jeff quadratic root 1

Percentage Accurate: 72.0% → 90.7%

Time: 9.3s

Alternatives: 4

Speedup: N/A×

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: 72.0% 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: 90.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\\ t_1 := \frac{2 \cdot c}{\mathsf{fma}\left(-1, b, t\_0\right)}\\ \mathbf{if}\;b \leq -1.72 \cdot 10^{+50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{-1 \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b + -1 \cdot b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+121}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fma (pow b 1.0) (pow b 1.0) (* -4.0 (* c a))) 0.5))
        (t_1 (/ (* 2.0 c) (fma -1.0 b t_0))))
   (if (<= b -1.72e+50)
     (if (>= b 0.0)
       (/ (+ b (sqrt (- (* b b) (* (* 4.0 a) c)))) (* -1.0 (* 2.0 a)))
       (/ (* 2.0 c) (+ (* -1.0 b) (* -1.0 b))))
     (if (<= b 7.6e+121)
       (if (>= b 0.0) (* (/ (+ b t_0) a) (* -1.0 0.5)) t_1)
       (if (>= b 0.0) (* (/ (+ b b) a) (* -1.0 0.5)) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fma(pow(b, 1.0), pow(b, 1.0), (-4.0 * (c * a))), 0.5);
	double t_1 = (2.0 * c) / fma(-1.0, b, t_0);
	double tmp_1;
	if (b <= -1.72e+50) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + sqrt(((b * b) - ((4.0 * a) * c)))) / (-1.0 * (2.0 * a));
		} else {
			tmp_2 = (2.0 * c) / ((-1.0 * b) + (-1.0 * b));
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.6e+121) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + t_0) / a) * (-1.0 * 0.5);
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * (-1.0 * 0.5);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma((b ^ 1.0), (b ^ 1.0), Float64(-4.0 * Float64(c * a))) ^ 0.5
	t_1 = Float64(Float64(2.0 * c) / fma(-1.0, b, t_0))
	tmp_1 = 0.0
	if (b <= -1.72e+50)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(-1.0 * Float64(2.0 * a)));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-1.0 * b) + Float64(-1.0 * b)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7.6e+121)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + t_0) / a) * Float64(-1.0 * 0.5));
		else
			tmp_3 = t_1;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * Float64(-1.0 * 0.5));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(N[Power[b, 1.0], $MachinePrecision] * N[Power[b, 1.0], $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * b + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.72e+50], 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[(-1.0 * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(N[(-1.0 * b), $MachinePrecision] + N[(-1.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 7.6e+121], If[GreaterEqual[b, 0.0], N[(N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.72e50

   1. Initial program 61.5%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if -1.72e50 < b < 7.6e121

   1. Initial program 81.1%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ } \end{array}} \]

   if 7.6e121 < b

   1. Initial program 48.7%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ } \end{array}} \]

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - b}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.72 \cdot 10^{+50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{-1 \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b + -1 \cdot b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+121}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\\ t_1 := \frac{2 \cdot c}{\mathsf{fma}\left(-1, b, t\_0\right)}\\ \mathbf{if}\;b \leq 7.6 \cdot 10^{+121}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fma (pow b 1.0) (pow b 1.0) (* -4.0 (* c a))) 0.5))
        (t_1 (/ (* 2.0 c) (fma -1.0 b t_0))))
   (if (<= b 7.6e+121)
     (if (>= b 0.0) (* (/ (+ b t_0) a) (* -1.0 0.5)) t_1)
     (if (>= b 0.0) (* (/ (+ b b) a) (* -1.0 0.5)) t_1))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fma(pow(b, 1.0), pow(b, 1.0), (-4.0 * (c * a))), 0.5);
	double t_1 = (2.0 * c) / fma(-1.0, b, t_0);
	double tmp_1;
	if (b <= 7.6e+121) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((b + t_0) / a) * (-1.0 * 0.5);
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * (-1.0 * 0.5);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma((b ^ 1.0), (b ^ 1.0), Float64(-4.0 * Float64(c * a))) ^ 0.5
	t_1 = Float64(Float64(2.0 * c) / fma(-1.0, b, t_0))
	tmp_1 = 0.0
	if (b <= 7.6e+121)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(b + t_0) / a) * Float64(-1.0 * 0.5));
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * Float64(-1.0 * 0.5));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(N[Power[b, 1.0], $MachinePrecision] * N[Power[b, 1.0], $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * b + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.6e+121], If[GreaterEqual[b, 0.0], N[(N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6e121

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ } \end{array}} \]

   if 7.6e121 < b

   1. Initial program 48.7%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ } \end{array}} \]

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - b}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{+121}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ t_1 := {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, t\_0\right)\right)}^{0.5}\\ t_2 := {\left(\mathsf{fma}\left(b, b, t\_0\right)\right)}^{0.5}\\ \mathbf{if}\;b \leq -7 \cdot 10^{-291} \lor \neg \left(b \leq 7.6 \cdot 10^{+121}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, t\_1\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_1}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{{\left(-1 \cdot b\right)}^{3} + {t\_2}^{3}}{\mathsf{fma}\left(-1 \cdot b, -1 \cdot b, {t\_2}^{2} - \left(-1 \cdot b\right) \cdot t\_2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -4.0 (* c a)))
        (t_1 (pow (fma (pow b 1.0) (pow b 1.0) t_0) 0.5))
        (t_2 (pow (fma b b t_0) 0.5)))
   (if (or (<= b -7e-291) (not (<= b 7.6e+121)))
     (if (>= b 0.0)
       (* (/ (+ b b) a) (* -1.0 0.5))
       (/ (* 2.0 c) (fma -1.0 b t_1)))
     (if (>= b 0.0)
       (* (/ (+ b t_1) a) (* -1.0 0.5))
       (/
        (* 2.0 c)
        (/
         (+ (pow (* -1.0 b) 3.0) (pow t_2 3.0))
         (fma (* -1.0 b) (* -1.0 b) (- (pow t_2 2.0) (* (* -1.0 b) t_2)))))))))

C

double code(double a, double b, double c) {
	double t_0 = -4.0 * (c * a);
	double t_1 = pow(fma(pow(b, 1.0), pow(b, 1.0), t_0), 0.5);
	double t_2 = pow(fma(b, b, t_0), 0.5);
	double tmp_1;
	if ((b <= -7e-291) || !(b <= 7.6e+121)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((b + b) / a) * (-1.0 * 0.5);
		} else {
			tmp_2 = (2.0 * c) / fma(-1.0, b, t_1);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((b + t_1) / a) * (-1.0 * 0.5);
	} else {
		tmp_1 = (2.0 * c) / ((pow((-1.0 * b), 3.0) + pow(t_2, 3.0)) / fma((-1.0 * b), (-1.0 * b), (pow(t_2, 2.0) - ((-1.0 * b) * t_2))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(-4.0 * Float64(c * a))
	t_1 = fma((b ^ 1.0), (b ^ 1.0), t_0) ^ 0.5
	t_2 = fma(b, b, t_0) ^ 0.5
	tmp_1 = 0.0
	if ((b <= -7e-291) || !(b <= 7.6e+121))
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(b + b) / a) * Float64(-1.0 * 0.5));
		else
			tmp_2 = Float64(Float64(2.0 * c) / fma(-1.0, b, t_1));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + t_1) / a) * Float64(-1.0 * 0.5));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64((Float64(-1.0 * b) ^ 3.0) + (t_2 ^ 3.0)) / fma(Float64(-1.0 * b), Float64(-1.0 * b), Float64((t_2 ^ 2.0) - Float64(Float64(-1.0 * b) * t_2)))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[Power[b, 1.0], $MachinePrecision] * N[Power[b, 1.0], $MachinePrecision] + t$95$0), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(b * b + t$95$0), $MachinePrecision], 0.5], $MachinePrecision]}, If[Or[LessEqual[b, -7e-291], N[Not[LessEqual[b, 7.6e+121]], $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * b + t$95$1), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + t$95$1), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(N[(N[Power[N[(-1.0 * b), $MachinePrecision], 3.0], $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 * b), $MachinePrecision] * N[(-1.0 * b), $MachinePrecision] + N[(N[Power[t$95$2, 2.0], $MachinePrecision] - N[(N[(-1.0 * b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.99999999999999991e-291 or 7.6e121 < b

   1. Initial program 64.4%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ } \end{array}} \]

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 76.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - b}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array} \]

   if -6.99999999999999991e-291 < b < 7.6e121

   1. Initial program 81.3%

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

   3. Taylor expanded in b around -inf

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

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{{\left(-1 \cdot b\right)}^{3} + {\left({\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\frac{1}{2}}\right)}^{3}}{\left(-1 \cdot b\right) \cdot \left(-1 \cdot b\right) + \left({\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\frac{1}{2}} \cdot {\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\frac{1}{2}} - \left(-1 \cdot b\right) \cdot {\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\frac{1}{2}}\right)}}\\ \end{array} \]

      9. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{{\left(-1 \cdot b\right)}^{3} + {\left({\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\frac{1}{2}}\right)}^{3}}{\left(-1 \cdot b\right) \cdot \left(-1 \cdot b\right) + \left({\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\frac{1}{2}} \cdot {\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\frac{1}{2}} - \left(-1 \cdot b\right) \cdot {\left({b}^{1} \cdot {b}^{1} + -4 \cdot \left(c \cdot a\right)\right)}^{\frac{1}{2}}\right)}}\\ \end{array} \]

   7. Applied rewrites 81.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{{\left(-1 \cdot b\right)}^{3} + {\left({\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot b, -1 \cdot b, {\left({\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}^{2} - \left(-1 \cdot b\right) \cdot {\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-291} \lor \neg \left(b \leq 7.6 \cdot 10^{+121}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{{\left(-1 \cdot b\right)}^{3} + {\left({\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot b, -1 \cdot b, {\left({\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}^{2} - \left(-1 \cdot b\right) \cdot {\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* (/ (+ b b) a) (* -1.0 0.5))
   (/
    (* 2.0 c)
    (fma -1.0 b (pow (fma (pow b 1.0) (pow b 1.0) (* -4.0 (* c a))) 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = ((b + b) / a) * (-1.0 * 0.5);
	} else {
		tmp = (2.0 * c) / fma(-1.0, b, pow(fma(pow(b, 1.0), pow(b, 1.0), (-4.0 * (c * a))), 0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(b + b) / a) * Float64(-1.0 * 0.5));
	else
		tmp = Float64(Float64(2.0 * c) / fma(-1.0, b, (fma((b ^ 1.0), (b ^ 1.0), Float64(-4.0 * Float64(c * a))) ^ 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * b + N[Power[N[(N[Power[b, 1.0], $MachinePrecision] * N[Power[b, 1.0], $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 69.5%

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

3. Taylor expanded in b around -inf

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

5. Applied rewrites 69.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ } \end{array}} \]

6. Taylor expanded in a around 0

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

8. Applied rewrites 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b - b}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array} \]

9. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot \left(-1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -4 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}\\ \end{array} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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)))))))