Cubic critical

Percentage Accurate: 52.8% → 85.3%

Time: 21.9s

Alternatives: 16

Speedup: 11.6×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-112}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+154)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (if (<= b 6.2e-112)
     (* (- b (sqrt (fma c (* a -3.0) (pow b 2.0)))) (/ 1.0 (* a -3.0)))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+154) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 6.2e-112) {
		tmp = (b - sqrt(fma(c, (a * -3.0), pow(b, 2.0)))) * (1.0 / (a * -3.0));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+154)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 6.2e-112)
		tmp = Float64(Float64(b - sqrt(fma(c, Float64(a * -3.0), (b ^ 2.0)))) * Float64(1.0 / Float64(a * -3.0)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+154], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-112], N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000007e154

   1. Initial program 45.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 45.8%

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

      2. sqr-neg 45.8%

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

      3. associate-*l* 45.8%

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

   3. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 99.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -2.00000000000000007e154 < b < 6.1999999999999995e-112

   1. Initial program 82.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 82.7%

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

      2. sqr-neg 82.7%

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

      3. associate-*l* 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 82.6%

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

      2. div-inv 82.6%

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

   6. Applied egg-rr 82.7%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. fma-undefine 82.7%

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

      2. unpow2 82.7%

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

      3. +-commutative 82.7%

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

      4. fma-define 82.7%

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

   8. Simplified 82.7%

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

   if 6.1999999999999995e-112 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 17.8%

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

      2. sqr-neg 17.8%

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

      3. associate-*l* 17.8%

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

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-112}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e+153)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (if (<= b 3.8e-117)
     (* (/ 1.0 (* a -3.0)) (- b (sqrt (fma b b (* c (* a -3.0))))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e+153) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 3.8e-117) {
		tmp = (1.0 / (a * -3.0)) * (b - sqrt(fma(b, b, (c * (a * -3.0)))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e+153)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 3.8e-117)
		tmp = Float64(Float64(1.0 / Float64(a * -3.0)) * Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e+153], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-117], N[(N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.20000000000000033e153

   1. Initial program 45.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 45.8%

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

      2. sqr-neg 45.8%

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

      3. associate-*l* 45.8%

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

   3. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 99.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -8.20000000000000033e153 < b < 3.79999999999999972e-117

   1. Initial program 82.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 82.7%

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

      2. sqr-neg 82.7%

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

      3. associate-*l* 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 82.6%

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

      2. div-inv 82.6%

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

   6. Applied egg-rr 82.7%

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

   if 3.79999999999999972e-117 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 17.8%

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

      2. sqr-neg 17.8%

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

      3. associate-*l* 17.8%

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

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+143}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+143)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (if (<= b 3.2e-117)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+143) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 3.2e-117) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / 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 <= (-4d+143)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + (0.5d0 * (c / b))
    else if (b <= 3.2d-117) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+143) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 3.2e-117) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e+143:
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b))
	elif b <= 3.2e-117:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+143)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 3.2e-117)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e+143)
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	elseif (b <= 3.2e-117)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+143], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-117], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.0000000000000001e143

   1. Initial program 48.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 48.3%

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

      2. sqr-neg 48.3%

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

      3. associate-*l* 48.3%

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

   3. Simplified 48.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 99.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -4.0000000000000001e143 < b < 3.19999999999999995e-117

   1. Initial program 82.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-*l* 82.3%

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

   3. Simplified 82.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   if 3.19999999999999995e-117 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 17.8%

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

      2. sqr-neg 17.8%

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

      3. associate-*l* 17.8%

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

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+143}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+142}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+142)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (if (<= b 1.02e-112)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+142) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 1.02e-112) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / 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 <= (-2.8d+142)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + (0.5d0 * (c / b))
    else if (b <= 1.02d-112) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+142) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 1.02e-112) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e+142:
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b))
	elif b <= 1.02e-112:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+142)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.02e-112)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e+142)
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	elseif (b <= 1.02e-112)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+142], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.02e-112], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8e142

   1. Initial program 48.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 48.3%

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

      2. sqr-neg 48.3%

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

      3. associate-*l* 48.3%

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

   3. Simplified 48.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 99.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -2.8e142 < b < 1.01999999999999996e-112

   1. Initial program 82.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   if 1.01999999999999996e-112 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 17.8%

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

      2. sqr-neg 17.8%

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

      3. associate-*l* 17.8%

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

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+142}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-112}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.3e-45)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (if (<= b 1.36e-112)
     (* 0.3333333333333333 (/ (+ b (sqrt (* -3.0 (* a c)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e-45) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 1.36e-112) {
		tmp = 0.3333333333333333 * ((b + sqrt((-3.0 * (a * c)))) / a);
	} else {
		tmp = (c * -0.5) / 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 <= (-1.3d-45)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + (0.5d0 * (c / b))
    else if (b <= 1.36d-112) then
        tmp = 0.3333333333333333d0 * ((b + sqrt(((-3.0d0) * (a * c)))) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e-45) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 1.36e-112) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((-3.0 * (a * c)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.3e-45:
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b))
	elif b <= 1.36e-112:
		tmp = 0.3333333333333333 * ((b + math.sqrt((-3.0 * (a * c)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.3e-45)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.36e-112)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(-3.0 * Float64(a * c)))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.3e-45)
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	elseif (b <= 1.36e-112)
		tmp = 0.3333333333333333 * ((b + sqrt((-3.0 * (a * c)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.3e-45], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.36e-112], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.29999999999999993e-45

   1. Initial program 67.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 67.7%

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

      2. sqr-neg 67.7%

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

      3. associate-*l* 67.7%

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

   3. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 89.5%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.6%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -1.29999999999999993e-45 < b < 1.3600000000000001e-112

   1. Initial program 77.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 77.9%

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

      2. sqr-neg 77.9%

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

      3. associate-*l* 77.9%

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

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-un-lft-identity 77.9%

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

      2. times-frac 77.9%

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

      3. metadata-eval 77.9%

         \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a} \]

      4. add-sqr-sqrt 53.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a} \]

      5. sqrt-unprod 76.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a} \]

      6. sqr-neg 76.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{b \cdot b}} + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a} \]

      7. sqrt-prod 25.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a} \]

      8. add-sqr-sqrt 68.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{b} + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a} \]

      9. fma-neg 68.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      10. associate-*r* 68.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(3 \cdot a\right) \cdot c}\right)}}{a} \]

      11. *-commutative 68.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)}}{a} \]

      12. distribute-rgt-neg-in 68.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)}}{a} \]

      13. *-commutative 68.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)}}{a} \]

      14. distribute-rgt-neg-in 68.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)}}{a} \]

      15. metadata-eval 68.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)}}{a} \]

   6. Applied egg-rr 68.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}{a}} \]

   7. Taylor expanded in b around 0 68.1%

      \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{a} \]

   if 1.3600000000000001e-112 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 17.8%

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

      2. sqr-neg 17.8%

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

      3. associate-*l* 17.8%

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

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-112}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e-43)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (if (<= b 4e-112)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-43) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 4e-112) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / 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 <= (-5.2d-43)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + (0.5d0 * (c / b))
    else if (b <= 4d-112) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-43) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 4e-112) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.2e-43:
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b))
	elif b <= 4e-112:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e-43)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 4e-112)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.2e-43)
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	elseif (b <= 4e-112)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e-43], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-112], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2e-43

   1. Initial program 67.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 67.7%

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

      2. sqr-neg 67.7%

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

      3. associate-*l* 67.7%

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

   3. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 89.5%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.6%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -5.2e-43 < b < 3.9999999999999998e-112

   1. Initial program 77.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 77.9%

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

      2. sqr-neg 77.9%

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

      3. associate-*l* 77.9%

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

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. prod-diff 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. *-commutative 77.6%

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

      5. distribute-rgt-neg-in 77.6%

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

      6. *-commutative 77.6%

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

      7. distribute-rgt-neg-in 77.6%

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

      8. metadata-eval 77.6%

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

      9. *-commutative 77.6%

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

      10. fma-undefine 77.7%

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

      11. distribute-lft-neg-in 77.7%

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

      12. *-commutative 77.7%

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

      13. associate-*r* 77.5%

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

      14. *-commutative 77.5%

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

      15. distribute-rgt-neg-in 77.5%

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

      16. *-commutative 77.5%

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

      17. distribute-rgt-neg-in 77.5%

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

      18. metadata-eval 77.5%

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

      19. associate-*r* 77.7%

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

      20. *-commutative 77.7%

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

      21. associate-*l* 77.6%

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

   6. Applied egg-rr 77.6%

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

   7. Taylor expanded in b around 0 70.0%

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

      1. mul-1-neg 70.0%

         \[\leadsto \frac{\sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)} + \color{blue}{\left(-b\right)}}{3 \cdot a} \]

      2. unsub-neg 70.0%

         \[\leadsto \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)} - b}}{3 \cdot a} \]

      3. distribute-rgt-out 70.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}} - b}{3 \cdot a} \]

      4. *-commutative 70.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)} - b}{3 \cdot a} \]

      5. metadata-eval 70.3%

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

      6. associate-*r* 70.4%

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

      7. *-commutative 70.4%

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

      8. associate-*l* 70.3%

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

      9. *-commutative 70.3%

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

   9. Simplified 70.3%

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

   if 3.9999999999999998e-112 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 17.8%

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

      2. sqr-neg 17.8%

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

      3. associate-*l* 17.8%

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

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-36}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.25e-36)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (if (<= b 6.2e-112)
     (/ (/ (- (sqrt (* c (* a -3.0))) b) a) 3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e-36) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 6.2e-112) {
		tmp = ((sqrt((c * (a * -3.0))) - b) / a) / 3.0;
	} else {
		tmp = (c * -0.5) / 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 <= (-2.25d-36)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + (0.5d0 * (c / b))
    else if (b <= 6.2d-112) then
        tmp = ((sqrt((c * (a * (-3.0d0)))) - b) / a) / 3.0d0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e-36) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 6.2e-112) {
		tmp = ((Math.sqrt((c * (a * -3.0))) - b) / a) / 3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.25e-36:
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b))
	elif b <= 6.2e-112:
		tmp = ((math.sqrt((c * (a * -3.0))) - b) / a) / 3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.25e-36)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 6.2e-112)
		tmp = Float64(Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / a) / 3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.25e-36)
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	elseif (b <= 6.2e-112)
		tmp = ((sqrt((c * (a * -3.0))) - b) / a) / 3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.25e-36], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-112], N[(N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.25000000000000012e-36

   1. Initial program 67.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 67.7%

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

      2. sqr-neg 67.7%

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

      3. associate-*l* 67.7%

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

   3. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 89.5%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.6%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -2.25000000000000012e-36 < b < 6.1999999999999995e-112

   1. Initial program 77.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 77.9%

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

      2. sqr-neg 77.9%

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

      3. associate-*l* 77.9%

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

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. prod-diff 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. *-commutative 77.6%

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

      5. distribute-rgt-neg-in 77.6%

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

      6. *-commutative 77.6%

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

      7. distribute-rgt-neg-in 77.6%

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

      8. metadata-eval 77.6%

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

      9. *-commutative 77.6%

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

      10. fma-undefine 77.7%

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

      11. distribute-lft-neg-in 77.7%

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

      12. *-commutative 77.7%

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

      13. associate-*r* 77.5%

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

      14. *-commutative 77.5%

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

      15. distribute-rgt-neg-in 77.5%

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

      16. *-commutative 77.5%

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

      17. distribute-rgt-neg-in 77.5%

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

      18. metadata-eval 77.5%

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

      19. associate-*r* 77.7%

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

      20. *-commutative 77.7%

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

      21. associate-*l* 77.6%

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

   6. Applied egg-rr 77.6%

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

   7. Taylor expanded in b around 0 70.0%

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

      1. mul-1-neg 70.0%

         \[\leadsto \frac{\sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)} + \color{blue}{\left(-b\right)}}{3 \cdot a} \]

      2. unsub-neg 70.0%

         \[\leadsto \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)} - b}}{3 \cdot a} \]

      3. distribute-rgt-out 70.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}} - b}{3 \cdot a} \]

      4. *-commutative 70.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)} - b}{3 \cdot a} \]

      5. metadata-eval 70.3%

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

      6. associate-*r* 70.4%

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

      7. *-commutative 70.4%

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

      8. associate-*l* 70.3%

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

      9. *-commutative 70.3%

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

   9. Simplified 70.3%

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

       1. add-cube-cbrt 69.3%

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

       2. *-commutative 69.3%

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

       3. times-frac 69.3%

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

       4. pow2 69.3%

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

       5. *-commutative 69.3%

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

       6. associate-*r* 69.3%

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

       7. *-commutative 69.3%

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

   11. Applied egg-rr 69.4%

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

       1. associate-*r/ 69.5%

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

       2. associate-*l/ 69.5%

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

       3. unpow2 69.5%

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

       4. rem-3cbrt-lft 70.4%

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

   13. Simplified 70.4%

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

   if 6.1999999999999995e-112 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 17.8%

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

      2. sqr-neg 17.8%

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

      3. associate-*l* 17.8%

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

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-36}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-112}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.4e-44)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (if (<= b 6.2e-112)
     (* 0.3333333333333333 (/ (sqrt (* a (* c -3.0))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e-44) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 6.2e-112) {
		tmp = 0.3333333333333333 * (sqrt((a * (c * -3.0))) / a);
	} else {
		tmp = (c * -0.5) / 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 <= (-6.4d-44)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + (0.5d0 * (c / b))
    else if (b <= 6.2d-112) then
        tmp = 0.3333333333333333d0 * (sqrt((a * (c * (-3.0d0)))) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e-44) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 6.2e-112) {
		tmp = 0.3333333333333333 * (Math.sqrt((a * (c * -3.0))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.4e-44:
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b))
	elif b <= 6.2e-112:
		tmp = 0.3333333333333333 * (math.sqrt((a * (c * -3.0))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.4e-44)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 6.2e-112)
		tmp = Float64(0.3333333333333333 * Float64(sqrt(Float64(a * Float64(c * -3.0))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.4e-44)
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	elseif (b <= 6.2e-112)
		tmp = 0.3333333333333333 * (sqrt((a * (c * -3.0))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.4e-44], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-112], N[(0.3333333333333333 * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.3999999999999999e-44

   1. Initial program 67.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 67.7%

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

      2. sqr-neg 67.7%

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

      3. associate-*l* 67.7%

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

   3. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 89.5%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.6%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -6.3999999999999999e-44 < b < 6.1999999999999995e-112

   1. Initial program 77.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 77.9%

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

      2. sqr-neg 77.9%

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

      3. associate-*l* 77.9%

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

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. prod-diff 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. *-commutative 77.6%

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

      5. distribute-rgt-neg-in 77.6%

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

      6. *-commutative 77.6%

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

      7. distribute-rgt-neg-in 77.6%

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

      8. metadata-eval 77.6%

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

      9. *-commutative 77.6%

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

      10. fma-undefine 77.7%

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

      11. distribute-lft-neg-in 77.7%

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

      12. *-commutative 77.7%

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

      13. associate-*r* 77.5%

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

      14. *-commutative 77.5%

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

      15. distribute-rgt-neg-in 77.5%

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

      16. *-commutative 77.5%

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

      17. distribute-rgt-neg-in 77.5%

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

      18. metadata-eval 77.5%

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

      19. associate-*r* 77.7%

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

      20. *-commutative 77.7%

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

      21. associate-*l* 77.6%

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

   6. Applied egg-rr 77.6%

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

   7. Taylor expanded in b around 0 67.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*l/ 67.8%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}{a}} \]

      2. *-lft-identity 67.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}}{a} \]

      3. distribute-rgt-out 68.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      4. *-commutative 68.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)}}{a} \]

      5. metadata-eval 68.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(c \cdot a\right) \cdot \color{blue}{-3}}}{a} \]

      6. associate-*r* 68.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]

      7. *-commutative 68.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot -3\right) \cdot c}}}{a} \]

      8. associate-*l* 68.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{a \cdot \left(-3 \cdot c\right)}}}{a} \]

      9. *-commutative 68.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{a \cdot \color{blue}{\left(c \cdot -3\right)}}}{a} \]

   9. Simplified 68.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}} \]

   if 6.1999999999999995e-112 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 17.8%

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

      2. sqr-neg 17.8%

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

      3. associate-*l* 17.8%

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

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-112}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-219}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{c \cdot -3}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.85e-219)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (if (<= b 1.02e-122)
     (* 0.3333333333333333 (sqrt (/ (* c -3.0) a)))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e-219) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 1.02e-122) {
		tmp = 0.3333333333333333 * sqrt(((c * -3.0) / a));
	} else {
		tmp = (c * -0.5) / 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 <= (-1.85d-219)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + (0.5d0 * (c / b))
    else if (b <= 1.02d-122) then
        tmp = 0.3333333333333333d0 * sqrt(((c * (-3.0d0)) / a))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e-219) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else if (b <= 1.02e-122) {
		tmp = 0.3333333333333333 * Math.sqrt(((c * -3.0) / a));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.85e-219:
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b))
	elif b <= 1.02e-122:
		tmp = 0.3333333333333333 * math.sqrt(((c * -3.0) / a))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.85e-219)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.02e-122)
		tmp = Float64(0.3333333333333333 * sqrt(Float64(Float64(c * -3.0) / a)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.85e-219)
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	elseif (b <= 1.02e-122)
		tmp = 0.3333333333333333 * sqrt(((c * -3.0) / a));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.85e-219], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.02e-122], N[(0.3333333333333333 * N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.85e-219

   1. Initial program 72.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 72.9%

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

      2. sqr-neg 72.9%

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

      3. associate-*l* 72.9%

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

   3. Simplified 72.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 74.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 74.3%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -1.85e-219 < b < 1.02000000000000002e-122

   1. Initial program 69.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 69.3%

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

      2. sqr-neg 69.3%

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

      3. associate-*l* 69.2%

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

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. prod-diff 68.8%

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

      2. *-commutative 68.8%

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

      3. associate-*r* 68.7%

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

      4. *-commutative 68.7%

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

      5. distribute-rgt-neg-in 68.7%

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

      6. *-commutative 68.7%

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

      7. distribute-rgt-neg-in 68.7%

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

      8. metadata-eval 68.7%

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

      9. *-commutative 68.7%

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

      10. fma-undefine 68.8%

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

      11. distribute-lft-neg-in 68.8%

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

      12. *-commutative 68.8%

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

      13. associate-*r* 68.6%

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

      14. *-commutative 68.6%

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

      15. distribute-rgt-neg-in 68.6%

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

      16. *-commutative 68.6%

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

      17. distribute-rgt-neg-in 68.6%

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

      18. metadata-eval 68.6%

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

      19. associate-*r* 68.8%

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

      20. *-commutative 68.8%

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

      21. associate-*l* 68.6%

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

   6. Applied egg-rr 68.6%

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

   7. Taylor expanded in a around inf 36.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{-6 \cdot c + 3 \cdot c}{a}}} \]
   8. Step-by-step derivation

      1. distribute-rgt-out 36.4%

         \[\leadsto 0.3333333333333333 \cdot \sqrt{\frac{\color{blue}{c \cdot \left(-6 + 3\right)}}{a}} \]

      2. metadata-eval 36.4%

         \[\leadsto 0.3333333333333333 \cdot \sqrt{\frac{c \cdot \color{blue}{-3}}{a}} \]

   9. Simplified 36.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{c \cdot -3}{a}}} \]

   if 1.02000000000000002e-122 < b

   1. Initial program 18.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 86.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-219}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{c \cdot -3}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311)
   (+ (* 0.5 (/ c b)) (* -0.6666666666666666 (/ b a)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	} else {
		tmp = (c * -0.5) / 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-311)) then
        tmp = (0.5d0 * (c / b)) + ((-0.6666666666666666d0) * (b / a))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(Float64(0.5 * Float64(c / b)) + Float64(-0.6666666666666666 * Float64(b / a)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 73.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 73.2%

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

      2. sqr-neg 73.2%

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

      3. associate-*l* 73.1%

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

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 68.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -5.00000000000023e-311 < b

   1. Initial program 26.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 26.8%

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

      2. sqr-neg 26.8%

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

      3. associate-*l* 26.8%

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

   3. Simplified 26.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 73.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 74.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311)
   (+ (/ (* b -0.6666666666666666) a) (* 0.5 (/ c b)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / 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-311)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + (0.5d0 * (c / b))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(0.5 * Float64(c / b)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = ((b * -0.6666666666666666) / a) + (0.5 * (c / b));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 73.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 73.2%

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

      2. sqr-neg 73.2%

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

      3. associate-*l* 73.1%

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

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 68.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 68.6%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   7. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -5.00000000000023e-311 < b

   1. Initial program 26.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 26.8%

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

      2. sqr-neg 26.8%

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

      3. associate-*l* 26.8%

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

   3. Simplified 26.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 73.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 74.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{1}{a \cdot -3} \cdot \left(b \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (* (/ 1.0 (* a -3.0)) (* b 2.0)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (1.0 / (a * -3.0)) * (b * 2.0);
	} else {
		tmp = (c * -0.5) / 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-311)) then
        tmp = (1.0d0 / (a * (-3.0d0))) * (b * 2.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (1.0 / (a * -3.0)) * (b * 2.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = (1.0 / (a * -3.0)) * (b * 2.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(Float64(1.0 / Float64(a * -3.0)) * Float64(b * 2.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = (1.0 / (a * -3.0)) * (b * 2.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision] * N[(b * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 73.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 73.2%

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

      2. sqr-neg 73.2%

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

      3. associate-*l* 73.1%

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

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 73.1%

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

      2. div-inv 73.1%

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

   6. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. fma-undefine 73.2%

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

      2. unpow2 73.2%

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

      3. +-commutative 73.2%

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

      4. fma-define 73.2%

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

   8. Simplified 73.2%

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

   9. Taylor expanded in b around -inf 68.0%

      \[\leadsto \color{blue}{\left(2 \cdot b\right)} \cdot \frac{1}{a \cdot -3} \]
   10. Step-by-step derivation

       1. *-commutative 68.0%

          \[\leadsto \color{blue}{\left(b \cdot 2\right)} \cdot \frac{1}{a \cdot -3} \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{\left(b \cdot 2\right)} \cdot \frac{1}{a \cdot -3} \]

   if -5.00000000000023e-311 < b

   1. Initial program 26.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 26.8%

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

      2. sqr-neg 26.8%

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

      3. associate-*l* 26.8%

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

   3. Simplified 26.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 73.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 74.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{1}{a \cdot -3} \cdot \left(b \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (* b (/ -0.6666666666666666 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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-311)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 73.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 73.2%

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

      2. sqr-neg 73.2%

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

      3. associate-*l* 73.1%

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

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 73.1%

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

      2. div-inv 73.1%

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

   6. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. fma-undefine 73.2%

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

      2. unpow2 73.2%

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

      3. +-commutative 73.2%

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

      4. fma-define 73.2%

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

   8. Simplified 73.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 73.2%

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

      2. fma-undefine 73.2%

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

      3. add-sqr-sqrt 57.4%

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

      4. unpow2 57.4%

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

      5. hypot-define 67.8%

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

   10. Applied egg-rr 67.8%

       \[\leadsto \left(b - \color{blue}{1 \cdot \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -3\right)}, b\right)}\right) \cdot \frac{1}{a \cdot -3} \]
   11. Step-by-step derivation

       1. *-lft-identity 67.8%

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

   12. Simplified 67.8%

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

   13. Taylor expanded in a around 0 67.6%

       \[\leadsto \left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -3\right)}, b\right)\right) \cdot \color{blue}{\frac{-0.3333333333333333}{a}} \]

   14. Taylor expanded in b around -inf 67.9%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   15. Step-by-step derivation

       1. associate-*r/ 68.0%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. associate-*l/ 68.0%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

       3. *-commutative 68.0%

          \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   16. Simplified 68.0%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   if -5.00000000000023e-311 < b

   1. Initial program 26.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 26.8%

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

      2. sqr-neg 26.8%

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

      3. associate-*l* 26.8%

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

   3. Simplified 26.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 73.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (/ (* b -0.6666666666666666) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	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-311)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 73.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 73.2%

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

      2. sqr-neg 73.2%

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

      3. associate-*l* 73.1%

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

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 73.1%

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

      2. div-inv 73.1%

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

   6. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. fma-undefine 73.2%

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

      2. unpow2 73.2%

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

      3. +-commutative 73.2%

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

      4. fma-define 73.2%

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

   8. Simplified 73.2%

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

   9. Taylor expanded in b around -inf 67.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 68.0%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

   if -5.00000000000023e-311 < b

   1. Initial program 26.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 26.8%

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

      2. sqr-neg 26.8%

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

      3. associate-*l* 26.8%

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

   3. Simplified 26.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 73.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / 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-311)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 73.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 73.2%

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

      2. sqr-neg 73.2%

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

      3. associate-*l* 73.1%

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

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 73.1%

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

      2. div-inv 73.1%

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

   6. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. fma-undefine 73.2%

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

      2. unpow2 73.2%

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

      3. +-commutative 73.2%

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

      4. fma-define 73.2%

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

   8. Simplified 73.2%

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

   9. Taylor expanded in b around -inf 67.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 68.0%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

   if -5.00000000000023e-311 < b

   1. Initial program 26.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 26.8%

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

      2. sqr-neg 26.8%

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

      3. associate-*l* 26.8%

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

   3. Simplified 26.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 73.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 74.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   7. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -0.5))

C

double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c / b) * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -0.5;
end

Wolfram

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

Derivation

1. Initial program 49.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. sqr-neg 49.3%

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

   2. sqr-neg 49.3%

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

   3. associate-*l* 49.3%

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

3. Simplified 49.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 39.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]

6. Final simplification 39.2%

   \[\leadsto \frac{c}{b} \cdot -0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))