jeff quadratic root 1

Percentage Accurate: 71.2% → 90.3%

Time: 4.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 90.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+136}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + t\_0\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* (* -4.0 a) c)))))
   (if (<= b -1.2e+136)
     (if (>= b 0.0)
       (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
       (/ (* 2.0 c) (* (- b) 2.0)))
     (if (<= b 2.7e+107)
       (if (>= b 0.0) (/ (* -0.5 (+ b t_0)) a) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (* (/ (+ b b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, ((-4.0 * a) * c)));
	double tmp_1;
	if (b <= -1.2e+136) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
		} else {
			tmp_2 = (2.0 * c) / (-b * 2.0);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.7e+107) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-0.5 * (b + t_0)) / a;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))
	tmp_1 = 0.0
	if (b <= -1.2e+136)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) * 2.0));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.7e+107)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-0.5 * Float64(b + t_0)) / a);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.2e+136], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) * 2.0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.7e+107], If[GreaterEqual[b, 0.0], N[(N[(-0.5 * N[(b + t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2e136

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-1 \cdot b\right) \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}\\ \end{array} \]

      2. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\\ \end{array} \]

      3. lift-neg.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}\\ \end{array} \]

      5. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \color{blue}{\left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + 2\right)}}\\ \end{array} \]

      6. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{2}} \cdot -2} + 2\right)}\\ \end{array} \]

      7. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{a \cdot c}{{b}^{2}}, -2, 2\right)}}\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{{b}^{2}}}, -2, 2\right)}\\ \end{array} \]

      9. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{{b}^{2}}}, -2, 2\right)}\\ \end{array} \]

      10. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \mathsf{fma}\left(a \cdot \color{blue}{\frac{c}{{b}^{2}}}, -2, 2\right)}\\ \end{array} \]

      11. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \mathsf{fma}\left(a \cdot \frac{c}{\color{blue}{b \cdot b}}, -2, 2\right)}\\ \end{array} \]

      12. lift-*.f64 69.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \mathsf{fma}\left(a \cdot \frac{c}{\color{blue}{b \cdot b}}, -2, 2\right)}\\ \end{array} \]

   4. Applied rewrites 69.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) \cdot \mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -2, 2\right)}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \color{blue}{2}}\\ \end{array} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot \color{blue}{2}}\\ \end{array} \]

   if -1.2e136 < b < 2.7000000000000001e107

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ } \end{array}} \]

   if 2.7000000000000001e107 < b

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(2 \cdot \frac{b}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. count-2-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lower-+.f64 57.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 57.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq 3.2 \cdot 10^{-302}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b 3.2e-302)
     (if (>= b 0.0)
       (* (- (/ (* c (sqrt (* -4.0 (/ a c)))) a)) -0.5)
       (/ (+ c c) (- t_0 b)))
     (if (<= b 2.7e+107)
       (if (>= b 0.0)
         (* (/ (+ t_0 b) a) -0.5)
         (/ -2.0 (* a (sqrt (/ -4.0 (* a c))))))
       (if (>= b 0.0)
         (* (/ (+ b b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= 3.2e-302) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -((c * sqrt((-4.0 * (a / c)))) / a) * -0.5;
		} else {
			tmp_2 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.7e+107) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((t_0 + b) / a) * -0.5;
		} else {
			tmp_3 = -2.0 / (a * sqrt((-4.0 / (a * c))));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= 3.2e-302)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-Float64(Float64(c * sqrt(Float64(-4.0 * Float64(a / c)))) / a)) * -0.5);
		else
			tmp_2 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.7e+107)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
		else
			tmp_3 = Float64(-2.0 / Float64(a * sqrt(Float64(-4.0 / Float64(a * c)))));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 3.2e-302], If[GreaterEqual[b, 0.0], N[((-N[(N[(c * N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]) * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.7e+107], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(-2.0 / N[(a * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.19999999999999978e-302

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-1 \cdot \frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\mathsf{neg}\left(\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right)\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lower-neg.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      3. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      6. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      7. lift-/.f64 42.5

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 42.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   if 3.19999999999999978e-302 < b < 2.7000000000000001e107

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]

      4. lift-/.f64 44.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]

   7. Applied rewrites 44.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]

   8. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \end{array} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \end{array} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \end{array} \]

      3. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \end{array} \]

      4. lower-*.f64 49.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \end{array} \]

   10. Applied rewrites 49.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \end{array} \]

   if 2.7000000000000001e107 < b

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(2 \cdot \frac{b}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. count-2-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lower-+.f64 57.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 57.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b 2.7e+107)
     (if (>= b 0.0) (* (/ (+ t_0 b) a) -0.5) (/ (+ c c) (- t_0 b)))
     (if (>= b 0.0) (* (/ (+ b b) a) -0.5) (/ (+ c c) (- (sqrt (* b b)) b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= 2.7e+107) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((t_0 + b) / a) * -0.5;
		} else {
			tmp_2 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= 2.7e+107)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
		else
			tmp_2 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 2.7e+107], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000001e107

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   if 2.7000000000000001e107 < b

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(2 \cdot \frac{b}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. count-2-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lower-+.f64 57.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 57.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-54}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{b \cdot b} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.3e-54)
   (if (>= b 0.0)
     (* (- (/ (* a (* c (sqrt (/ -4.0 (* a c))))) a)) -0.5)
     (/ (+ c c) (- (sqrt (fma (* -4.0 a) c (* b b))) b)))
   (if (>= b 0.0)
     (/ (* b (- (* a (/ c (* b b))) 1.0)) a)
     (/ (+ c c) (- (sqrt (fma b b (* (* -4.0 a) c))) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 1.3e-54) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -((a * (c * sqrt((-4.0 / (a * c))))) / a) * -0.5;
		} else {
			tmp_2 = (c + c) / (sqrt(fma((-4.0 * a), c, (b * b))) - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b * ((a * (c / (b * b))) - 1.0)) / a;
	} else {
		tmp_1 = (c + c) / (sqrt(fma(b, b, ((-4.0 * a) * c))) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 1.3e-54)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-Float64(Float64(a * Float64(c * sqrt(Float64(-4.0 / Float64(a * c))))) / a)) * -0.5);
		else
			tmp_2 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(b * Float64(Float64(a * Float64(c / Float64(b * b))) - 1.0)) / a);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c))) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.3e-54], If[GreaterEqual[b, 0.0], N[((-N[(N[(a * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]) * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(b * N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.30000000000000001e-54

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-1 \cdot \frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\mathsf{neg}\left(\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right)\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lower-neg.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      3. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      6. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      7. lift-/.f64 42.5

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 42.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      4. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      5. lower-*.f64 50.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   10. Applied rewrites 50.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   if 1.30000000000000001e-54 < b

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ } \end{array}} \]

   8. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(\frac{a \cdot c}{{b}^{2}} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(\frac{a \cdot c}{{b}^{2}} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \]

      2. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(\frac{a \cdot c}{{b}^{2}} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{{b}^{2}} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{{b}^{2}} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \]

      5. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{{b}^{2}} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \]

      6. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{b \cdot b} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \]

      7. lift-*.f64 68.8

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{b \cdot b} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \]

   10. Applied rewrites 68.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{b \cdot b} - 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \mathbf{if}\;b \leq 3.2 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (fma (* -4.0 a) c (* b b))) b))))
   (if (<= b 3.2e-55)
     (if (>= b 0.0) (* (- (/ (* a (* c (sqrt (/ -4.0 (* a c))))) a)) -0.5) t_0)
     (if (>= b 0.0) (* (/ (+ b b) a) -0.5) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt(fma((-4.0 * a), c, (b * b))) - b);
	double tmp_1;
	if (b <= 3.2e-55) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -((a * (c * sqrt((-4.0 / (a * c))))) / a) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b))
	tmp_1 = 0.0
	if (b <= 3.2e-55)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-Float64(Float64(a * Float64(c * sqrt(Float64(-4.0 / Float64(a * c))))) / a)) * -0.5);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.2e-55], If[GreaterEqual[b, 0.0], N[((-N[(N[(a * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]) * -0.5), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.2000000000000001e-55

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-1 \cdot \frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\mathsf{neg}\left(\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right)\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lower-neg.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      3. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      6. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      7. lift-/.f64 42.5

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 42.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      4. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      5. lower-*.f64 50.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   10. Applied rewrites 50.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   if 3.2000000000000001e-55 < b

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lower-+.f64 69.2

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 69.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.2e-55)
   (if (>= b 0.0)
     (* (- (/ (* a (* c (sqrt (/ -4.0 (* a c))))) a)) -0.5)
     (/ (+ c c) (- (sqrt (fma (* -4.0 a) c (* b b))) b)))
   (if (>= b 0.0) (* (/ (+ b b) a) -0.5) (/ (+ c c) (- (sqrt (* b b)) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 3.2e-55) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -((a * (c * sqrt((-4.0 / (a * c))))) / a) * -0.5;
		} else {
			tmp_2 = (c + c) / (sqrt(fma((-4.0 * a), c, (b * b))) - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 3.2e-55)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-Float64(Float64(a * Float64(c * sqrt(Float64(-4.0 / Float64(a * c))))) / a)) * -0.5);
		else
			tmp_2 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.2e-55], If[GreaterEqual[b, 0.0], N[((-N[(N[(a * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]) * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.2000000000000001e-55

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-1 \cdot \frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\mathsf{neg}\left(\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right)\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lower-neg.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      3. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      6. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      7. lift-/.f64 42.5

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 42.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      4. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      5. lower-*.f64 50.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   10. Applied rewrites 50.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{a \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   if 3.2000000000000001e-55 < b

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(2 \cdot \frac{b}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. count-2-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lower-+.f64 57.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 57.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{b \cdot b} - b}\\ t_1 := \sqrt{\left(-4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-42}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_1 + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (* b b)) b))) (t_1 (sqrt (* (* -4.0 a) c))))
   (if (<= b -3.5e-42)
     (if (>= b 0.0) (* (- (sqrt (* -4.0 (/ c a)))) -0.5) t_0)
     (if (<= b 3.8e-55)
       (if (>= b 0.0) (* (/ (+ t_1 b) a) -0.5) (/ (+ c c) (- t_1 b)))
       (if (>= b 0.0) (* (/ (+ b b) a) -0.5) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt((b * b)) - b);
	double t_1 = sqrt(((-4.0 * a) * c));
	double tmp_1;
	if (b <= -3.5e-42) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.8e-55) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((t_1 + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = (c + c) / (sqrt((b * b)) - b)
    t_1 = sqrt((((-4.0d0) * a) * c))
    if (b <= (-3.5d-42)) then
        if (b >= 0.0d0) then
            tmp_2 = -sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= 3.8d-55) then
        if (b >= 0.0d0) then
            tmp_3 = ((t_1 + b) / a) * (-0.5d0)
        else
            tmp_3 = (c + c) / (t_1 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = ((b + b) / a) * (-0.5d0)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c + c) / (Math.sqrt((b * b)) - b);
	double t_1 = Math.sqrt(((-4.0 * a) * c));
	double tmp_1;
	if (b <= -3.5e-42) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -Math.sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.8e-55) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((t_1 + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c + c) / (math.sqrt((b * b)) - b)
	t_1 = math.sqrt(((-4.0 * a) * c))
	tmp_1 = 0
	if b <= -3.5e-42:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -math.sqrt((-4.0 * (c / a))) * -0.5
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= 3.8e-55:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = ((t_1 + b) / a) * -0.5
		else:
			tmp_3 = (c + c) / (t_1 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = ((b + b) / a) * -0.5
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b))
	t_1 = sqrt(Float64(Float64(-4.0 * a) * c))
	tmp_1 = 0.0
	if (b <= -3.5e-42)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-sqrt(Float64(-4.0 * Float64(c / a)))) * -0.5);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.8e-55)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(t_1 + b) / a) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (c + c) / (sqrt((b * b)) - b);
	t_1 = sqrt(((-4.0 * a) * c));
	tmp_2 = 0.0;
	if (b <= -3.5e-42)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -sqrt((-4.0 * (c / a))) * -0.5;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= 3.8e-55)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = ((t_1 + b) / a) * -0.5;
		else
			tmp_4 = (c + c) / (t_1 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = ((b + b) / a) * -0.5;
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3.5e-42], If[GreaterEqual[b, 0.0], N[((-N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * -0.5), $MachinePrecision], t$95$0], If[LessEqual[b, 3.8e-55], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$1 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5000000000000002e-42

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in a around -inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-1 \cdot \sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\mathsf{neg}\left(\sqrt{-4 \cdot \frac{c}{a}}\right)\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       5. lift-sqrt.f64 31.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 31.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   if -3.5000000000000002e-42 < b < 3.7999999999999997e-55

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      3. lift-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}\\ \end{array} \]

      3. lift-*.f64 40.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}\\ \end{array} \]

   10. Applied rewrites 40.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}\\ \end{array} \]

   if 3.7999999999999997e-55 < b

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(2 \cdot \frac{b}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. count-2-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lower-+.f64 57.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 57.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{b \cdot b} - b}\\ \mathbf{if}\;b \leq -4.3 \cdot 10^{-80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (* b b)) b))))
   (if (<= b -4.3e-80)
     (if (>= b 0.0) (* (- (sqrt (* -4.0 (/ c a)))) -0.5) t_0)
     (if (<= b 3.2e-55)
       (if (>= b 0.0)
         (* -0.5 (/ (sqrt (* (* -4.0 a) c)) a))
         (/ (* 2.0 c) (sqrt (* -4.0 (* a c)))))
       (if (>= b 0.0) (* (/ (+ b b) a) -0.5) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt((b * b)) - b);
	double tmp_1;
	if (b <= -4.3e-80) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.2e-55) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * (sqrt(((-4.0 * a) * c)) / a);
		} else {
			tmp_3 = (2.0 * c) / sqrt((-4.0 * (a * c)));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = (c + c) / (sqrt((b * b)) - b)
    if (b <= (-4.3d-80)) then
        if (b >= 0.0d0) then
            tmp_2 = -sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= 3.2d-55) then
        if (b >= 0.0d0) then
            tmp_3 = (-0.5d0) * (sqrt((((-4.0d0) * a) * c)) / a)
        else
            tmp_3 = (2.0d0 * c) / sqrt(((-4.0d0) * (a * c)))
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = ((b + b) / a) * (-0.5d0)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c + c) / (Math.sqrt((b * b)) - b);
	double tmp_1;
	if (b <= -4.3e-80) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -Math.sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.2e-55) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * (Math.sqrt(((-4.0 * a) * c)) / a);
		} else {
			tmp_3 = (2.0 * c) / Math.sqrt((-4.0 * (a * c)));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c + c) / (math.sqrt((b * b)) - b)
	tmp_1 = 0
	if b <= -4.3e-80:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -math.sqrt((-4.0 * (c / a))) * -0.5
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= 3.2e-55:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = -0.5 * (math.sqrt(((-4.0 * a) * c)) / a)
		else:
			tmp_3 = (2.0 * c) / math.sqrt((-4.0 * (a * c)))
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = ((b + b) / a) * -0.5
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b))
	tmp_1 = 0.0
	if (b <= -4.3e-80)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-sqrt(Float64(-4.0 * Float64(c / a)))) * -0.5);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.2e-55)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-0.5 * Float64(sqrt(Float64(Float64(-4.0 * a) * c)) / a));
		else
			tmp_3 = Float64(Float64(2.0 * c) / sqrt(Float64(-4.0 * Float64(a * c))));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (c + c) / (sqrt((b * b)) - b);
	tmp_2 = 0.0;
	if (b <= -4.3e-80)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -sqrt((-4.0 * (c / a))) * -0.5;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= 3.2e-55)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = -0.5 * (sqrt(((-4.0 * a) * c)) / a);
		else
			tmp_4 = (2.0 * c) / sqrt((-4.0 * (a * c)));
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = ((b + b) / a) * -0.5;
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.3e-80], If[GreaterEqual[b, 0.0], N[((-N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * -0.5), $MachinePrecision], t$95$0], If[LessEqual[b, 3.2e-55], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.3000000000000001e-80

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in a around -inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-1 \cdot \sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\mathsf{neg}\left(\sqrt{-4 \cdot \frac{c}{a}}\right)\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       5. lift-sqrt.f64 31.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 31.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   if -4.3000000000000001e-80 < b < 3.2000000000000001e-55

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot \color{blue}{b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot \color{blue}{b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   4. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{{b}^{2}}}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

   7. Applied rewrites 47.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b}}\\ \end{array} \]

   8. Taylor expanded in b around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      3. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      4. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      5. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      6. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      8. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      9. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

      10. lift-*.f64 37.0

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

   10. Applied rewrites 37.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-0.5 \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b}}\\ \end{array} \]

   11. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \end{array} \]
   12. Step-by-step derivation

       1. distribute-lft-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       2. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\color{blue}{-4} \cdot \left(a \cdot c\right)}}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       4. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       5. lift-*.f64 26.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \end{array} \]

   13. Applied rewrites 26.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

   if 3.2000000000000001e-55 < b

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(2 \cdot \frac{b}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. count-2-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lower-+.f64 57.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 57.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{b \cdot b} - b}\\ \mathbf{if}\;c \leq -4.8 \cdot 10^{+75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (* b b)) b))))
   (if (<= c -4.8e+75)
     (if (>= b 0.0) (* (sqrt (* -4.0 (/ c a))) -0.5) t_0)
     (if (>= b 0.0) (* (/ (+ b b) a) -0.5) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt((b * b)) - b);
	double tmp_1;
	if (c <= -4.8e+75) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (c + c) / (sqrt((b * b)) - b)
    if (c <= (-4.8d+75)) then
        if (b >= 0.0d0) then
            tmp_2 = sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = ((b + b) / a) * (-0.5d0)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c + c) / (Math.sqrt((b * b)) - b);
	double tmp_1;
	if (c <= -4.8e+75) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = Math.sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c + c) / (math.sqrt((b * b)) - b)
	tmp_1 = 0
	if c <= -4.8e+75:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = math.sqrt((-4.0 * (c / a))) * -0.5
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = ((b + b) / a) * -0.5
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b))
	tmp_1 = 0.0
	if (c <= -4.8e+75)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(sqrt(Float64(-4.0 * Float64(c / a))) * -0.5);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (c + c) / (sqrt((b * b)) - b);
	tmp_2 = 0.0;
	if (c <= -4.8e+75)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = sqrt((-4.0 * (c / a))) * -0.5;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = ((b + b) / a) * -0.5;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.8e+75], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.8e75

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in a around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. lift-sqrt.f64 31.8

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 31.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   if -4.8e75 < c

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(2 \cdot \frac{b}{a}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. count-2-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lower-+.f64 57.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 57.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 39.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{b \cdot b} - b}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_2 := \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array} \leq -7 \cdot 10^{-178}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2 \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(-t\_2\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (* b b)) b)))
        (t_1 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_2 (sqrt (* -4.0 (/ c a)))))
   (if (<=
        (if (>= b 0.0) (/ (- (- b) t_1) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_1)))
        -7e-178)
     (if (>= b 0.0) (* t_2 -0.5) t_0)
     (if (>= b 0.0) (* (- t_2) -0.5) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt((b * b)) - b);
	double t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	double t_2 = sqrt((-4.0 * (c / a)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_1) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_1);
	}
	double tmp_2;
	if (tmp <= -7e-178) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2 * -0.5;
		} else {
			tmp_3 = t_0;
		}
		tmp_2 = tmp_3;
	} else if (b >= 0.0) {
		tmp_2 = -t_2 * -0.5;
	} else {
		tmp_2 = t_0;
	}
	return tmp_2;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = (c + c) / (sqrt((b * b)) - b)
    t_1 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    t_2 = sqrt(((-4.0d0) * (c / a)))
    if (b >= 0.0d0) then
        tmp = (-b - t_1) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_1)
    end if
    if (tmp <= (-7d-178)) then
        if (b >= 0.0d0) then
            tmp_3 = t_2 * (-0.5d0)
        else
            tmp_3 = t_0
        end if
        tmp_2 = tmp_3
    else if (b >= 0.0d0) then
        tmp_2 = -t_2 * (-0.5d0)
    else
        tmp_2 = t_0
    end if
    code = tmp_2
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c + c) / (Math.sqrt((b * b)) - b);
	double t_1 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double t_2 = Math.sqrt((-4.0 * (c / a)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_1) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_1);
	}
	double tmp_2;
	if (tmp <= -7e-178) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2 * -0.5;
		} else {
			tmp_3 = t_0;
		}
		tmp_2 = tmp_3;
	} else if (b >= 0.0) {
		tmp_2 = -t_2 * -0.5;
	} else {
		tmp_2 = t_0;
	}
	return tmp_2;
}

Python

def code(a, b, c):
	t_0 = (c + c) / (math.sqrt((b * b)) - b)
	t_1 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	t_2 = math.sqrt((-4.0 * (c / a)))
	tmp = 0
	if b >= 0.0:
		tmp = (-b - t_1) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_1)
	tmp_2 = 0
	if tmp <= -7e-178:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_2 * -0.5
		else:
			tmp_3 = t_0
		tmp_2 = tmp_3
	elif b >= 0.0:
		tmp_2 = -t_2 * -0.5
	else:
		tmp_2 = t_0
	return tmp_2

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	t_2 = sqrt(Float64(-4.0 * Float64(c / a)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
	end
	tmp_2 = 0.0
	if (tmp <= -7e-178)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(t_2 * -0.5);
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = Float64(Float64(-t_2) * -0.5);
	else
		tmp_2 = t_0;
	end
	return tmp_2
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (c + c) / (sqrt((b * b)) - b);
	t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	t_2 = sqrt((-4.0 * (c / a)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - t_1) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_1);
	end
	tmp_3 = 0.0;
	if (tmp <= -7e-178)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_2 * -0.5;
		else
			tmp_4 = t_0;
		end
		tmp_3 = tmp_4;
	elseif (b >= 0.0)
		tmp_3 = -t_2 * -0.5;
	else
		tmp_3 = t_0;
	end
	tmp_5 = tmp_3;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]], -7e-178], If[GreaterEqual[b, 0.0], N[(t$95$2 * -0.5), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[((-t$95$2) * -0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if (if (>=.f64 b #s(literal 0 binary64)) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) (/.f64 (*.f64 #s(literal 2 binary64) c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))))) < -6.99999999999999966e-178

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in a around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. lift-sqrt.f64 31.8

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 31.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   if -6.99999999999999966e-178 < (if (>=.f64 b #s(literal 0 binary64)) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) (/.f64 (*.f64 #s(literal 2 binary64) c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c))))))

   1. Initial program 71.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

      2. lift-*.f64 59.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   7. Applied rewrites 59.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

      2. lift-*.f64 47.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   10. Applied rewrites 47.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   11. Taylor expanded in a around -inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-1 \cdot \sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\mathsf{neg}\left(\sqrt{-4 \cdot \frac{c}{a}}\right)\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       2. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       3. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       4. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

       5. lift-sqrt.f64 31.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   13. Applied rewrites 31.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 31.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* (- (sqrt (* -4.0 (/ c a)))) -0.5)
   (/ (+ c c) (- (sqrt (* b b)) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -sqrt((-4.0 * (c / a))) * -0.5;
	} else {
		tmp = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = -sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
    else
        tmp = (c + c) / (sqrt((b * b)) - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -Math.sqrt((-4.0 * (c / a))) * -0.5;
	} else {
		tmp = (c + c) / (Math.sqrt((b * b)) - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -math.sqrt((-4.0 * (c / a))) * -0.5
	else:
		tmp = (c + c) / (math.sqrt((b * b)) - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(-4.0 * Float64(c / a)))) * -0.5);
	else
		tmp = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -sqrt((-4.0 * (c / a))) * -0.5;
	else
		tmp = (c + c) / (sqrt((b * b)) - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[((-N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.2%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
3. Step-by-step derivation

4. Applied rewrites 71.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ } \end{array}} \]

5. Taylor expanded in a around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{{b}^{2}} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]
6. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

   2. lift-*.f64 59.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

7. Applied rewrites 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array} \]

8. Taylor expanded in a around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{{b}^{2}} - b}\\ \end{array} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

   2. lift-*.f64 47.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

10. Applied rewrites 47.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{b \cdot b} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

11. Taylor expanded in a around -inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-1 \cdot \sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\mathsf{neg}\left(\sqrt{-4 \cdot \frac{c}{a}}\right)\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

    2. lower-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

    3. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

    4. lift-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

    5. lift-sqrt.f64 31.5

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]

13. Applied rewrites 31.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\sqrt{-4 \cdot \frac{c}{a}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025138 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))