jeff quadratic root 1

Percentage Accurate: 72.3% → 90.4%

Time: 26.7s

Alternatives: 4

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

real(8) function code(a, b, c)
    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: 72.3% 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

real(8) function code(a, b, c)
    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: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{+151}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-2 \cdot \frac{a \cdot c}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -2.4e+151)
     (if (>= b 0.0) (* -0.5 (/ (+ b b) a)) (/ c (- b)))
     (if (<= b 4e+72)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0)
         (/ (* 2.0 (- (* a (/ c b)) b)) (* a 2.0))
         (/ (* c 2.0) (* -2.0 (/ (* a c) b))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -2.4e+151) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * ((b + b) / a);
		} else {
			tmp_2 = c / -b;
		}
		tmp_1 = tmp_2;
	} else if (b <= 4e+72) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	} else {
		tmp_1 = (c * 2.0) / (-2.0 * ((a * c) / b));
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    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 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-2.4d+151)) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * ((b + b) / a)
        else
            tmp_2 = c / -b
        end if
        tmp_1 = tmp_2
    else if (b <= 4d+72) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * ((a * (c / b)) - b)) / (a * 2.0d0)
    else
        tmp_1 = (c * 2.0d0) / ((-2.0d0) * ((a * c) / b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -2.4e+151) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * ((b + b) / a);
		} else {
			tmp_2 = c / -b;
		}
		tmp_1 = tmp_2;
	} else if (b <= 4e+72) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	} else {
		tmp_1 = (c * 2.0) / (-2.0 * ((a * c) / b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -2.4e+151:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * ((b + b) / a)
		else:
			tmp_2 = c / -b
		tmp_1 = tmp_2
	elif b <= 4e+72:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (a * 2.0)
		else:
			tmp_3 = (c * 2.0) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0)
	else:
		tmp_1 = (c * 2.0) / (-2.0 * ((a * c) / b))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -2.4e+151)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(Float64(b + b) / a));
		else
			tmp_2 = Float64(c / Float64(-b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4e+72)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)) / Float64(a * 2.0));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(-2.0 * Float64(Float64(a * c) / b)));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -2.4e+151)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * ((b + b) / a);
		else
			tmp_3 = c / -b;
		end
		tmp_2 = tmp_3;
	elseif (b <= 4e+72)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = (c * 2.0) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	else
		tmp_2 = (c * 2.0) / (-2.0 * ((a * c) / b));
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.4e+151], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]], If[LessEqual[b, 4e+72], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(-2.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4000000000000001e151

   1. Initial program 46.7%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in b around -inf 95.3%

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

   6. Taylor expanded in c around 0 95.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{-1 \cdot b - b}\\ \end{array} \]

   7. Taylor expanded in c around 0 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-*r/ 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

      2. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      3. sub-neg 95.5%

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

      4. mul-1-neg 95.5%

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

      5. distribute-neg-out 95.5%

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

      6. *-rgt-identity 95.5%

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

      7. *-rgt-identity 95.5%

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

      8. distribute-lft-out 95.5%

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

      9. metadata-eval 95.5%

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

      10. *-commutative 95.5%

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

      11. distribute-neg-frac2 95.5%

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

      12. *-commutative 95.5%

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

      13. times-frac 95.5%

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

      14. metadata-eval 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b} \cdot 1\\ \end{array} \]

      15. *-rgt-identity 95.5%

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

      16. distribute-neg-frac2 95.5%

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

   9. Simplified 95.5%

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

   if -2.4000000000000001e151 < b < 3.99999999999999978e72

   1. Initial program 90.1%

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

   if 3.99999999999999978e72 < b

   1. Initial program 56.2%

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

   3. Taylor expanded in a around 0 90.2%

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

      1. distribute-lft-out-- 90.2%

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

      2. associate-/l* 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in b around inf 97.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+151}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-2 \cdot \frac{a \cdot c}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+155)
   (if (>= b 0.0) (* -0.5 (/ (+ b b) a)) (/ c (- b)))
   (if (>= b 0.0)
     (/ (* 2.0 (- (* a (/ c b)) b)) (* a 2.0))
     (/ (* c 2.0) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2e+155) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * ((b + b) / a);
		} else {
			tmp_2 = c / -b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	} else {
		tmp_1 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - b);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-2d+155)) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * ((b + b) / a)
        else
            tmp_2 = c / -b
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * ((a * (c / b)) - b)) / (a * 2.0d0)
    else
        tmp_1 = (c * 2.0d0) / (sqrt(((b * b) - (c * (a * 4.0d0)))) - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2e+155) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * ((b + b) / a);
		} else {
			tmp_2 = c / -b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	} else {
		tmp_1 = (c * 2.0) / (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -2e+155:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * ((b + b) / a)
		else:
			tmp_2 = c / -b
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0)
	else:
		tmp_1 = (c * 2.0) / (math.sqrt(((b * b) - (c * (a * 4.0)))) - b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -2e+155)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(Float64(b + b) / a));
		else
			tmp_2 = Float64(c / Float64(-b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)) / Float64(a * 2.0));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -2e+155)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * ((b + b) / a);
		else
			tmp_3 = c / -b;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	else
		tmp_2 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+155], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.00000000000000001e155

   1. Initial program 46.7%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in b around -inf 95.3%

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

   6. Taylor expanded in c around 0 95.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{-1 \cdot b - b}\\ \end{array} \]

   7. Taylor expanded in c around 0 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-*r/ 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

      2. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      3. sub-neg 95.5%

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

      4. mul-1-neg 95.5%

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

      5. distribute-neg-out 95.5%

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

      6. *-rgt-identity 95.5%

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

      7. *-rgt-identity 95.5%

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

      8. distribute-lft-out 95.5%

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

      9. metadata-eval 95.5%

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

      10. *-commutative 95.5%

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

      11. distribute-neg-frac2 95.5%

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

      12. *-commutative 95.5%

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

      13. times-frac 95.5%

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

      14. metadata-eval 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b} \cdot 1\\ \end{array} \]

      15. *-rgt-identity 95.5%

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

      16. distribute-neg-frac2 95.5%

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

   9. Simplified 95.5%

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

   if -2.00000000000000001e155 < b

   1. Initial program 79.1%

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

   3. Taylor expanded in a around 0 78.0%

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

      1. distribute-lft-out-- 78.0%

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

      2. associate-/l* 80.3%

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

   5. Simplified 80.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 6.7× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (* 2.0 (- (* a (/ c b)) b)) (* a 2.0))
   (/ (* c 2.0) (- (- b) b))))

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (2.0d0 * ((a * (c / b)) - b)) / (a * 2.0d0)
    else
        tmp = (c * 2.0d0) / (-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 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	} else {
		tmp = (c * 2.0) / (-b - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * ((a * (c / b)) - b)) / (a * 2.0)
	else:
		tmp = (c * 2.0) / (-b - b)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in a around 0 73.0%

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

   1. distribute-lft-out-- 73.0%

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

   2. associate-/l* 74.9%

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

5. Simplified 74.9%

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

6. Taylor expanded in b around -inf 69.2%

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

7. Final simplification 69.2%

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

Alternative 4: 68.1% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-0.5d0) * ((b + b) / a)
    else
        tmp = c / -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 = -0.5 * ((b + b) / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

3. Simplified 73.8%

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

5. Taylor expanded in b around -inf 68.1%

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

6. Taylor expanded in c around 0 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{-1 \cdot b - b}\\ \end{array} \]

7. Taylor expanded in c around 0 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
8. Step-by-step derivation

   1. associate-*r/ 69.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

   2. *-commutative 69.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   3. sub-neg 69.2%

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

   4. mul-1-neg 69.2%

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

   5. distribute-neg-out 69.2%

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

   6. *-rgt-identity 69.2%

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

   7. *-rgt-identity 69.2%

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

   8. distribute-lft-out 69.2%

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

   9. metadata-eval 69.2%

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

   10. *-commutative 69.2%

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

   11. distribute-neg-frac2 69.2%

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

   12. *-commutative 69.2%

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

   13. times-frac 69.2%

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

   14. metadata-eval 69.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b} \cdot 1\\ \end{array} \]

   15. *-rgt-identity 69.2%

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

   16. distribute-neg-frac2 69.2%

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

9. Simplified 69.2%

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

Reproduce

herbie shell --seed 2024085 
(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)))))))