (b <= -16500000000000.0) {
		tmp = (c / b) * -1.0;
	} else if (b <= -4.6e-142) {
		tmp = (c / a) / fma((b / a), -0.5, (pow(fma(t_0, t_0, ((c * a) * -4.0)), 0.5) / (a * 2.0)));
	} else if (b <= 2e+129) {
		tmp = ((-1.0 * b) / (2.0 * a)) - (pow(fma(pow(b, 1.0), pow(b, 1.0), (-4.0 * (c * a))), 0.5) / (2.0 * a));
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = abs(b) ^ 1.0
	tmp = 0.0
	if (b <= -16500000000000.0)
		tmp = Float64(Float64(c / b) * -1.0);
	elseif (b <= -4.6e-142)
		tmp = Float64(Float64(c / a) / fma(Float64(b / a), -0.5, Float64((fma(t_0, t_0, Float64(Float64(c * a) * -4.0)) ^ 0.5) / Float64(a * 2.0))));
	elseif (b <= 2e+129)
		tmp = Float64(Float64(Float64(-1.0 * b) / Float64(2.0 * a)) - Float64((fma((b ^ 1.0), (b ^ 1.0), Float64(-4.0 * Float64(c * a))) ^ 0.5) / Float64(2.0 * a)));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 1.0], $MachinePrecision]}, If[LessEqual[b, -16500000000000.0], N[(N[(c / b), $MachinePrecision] * -1.0), $MachinePrecision], If[LessEqual[b, -4.6e-142], N[(N[(c / a), $MachinePrecision] / N[(N[(b / a), $MachinePrecision] * -0.5 + N[(N[Power[N[(t$95$0 * t$95$0 + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+129], N[(N[(N[(-1.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[(N[Power[b, 1.0], $MachinePrecision] * N[Power[b, 1.0], $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.65e13

   1. Initial program 14.0%

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

   2. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if -1.65e13 < b < -4.60000000000000005e-142

   1. Initial program 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{\color{blue}{2 \cdot a}} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

   3. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{2 \cdot a}} \]

   5. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\frac{\left(\frac{b}{a} \cdot -0.5\right) \cdot \left(\frac{b}{a} \cdot -0.5\right) - \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2} \cdot \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}}{\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}\right)}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\frac{c}{a}}}{\mathsf{fma}\left(\frac{b}{a}, \frac{-1}{2}, \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{\frac{1}{2}}}{a \cdot 2}\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 69.7

         \[\leadsto \frac{\frac{c}{\color{blue}{a}}}{\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}\right)} \]

   8. Applied rewrites 69.7%

      \[\leadsto \frac{\color{blue}{\frac{c}{a}}}{\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.5}}{a \cdot 2}\right)} \]

   if -4.60000000000000005e-142 < b < 2e129

   1. Initial program 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{\color{blue}{2 \cdot a}} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

   3. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{2 \cdot a}} \]

   if 2e129 < b

   1. Initial program 48.1%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-120)
   (* (/ c b) -1.0)
   (if (<= b 2e+129)
     (-
      (/ (* -1.0 b) (* 2.0 a))
      (/ (pow (fma (pow b 1.0) (pow b 1.0) (* -4.0 (* c a))) 0.5) (* 2.0 a)))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-120) {
		tmp = (c / b) * -1.0;
	} else if (b <= 2e+129) {
		tmp = ((-1.0 * b) / (2.0 * a)) - (pow(fma(pow(b, 1.0), pow(b, 1.0), (-4.0 * (c * a))), 0.5) / (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 <= -5.5e-120)
		tmp = Float64(Float64(c / b) * -1.0);
	elseif (b <= 2e+129)
		tmp = Float64(Float64(Float64(-1.0 * b) / Float64(2.0 * a)) - Float64((fma((b ^ 1.0), (b ^ 1.0), Float64(-4.0 * Float64(c * a))) ^ 0.5) / 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, -5.5e-120], N[(N[(c / b), $MachinePrecision] * -1.0), $MachinePrecision], If[LessEqual[b, 2e+129], N[(N[(N[(-1.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[(N[Power[b, 1.0], $MachinePrecision] * N[Power[b, 1.0], $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5000000000000001e-120

   1. Initial program 22.2%

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

   2. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.6

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

   4. Applied rewrites 80.6%

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

   if -5.5000000000000001e-120 < b < 2e129

   1. Initial program 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{\color{blue}{2 \cdot a}} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{2 \cdot a}} \]

   if 2e129 < b

   1. Initial program 48.1%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

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

Alternative 4: 84.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{1}\\ t_1 := {\left(\mathsf{fma}\left(t\_0, t\_0, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.25}\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{-1 \cdot b}{2 \cdot a} - \frac{t\_1 \cdot t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 1.0))
        (t_1 (pow (fma t_0 t_0 (* (* c a) -4.0)) 0.25)))
   (if (<= b -5.5e-120)
     (* (/ c b) -1.0)
     (if (<= b 1.8e+129)
       (- (/ (* -1.0 b) (* 2.0 a)) (/ (* t_1 t_1) (* 2.0 a)))
       (fma (/ b a) -1.0 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fabs(b), 1.0);
	double t_1 = pow(fma(t_0, t_0, ((c * a) * -4.0)), 0.25);
	double tmp;
	if (b <= -5.5e-120) {
		tmp = (c / b) * -1.0;
	} else if (b <= 1.8e+129) {
		tmp = ((-1.0 * b) / (2.0 * a)) - ((t_1 * t_1) / (2.0 * a));
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = abs(b) ^ 1.0
	t_1 = fma(t_0, t_0, Float64(Float64(c * a) * -4.0)) ^ 0.25
	tmp = 0.0
	if (b <= -5.5e-120)
		tmp = Float64(Float64(c / b) * -1.0);
	elseif (b <= 1.8e+129)
		tmp = Float64(Float64(Float64(-1.0 * b) / Float64(2.0 * a)) - Float64(Float64(t_1 * t_1) / Float64(2.0 * a)));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 1.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(t$95$0 * t$95$0 + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]}, If[LessEqual[b, -5.5e-120], N[(N[(c / b), $MachinePrecision] * -1.0), $MachinePrecision], If[LessEqual[b, 1.8e+129], N[(N[(N[(-1.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5000000000000001e-120

   1. Initial program 22.2%

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

   2. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.6

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

   4. Applied rewrites 80.6%

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

   if -5.5000000000000001e-120 < b < 1.8000000000000001e129

   1. Initial program 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{\color{blue}{2 \cdot a}} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{2 \cdot a}} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}}{2 \cdot a} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left(\color{blue}{{b}^{1}}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{2 \cdot a} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, \color{blue}{{b}^{1}}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{2 \cdot a} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\color{blue}{\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}}^{\frac{1}{2}}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\left({b}^{1} \cdot {b}^{1} + -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{2 \cdot a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\left({b}^{1} \cdot {b}^{1} + \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{2 \cdot a} \]

      7. sqr-pow N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{\color{blue}{{\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{2 \cdot a} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{\color{blue}{{\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{2 \cdot a} \]

   5. Applied rewrites 83.1%

      \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.25}}}{2 \cdot a} \]

   if 1.8000000000000001e129 < b

   1. Initial program 48.2%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

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

Alternative 5: 84.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.25}\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{-1 \cdot b}{2 \cdot a} - \frac{t\_0}{2} \cdot \frac{t\_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fma b b (* (* c a) -4.0)) 0.25)))
   (if (<= b -5.5e-120)
     (* (/ c b) -1.0)
     (if (<= b 1.8e+129)
       (- (/ (* -1.0 b) (* 2.0 a)) (* (/ t_0 2.0) (/ t_0 a)))
       (fma (/ b a) -1.0 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fma(b, b, ((c * a) * -4.0)), 0.25);
	double tmp;
	if (b <= -5.5e-120) {
		tmp = (c / b) * -1.0;
	} else if (b <= 1.8e+129) {
		tmp = ((-1.0 * b) / (2.0 * a)) - ((t_0 / 2.0) * (t_0 / a));
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(c * a) * -4.0)) ^ 0.25
	tmp = 0.0
	if (b <= -5.5e-120)
		tmp = Float64(Float64(c / b) * -1.0);
	elseif (b <= 1.8e+129)
		tmp = Float64(Float64(Float64(-1.0 * b) / Float64(2.0 * a)) - Float64(Float64(t_0 / 2.0) * Float64(t_0 / a)));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]}, If[LessEqual[b, -5.5e-120], N[(N[(c / b), $MachinePrecision] * -1.0), $MachinePrecision], If[LessEqual[b, 1.8e+129], N[(N[(N[(-1.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5000000000000001e-120

   1. Initial program 22.2%

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

   2. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.6

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

   4. Applied rewrites 80.6%

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

   if -5.5000000000000001e-120 < b < 1.8000000000000001e129

   1. Initial program 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{\color{blue}{2 \cdot a}} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{2 \cdot a}} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}}{2 \cdot a} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left(\color{blue}{{b}^{1}}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{2 \cdot a} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({b}^{1}, \color{blue}{{b}^{1}}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{2 \cdot a} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\color{blue}{\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}}^{\frac{1}{2}}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\left({b}^{1} \cdot {b}^{1} + -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{2 \cdot a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\left({b}^{1} \cdot {b}^{1} + \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{2 \cdot a} \]

      7. sqr-pow N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{\color{blue}{{\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{2 \cdot a} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{\color{blue}{{\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{2 \cdot a} \]

   5. Applied rewrites 83.1%

      \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.25}}}{2 \cdot a} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{\frac{1}{4}}}{\color{blue}{2 \cdot a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \color{blue}{\frac{{\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left({\left(\left|b\right|\right)}^{1}, {\left(\left|b\right|\right)}^{1}, \left(c \cdot a\right) \cdot -4\right)\right)}^{\frac{1}{4}}}{2 \cdot a}} \]

   7. Applied rewrites 83.1%

      \[\leadsto \frac{-1 \cdot b}{2 \cdot a} - \color{blue}{\frac{{\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.25}}{2} \cdot \frac{{\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{0.25}}{a}} \]

   if 1.8000000000000001e129 < b

   1. Initial program 48.2%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

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

Alternative 6: 66.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* (/ c b) -1.0) (fma (/ b a) -1.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) * -1.0;
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(c / b) * -1.0);
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(c / b), $MachinePrecision] * -1.0), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 33.2%

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

   2. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 65.7

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

   4. Applied rewrites 65.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 73.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 67.2

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

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 33.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -1 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -1.0))

C

double code(double a, double b, double c) {
	return (c / b) * -1.0;
}

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) * (-1.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -1.0;
}

Python

def code(a, b, c):
	return (c / b) * -1.0

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -1.0)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -1.0;
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * -1.0), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

2. Taylor expanded in b around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 33.3

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

4. Applied rewrites 33.3%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10
  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))