quadp (p42, positive)

Percentage Accurate: 52.2% → 84.6%

Time: 14.1s

Alternatives: 7

Speedup: 12.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{b}{a \cdot -2} - \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)} \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+152)
   (/ b (- a))
   (if (<= b 1.75e-82)
     (- (/ b (* a -2.0)) (/ (* (sqrt (fma (* c -4.0) a (pow b 2.0))) -0.5) a))
     (* c (- (/ -1.0 b) (* a (/ c (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+152) {
		tmp = b / -a;
	} else if (b <= 1.75e-82) {
		tmp = (b / (a * -2.0)) - ((sqrt(fma((c * -4.0), a, pow(b, 2.0))) * -0.5) / a);
	} else {
		tmp = c * ((-1.0 / b) - (a * (c / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e+152)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.75e-82)
		tmp = Float64(Float64(b / Float64(a * -2.0)) - Float64(Float64(sqrt(fma(Float64(c * -4.0), a, (b ^ 2.0))) * -0.5) / a));
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(a * Float64(c / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+152], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.75e-82], N[(N[(b / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.99999999999999991e152

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

   3. Simplified 39.0%

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

   5. Taylor expanded in b around -inf 99.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.7%

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

      2. distribute-neg-frac2 99.7%

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

   7. Simplified 99.7%

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

   if -2.99999999999999991e152 < b < 1.7499999999999999e-82

   1. Initial program 88.5%

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

      1. *-commutative 88.5%

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

   3. Simplified 88.5%

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

      1. frac-2neg 88.5%

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

      2. div-inv 88.2%

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

   6. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot -4}, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2} \]

   8. Simplified 88.2%

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

      1. add-cube-cbrt 87.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}} \cdot \sqrt[3]{\left(b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}}\right) \cdot \sqrt[3]{\left(b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}}} \]

      2. pow3 87.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}}\right)}^{3}} \]

      3. un-div-inv 87.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a \cdot -2}}}\right)}^{3} \]

   10. Applied egg-rr 87.0%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a \cdot -2}}\right)}^{3}} \]
   11. Step-by-step derivation

       1. div-sub 87.0%

          \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{b}{a \cdot -2} - \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a \cdot -2}}}\right)}^{3} \]

   12. Applied egg-rr 87.0%

       \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{b}{a \cdot -2} - \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a \cdot -2}}}\right)}^{3} \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 88.5%

          \[\leadsto \color{blue}{\frac{b}{a \cdot -2} - \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a \cdot -2}} \]

       2. associate-/r* 88.5%

          \[\leadsto \frac{b}{a \cdot -2} - \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a}}{-2}} \]

       3. div-inv 88.5%

          \[\leadsto \frac{b}{a \cdot -2} - \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a} \cdot \frac{1}{-2}} \]

       4. metadata-eval 88.5%

          \[\leadsto \frac{b}{a \cdot -2} - \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a} \cdot \color{blue}{-0.5} \]

   14. Applied egg-rr 88.5%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -2} - \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a} \cdot -0.5} \]
   15. Step-by-step derivation

       1. associate-*l/ 88.5%

          \[\leadsto \frac{b}{a \cdot -2} - \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)} \cdot -0.5}{a}} \]

   16. Simplified 88.5%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -2} - \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)} \cdot -0.5}{a}} \]

   if 1.7499999999999999e-82 < b

   1. Initial program 13.4%

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

      1. *-commutative 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in c around 0 92.1%

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 92.1%

         \[\leadsto c \cdot \left(\color{blue}{\left(-\frac{a \cdot c}{{b}^{3}}\right)} - \frac{1}{b}\right) \]

      2. associate-/l* 93.4%

         \[\leadsto c \cdot \left(\left(-\color{blue}{a \cdot \frac{c}{{b}^{3}}}\right) - \frac{1}{b}\right) \]

   7. Simplified 93.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{b}{a \cdot -2} - \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)} \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+150)
   (/ b (- a))
   (if (<= b 1.3e-82)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (* c (- (/ -1.0 b) (* a (/ c (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+150) {
		tmp = b / -a;
	} else if (b <= 1.3e-82) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - (a * (c / pow(b, 3.0))));
	}
	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) :: tmp
    if (b <= (-4.5d+150)) then
        tmp = b / -a
    else if (b <= 1.3d-82) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = c * (((-1.0d0) / b) - (a * (c / (b ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+150) {
		tmp = b / -a;
	} else if (b <= 1.3e-82) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - (a * (c / Math.pow(b, 3.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e+150:
		tmp = b / -a
	elif b <= 1.3e-82:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = c * ((-1.0 / b) - (a * (c / math.pow(b, 3.0))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+150)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.3e-82)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(a * Float64(c / (b ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e+150)
		tmp = b / -a;
	elseif (b <= 1.3e-82)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = c * ((-1.0 / b) - (a * (c / (b ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+150], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.3e-82], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5e150

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

   3. Simplified 39.0%

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

   5. Taylor expanded in b around -inf 99.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.7%

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

      2. distribute-neg-frac2 99.7%

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

   7. Simplified 99.7%

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

   if -4.5e150 < b < 1.3e-82

   1. Initial program 88.5%

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

   if 1.3e-82 < b

   1. Initial program 13.4%

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

      1. *-commutative 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in c around 0 92.1%

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 92.1%

         \[\leadsto c \cdot \left(\color{blue}{\left(-\frac{a \cdot c}{{b}^{3}}\right)} - \frac{1}{b}\right) \]

      2. associate-/l* 93.4%

         \[\leadsto c \cdot \left(\left(-\color{blue}{a \cdot \frac{c}{{b}^{3}}}\right) - \frac{1}{b}\right) \]

   7. Simplified 93.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-78)
   (/ b (- a))
   (if (<= b 2.6e-83)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (* c (- (/ -1.0 b) (* a (/ c (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-78) {
		tmp = b / -a;
	} else if (b <= 2.6e-83) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - (a * (c / pow(b, 3.0))));
	}
	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) :: tmp
    if (b <= (-3.7d-78)) then
        tmp = b / -a
    else if (b <= 2.6d-83) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c * (((-1.0d0) / b) - (a * (c / (b ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-78) {
		tmp = b / -a;
	} else if (b <= 2.6e-83) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - (a * (c / Math.pow(b, 3.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-78:
		tmp = b / -a
	elif b <= 2.6e-83:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c * ((-1.0 / b) - (a * (c / math.pow(b, 3.0))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-78)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.6e-83)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(a * Float64(c / (b ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.7e-78)
		tmp = b / -a;
	elseif (b <= 2.6e-83)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = c * ((-1.0 / b) - (a * (c / (b ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-78], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.6e-83], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.70000000000000006e-78

   1. Initial program 67.8%

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

      1. *-commutative 67.8%

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

   3. Simplified 67.8%

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

   5. Taylor expanded in b around -inf 89.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg 89.1%

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

      2. distribute-neg-frac2 89.1%

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

   7. Simplified 89.1%

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

   if -3.70000000000000006e-78 < b < 2.60000000000000009e-83

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in b around 0 78.6%

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

      1. *-commutative 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-*l* 78.6%

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

   7. Simplified 78.6%

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

   if 2.60000000000000009e-83 < b

   1. Initial program 13.4%

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

      1. *-commutative 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in c around 0 92.1%

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 92.1%

         \[\leadsto c \cdot \left(\color{blue}{\left(-\frac{a \cdot c}{{b}^{3}}\right)} - \frac{1}{b}\right) \]

      2. associate-/l* 93.4%

         \[\leadsto c \cdot \left(\left(-\color{blue}{a \cdot \frac{c}{{b}^{3}}}\right) - \frac{1}{b}\right) \]

   7. Simplified 93.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-75)
   (/ b (- a))
   (if (<= b 1.04e-82)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-75) {
		tmp = b / -a;
	} else if (b <= 1.04e-82) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} 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 <= (-2.3d-75)) then
        tmp = b / -a
    else if (b <= 1.04d-82) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    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 <= -2.3e-75) {
		tmp = b / -a;
	} else if (b <= 1.04e-82) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-75:
		tmp = b / -a
	elif b <= 1.04e-82:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-75)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.04e-82)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e-75)
		tmp = b / -a;
	elseif (b <= 1.04e-82)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-75], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.04e-82], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.3e-75

   1. Initial program 67.8%

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

      1. *-commutative 67.8%

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

   3. Simplified 67.8%

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

   5. Taylor expanded in b around -inf 89.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg 89.1%

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

      2. distribute-neg-frac2 89.1%

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

   7. Simplified 89.1%

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

   if -2.3e-75 < b < 1.04000000000000004e-82

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in b around 0 78.6%

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

      1. *-commutative 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-*l* 78.6%

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

   7. Simplified 78.6%

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

   if 1.04000000000000004e-82 < b

   1. Initial program 13.4%

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

      1. *-commutative 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 92.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 92.8%

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

      2. neg-mul-1 92.8%

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

   7. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 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 <= (-5d-310)) then
        tmp = 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 <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in b around -inf 67.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg 67.7%

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

      2. distribute-neg-frac2 67.7%

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

   7. Simplified 67.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 32.1%

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

      1. *-commutative 32.1%

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

   3. Simplified 32.1%

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

   5. Taylor expanded in b around inf 70.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 70.6%

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

      2. neg-mul-1 70.6%

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

   7. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \frac{b}{-a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b (- a)))

C

double code(double a, double b, double c) {
	return b / -a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / -a
end function

Java

public static double code(double a, double b, double c) {
	return b / -a;
}

Python

def code(a, b, c):
	return b / -a

Julia

function code(a, b, c)
	return Float64(b / Float64(-a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / -a;
end

Wolfram

code[a_, b_, c_] := N[(b / (-a)), $MachinePrecision]

Derivation

1. Initial program 53.9%

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

   1. *-commutative 53.9%

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

3. Simplified 53.9%

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

5. Taylor expanded in b around -inf 36.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
6. Step-by-step derivation

   1. mul-1-neg 36.9%

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

   2. distribute-neg-frac2 36.9%

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

7. Simplified 36.9%

   \[\leadsto \color{blue}{\frac{b}{-a}} \]
8. Add Preprocessing

Alternative 7: 2.6% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b a))

C

double code(double a, double b, double c) {
	return b / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / a
end function

Java

public static double code(double a, double b, double c) {
	return b / a;
}

Python

def code(a, b, c):
	return b / a

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / a;
end

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 53.9%

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

   1. *-commutative 53.9%

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

3. Simplified 53.9%

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

   1. frac-2neg 53.9%

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

   2. div-inv 53.8%

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

6. Applied egg-rr 53.8%

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

   1. *-commutative 53.8%

      \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot -4}, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2} \]

8. Simplified 53.8%

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

   1. add-cube-cbrt 53.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left(b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}} \cdot \sqrt[3]{\left(b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}}\right) \cdot \sqrt[3]{\left(b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}}} \]

   2. pow3 53.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}}\right)}^{3}} \]

   3. un-div-inv 53.2%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a \cdot -2}}}\right)}^{3} \]

10. Applied egg-rr 53.2%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, {b}^{2}\right)}}{a \cdot -2}}\right)}^{3}} \]

11. Taylor expanded in b around -inf 36.5%

    \[\leadsto {\left(\sqrt[3]{\color{blue}{-1 \cdot \frac{b}{a}}}\right)}^{3} \]
12. Step-by-step derivation

    1. mul-1-neg 36.5%

       \[\leadsto {\left(\sqrt[3]{\color{blue}{-\frac{b}{a}}}\right)}^{3} \]

13. Simplified 36.5%

    \[\leadsto {\left(\sqrt[3]{\color{blue}{-\frac{b}{a}}}\right)}^{3} \]
14. Step-by-step derivation

    1. rem-cube-cbrt 36.9%

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

    2. neg-sub0 36.9%

       \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

    3. sub-neg 36.9%

       \[\leadsto \color{blue}{0 + \left(-\frac{b}{a}\right)} \]

    4. add-cube-cbrt 36.5%

       \[\leadsto 0 + \left(-\color{blue}{\left(\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}\right) \cdot \sqrt[3]{\frac{b}{a}}}\right) \]

    5. pow3 36.5%

       \[\leadsto 0 + \left(-\color{blue}{{\left(\sqrt[3]{\frac{b}{a}}\right)}^{3}}\right) \]

    6. *-rgt-identity 36.5%

       \[\leadsto 0 + \left(-{\color{blue}{\left(\sqrt[3]{\frac{b}{a}} \cdot 1\right)}}^{3}\right) \]

    7. cube-neg 36.5%

       \[\leadsto 0 + \color{blue}{{\left(-\sqrt[3]{\frac{b}{a}} \cdot 1\right)}^{3}} \]

    8. distribute-lft-neg-out 36.5%

       \[\leadsto 0 + {\color{blue}{\left(\left(-\sqrt[3]{\frac{b}{a}}\right) \cdot 1\right)}}^{3} \]

    9. metadata-eval 36.5%

       \[\leadsto 0 + {\left(\left(-\sqrt[3]{\frac{b}{a}}\right) \cdot \color{blue}{\sqrt[3]{1}}\right)}^{3} \]

    10. metadata-eval 36.5%

        \[\leadsto 0 + {\left(\left(-\sqrt[3]{\frac{b}{a}}\right) \cdot \sqrt[3]{\color{blue}{2 \cdot 0.5}}\right)}^{3} \]

    11. cbrt-unprod 36.1%

        \[\leadsto 0 + {\left(\left(-\sqrt[3]{\frac{b}{a}}\right) \cdot \color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{0.5}\right)}\right)}^{3} \]

    12. add-sqr-sqrt 21.5%

        \[\leadsto 0 + \color{blue}{\sqrt{{\left(\left(-\sqrt[3]{\frac{b}{a}}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{0.5}\right)\right)}^{3}} \cdot \sqrt{{\left(\left(-\sqrt[3]{\frac{b}{a}}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{0.5}\right)\right)}^{3}}} \]

    13. sqrt-unprod 16.2%

        \[\leadsto 0 + \color{blue}{\sqrt{{\left(\left(-\sqrt[3]{\frac{b}{a}}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{0.5}\right)\right)}^{3} \cdot {\left(\left(-\sqrt[3]{\frac{b}{a}}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{0.5}\right)\right)}^{3}}} \]

    14. pow2 16.2%

        \[\leadsto 0 + \sqrt{\color{blue}{{\left({\left(\left(-\sqrt[3]{\frac{b}{a}}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{0.5}\right)\right)}^{3}\right)}^{2}}} \]

15. Applied egg-rr 2.4%

    \[\leadsto \color{blue}{0 + \frac{b}{a}} \]
16. Step-by-step derivation

    1. +-lft-identity 2.4%

       \[\leadsto \color{blue}{\frac{b}{a}} \]

17. Simplified 2.4%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
18. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024097 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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