The quadratic formula (r2)

Percentage Accurate: 51.3% → 85.5%

Time: 4.3s

Alternatives: 5

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: 51.3% 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: 85.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-96)
   (/ (- c) b)
   (if (<= b 3.1e+88)
     (/ (+ 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 tmp;
	if (b <= -3.2e-96) {
		tmp = -c / b;
	} else if (b <= 3.1e+88) {
		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)
	tmp = 0.0
	if (b <= -3.2e-96)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 3.1e+88)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * Float64(-a)));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-96], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 3.1e+88], 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 3 regimes

2. if b < -3.20000000000000012e-96

   1. Initial program 12.3%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -3.20000000000000012e-96 < b < 3.1000000000000001e88

   1. Initial program 77.1%

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

   if 3.1000000000000001e88 < b

   1. Initial program 62.6%

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

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

      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 92.1

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

   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 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{a}}{c}, 0.5, -\sqrt{\frac{{a}^{-1}}{c} \cdot -1}\right) \cdot \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-96)
   (/ (- c) b)
   (if (<= b 1.25e-140)
     (* (fma (/ (/ b a) c) 0.5 (- (sqrt (* (/ (pow a -1.0) c) -1.0)))) (- c))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-96) {
		tmp = -c / b;
	} else if (b <= 1.25e-140) {
		tmp = fma(((b / a) / c), 0.5, -sqrt(((pow(a, -1.0) / c) * -1.0))) * -c;
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-96)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.25e-140)
		tmp = Float64(fma(Float64(Float64(b / a) / c), 0.5, Float64(-sqrt(Float64(Float64((a ^ -1.0) / c) * -1.0)))) * Float64(-c));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-96], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.25e-140], N[(N[(N[(N[(b / a), $MachinePrecision] / c), $MachinePrecision] * 0.5 + (-N[Sqrt[N[(N[(N[Power[a, -1.0], $MachinePrecision] / c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * (-c)), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.20000000000000012e-96

   1. Initial program 12.3%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -3.20000000000000012e-96 < b < 1.25000000000000004e-140

   1. Initial program 75.7%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 73.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 75.4

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

   7. Applied rewrites 75.4%

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

   if 1.25000000000000004e-140 < b

   1. Initial program 69.7%

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

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

      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 74.9

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

   5. Applied rewrites 74.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{a}}{c}, 0.5, -\sqrt{\frac{{a}^{-1}}{c} \cdot -1}\right) \cdot \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 14.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{a}}{c}, 0.5, -\sqrt{\frac{{a}^{-1}}{c} \cdot -1}\right) \cdot \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{a}, -\sqrt{\frac{-c}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-96)
   (/ (- c) b)
   (if (<= b 14.5)
     (* (fma (/ (/ b a) c) 0.5 (- (sqrt (* (/ (pow a -1.0) c) -1.0)))) (- c))
     (fma -0.5 (/ b a) (- (sqrt (/ (- c) a)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-96) {
		tmp = -c / b;
	} else if (b <= 14.5) {
		tmp = fma(((b / a) / c), 0.5, -sqrt(((pow(a, -1.0) / c) * -1.0))) * -c;
	} else {
		tmp = fma(-0.5, (b / a), -sqrt((-c / a)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-96)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 14.5)
		tmp = Float64(fma(Float64(Float64(b / a) / c), 0.5, Float64(-sqrt(Float64(Float64((a ^ -1.0) / c) * -1.0)))) * Float64(-c));
	else
		tmp = fma(-0.5, Float64(b / a), Float64(-sqrt(Float64(Float64(-c) / a))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-96], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 14.5], N[(N[(N[(N[(b / a), $MachinePrecision] / c), $MachinePrecision] * 0.5 + (-N[Sqrt[N[(N[(N[Power[a, -1.0], $MachinePrecision] / c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * (-c)), $MachinePrecision], N[(-0.5 * N[(b / a), $MachinePrecision] + (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.20000000000000012e-96

   1. Initial program 12.3%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -3.20000000000000012e-96 < b < 14.5

   1. Initial program 77.6%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 63.3

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

   7. Applied rewrites 63.3%

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

   if 14.5 < b

   1. Initial program 66.4%

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

   3. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 27.2

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

   5. Applied rewrites 27.2%

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

   6. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 31.2

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

   8. Applied rewrites 31.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 14.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{a}}{c}, 0.5, -\sqrt{\frac{{a}^{-1}}{c} \cdot -1}\right) \cdot \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{a}, -\sqrt{\frac{-c}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{a}}{c}, 0.5, -\sqrt{\frac{{a}^{-1}}{c} \cdot -1}\right) \cdot \left(-c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-96)
   (/ (- c) b)
   (* (fma (/ (/ b a) c) 0.5 (- (sqrt (* (/ (pow a -1.0) c) -1.0)))) (- c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-96) {
		tmp = -c / b;
	} else {
		tmp = fma(((b / a) / c), 0.5, -sqrt(((pow(a, -1.0) / c) * -1.0))) * -c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-96)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(fma(Float64(Float64(b / a) / c), 0.5, Float64(-sqrt(Float64(Float64((a ^ -1.0) / c) * -1.0)))) * Float64(-c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-96], N[((-c) / b), $MachinePrecision], N[(N[(N[(N[(b / a), $MachinePrecision] / c), $MachinePrecision] * 0.5 + (-N[Sqrt[N[(N[(N[Power[a, -1.0], $MachinePrecision] / c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * (-c)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.20000000000000012e-96

   1. Initial program 12.3%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -3.20000000000000012e-96 < b

   1. Initial program 71.7%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 43.5

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

   7. Applied rewrites 43.5%

      \[\leadsto -\mathsf{fma}\left(\frac{\frac{b}{a}}{c}, 0.5, -\sqrt{\frac{{a}^{-1}}{c} \cdot -1}\right) \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{a}}{c}, 0.5, -\sqrt{\frac{{a}^{-1}}{c} \cdot -1}\right) \cdot \left(-c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 31.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\frac{b}{a}}{c}, 0.5, -\sqrt{\frac{{a}^{-1}}{c} \cdot -1}\right) \cdot \left(-c\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (fma (/ (/ b a) c) 0.5 (- (sqrt (* (/ (pow a -1.0) c) -1.0)))) (- c)))

C

double code(double a, double b, double c) {
	return fma(((b / a) / c), 0.5, -sqrt(((pow(a, -1.0) / c) * -1.0))) * -c;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(b / a) / c), 0.5, Float64(-sqrt(Float64(Float64((a ^ -1.0) / c) * -1.0)))) * Float64(-c))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(b / a), $MachinePrecision] / c), $MachinePrecision] * 0.5 + (-N[Sqrt[N[(N[(N[Power[a, -1.0], $MachinePrecision] / c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * (-c)), $MachinePrecision]

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in c around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 29.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow-1 N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. inv-pow N/A

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

   8. lower-pow.f64 29.7

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

7. Applied rewrites 29.7%

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

8. Final simplification 29.7%

   \[\leadsto \mathsf{fma}\left(\frac{\frac{b}{a}}{c}, 0.5, -\sqrt{\frac{{a}^{-1}}{c} \cdot -1}\right) \cdot \left(-c\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

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