jeff quadratic root 2

Percentage Accurate: 72.6% → 90.1%

Time: 14.8s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	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(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	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 = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	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[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	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(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	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 = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	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[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+49}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot -2\right) \cdot 0.5}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c \cdot a}{b} + b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -4.5e+49)
     (if (>= b 0.0) (/ 2.0 (/ (- (- b) b) c)) (/ (* (* b -2.0) 0.5) a))
     (if (<= b 6e+153)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0)
         (* -2.0 (/ c (+ b (+ b (* -2.0 (/ c (/ b a)))))))
         (* (+ (* -2.0 (/ (* c a) b)) (* b 2.0)) (/ -0.5 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -4.5e+49) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / ((-b - b) / c);
		} else {
			tmp_2 = ((b * -2.0) * 0.5) / a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 6e+153) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
	} else {
		tmp_1 = ((-2.0 * ((c * a) / b)) + (b * 2.0)) * (-0.5 / a);
	}
	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 <= (-4.5d+49)) then
        if (b >= 0.0d0) then
            tmp_2 = 2.0d0 / ((-b - b) / c)
        else
            tmp_2 = ((b * (-2.0d0)) * 0.5d0) / a
        end if
        tmp_1 = tmp_2
    else if (b <= 6d+153) then
        if (b >= 0.0d0) then
            tmp_3 = (2.0d0 * c) / (-b - t_0)
        else
            tmp_3 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (-2.0d0) * (c / (b + (b + ((-2.0d0) * (c / (b / a))))))
    else
        tmp_1 = (((-2.0d0) * ((c * a) / b)) + (b * 2.0d0)) * ((-0.5d0) / a)
    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 <= -4.5e+49) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / ((-b - b) / c);
		} else {
			tmp_2 = ((b * -2.0) * 0.5) / a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 6e+153) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
	} else {
		tmp_1 = ((-2.0 * ((c * a) / b)) + (b * 2.0)) * (-0.5 / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -4.5e+49:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 2.0 / ((-b - b) / c)
		else:
			tmp_2 = ((b * -2.0) * 0.5) / a
		tmp_1 = tmp_2
	elif b <= 6e+153:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (-b - t_0)
		else:
			tmp_3 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))))
	else:
		tmp_1 = ((-2.0 * ((c * a) / b)) + (b * 2.0)) * (-0.5 / a)
	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 <= -4.5e+49)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(2.0 / Float64(Float64(Float64(-b) - b) / c));
		else
			tmp_2 = Float64(Float64(Float64(b * -2.0) * 0.5) / a);
		end
		tmp_1 = tmp_2;
	elseif (b <= 6e+153)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a)))))));
	else
		tmp_1 = Float64(Float64(Float64(-2.0 * Float64(Float64(c * a) / b)) + Float64(b * 2.0)) * Float64(-0.5 / a));
	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 <= -4.5e+49)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 2.0 / ((-b - b) / c);
		else
			tmp_3 = ((b * -2.0) * 0.5) / a;
		end
		tmp_2 = tmp_3;
	elseif (b <= 6e+153)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (-b - t_0);
		else
			tmp_4 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
	else
		tmp_2 = ((-2.0 * ((c * a) / b)) + (b * 2.0)) * (-0.5 / a);
	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, -4.5e+49], If[GreaterEqual[b, 0.0], N[(2.0 / N[(N[((-b) - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * -2.0), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision]], If[LessEqual[b, 6e+153], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.49999999999999982e49

   1. Initial program 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in b around inf 60.4%

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

      1. div-sub 60.4%

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

      2. fma-udef 60.4%

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

      3. add-sqr-sqrt 32.8%

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

      4. hypot-def 51.0%

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

      5. *-commutative 51.0%

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

      6. *-commutative 51.0%

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

   6. Applied egg-rr 51.0%

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

      1. div-sub 51.0%

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

      2. *-rgt-identity 51.0%

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

      3. associate-*r/ 50.8%

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

      4. *-commutative 50.8%

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

      5. associate-*l* 50.8%

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

      6. *-commutative 50.8%

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

      7. associate-/r* 50.8%

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

      8. metadata-eval 50.8%

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

   8. Simplified 50.8%

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

   9. Taylor expanded in b around -inf 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
   10. Step-by-step derivation

       1. *-commutative 95.5%

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

   11. Simplified 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot -2\right) \cdot \frac{0.5}{a}\\ \end{array} \]
   12. Step-by-step derivation

       1. associate-*r/ 95.9%

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

   13. Applied egg-rr 95.9%

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

   if -4.49999999999999982e49 < b < 6.00000000000000037e153

   1. Initial program 85.3%

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

   if 6.00000000000000037e153 < b

   1. Initial program 32.6%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in b around inf 86.7%

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

      1. associate-/l* 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in b around -inf 96.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+49}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot -2\right) \cdot 0.5}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c \cdot a}{b} + b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

Alternative 2: 67.9% accurate, 6.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
	} else {
		tmp = (-0.5 / a) * (b + (b - (a * ((2.0 * 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 = (-2.0d0) * (c / (b + (b + ((-2.0d0) * (c / (b / a))))))
    else
        tmp = ((-0.5d0) / a) * (b + (b - (a * ((2.0d0 * 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 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
	} else {
		tmp = (-0.5 / a) * (b + (b - (a * ((2.0 * c) / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))))
	else:
		tmp = (-0.5 / a) * (b + (b - (a * ((2.0 * c) / b))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
	else
		tmp = (-0.5 / a) * (b + (b - (a * ((2.0 * c) / b))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 68.3%

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

3. Simplified 68.2%

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

4. Taylor expanded in b around inf 68.9%

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

   1. associate-/l* 70.9%

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

6. Simplified 70.9%

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

7. Taylor expanded in b around -inf 69.0%

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

   1. neg-mul-1 69.0%

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

   2. unsub-neg 69.0%

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

   3. associate-/l* 70.6%

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

   4. associate-*r/ 70.6%

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

9. Simplified 70.6%

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

    1. associate-/r/ 70.6%

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

11. Applied egg-rr 70.6%

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

12. Final simplification 70.6%

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

Alternative 3: 67.8% accurate, 11.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -c / b;
	} else {
		tmp = (-b - b) / (2.0 * a);
	}
	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 = -c / b
    else
        tmp = (-b - b) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

3. Simplified 68.0%

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

4. Taylor expanded in b around inf 70.4%

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

5. Taylor expanded in b around -inf 70.1%

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

   1. neg-mul-1 70.1%

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

7. Simplified 70.1%

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

8. Taylor expanded in b around 0 70.5%

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

   1. associate-*r/ 70.3%

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

   2. mul-1-neg 70.3%

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

10. Simplified 70.5%

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

11. Final simplification 70.5%

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

Alternative 4: 67.7% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (/ (- c) b) (* (* b -2.0) (/ 0.5 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -c / b;
	} else {
		tmp = (b * -2.0) * (0.5 / a);
	}
	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 = -c / b
    else
        tmp = (b * (-2.0d0)) * (0.5d0 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -c / b;
	} else {
		tmp = (b * -2.0) * (0.5 / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -c / b
	else:
		tmp = (b * -2.0) * (0.5 / a)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -c / b;
	else
		tmp = (b * -2.0) * (0.5 / a);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 68.3%

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

3. Simplified 68.0%

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

4. Taylor expanded in b around inf 70.4%

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

   1. div-sub 70.4%

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

   2. fma-udef 70.4%

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

   3. add-sqr-sqrt 60.1%

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

   4. hypot-def 65.7%

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

   5. *-commutative 65.7%

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

   6. *-commutative 65.7%

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

6. Applied egg-rr 65.7%

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

   1. div-sub 65.7%

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

   2. *-rgt-identity 65.7%

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

   3. associate-*r/ 65.7%

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

   4. *-commutative 65.7%

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

   5. associate-*l* 65.7%

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

   6. *-commutative 65.7%

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

   7. associate-/r* 65.7%

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

   8. metadata-eval 65.7%

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

8. Simplified 65.7%

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

9. Taylor expanded in b around -inf 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
10. Step-by-step derivation

    1. *-commutative 70.0%

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

11. Simplified 70.0%

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

12. Taylor expanded in b around 0 70.3%

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

    1. associate-*r/ 70.3%

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

    2. mul-1-neg 70.3%

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

14. Simplified 70.3%

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

15. Final simplification 70.3%

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

Reproduce

herbie shell --seed 2023229 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))