Cubic critical

Percentage Accurate: 51.6% → 85.3%

Time: 19.5s

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: 51.6% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+76)
   (/ (fma b -2.0 (/ (* 1.5 a) (/ b c))) (* a 3.0))
   (if (<= b 8e-43)
     (/ (- (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 <= -1e+76) {
		tmp = fma(b, -2.0, ((1.5 * a) / (b / c))) / (a * 3.0);
	} else if (b <= 8e-43) {
		tmp = (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 <= -1e+76)
		tmp = Float64(fma(b, -2.0, Float64(Float64(1.5 * a) / Float64(b / c))) / Float64(a * 3.0));
	elseif (b <= 8e-43)
		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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+76], N[(N[(b * -2.0 + N[(N[(1.5 * a), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-43], 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 < -1e76

   1. Initial program 62.1%

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

   3. Taylor expanded in b around -inf 88.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 88.0%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 88.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 95.6%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, 1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

      4. associate-*r/ 95.6%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

   5. Simplified 95.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}}{3 \cdot a} \]

   if -1e76 < b < 8.00000000000000062e-43

   1. Initial program 88.6%

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

   if 8.00000000000000062e-43 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 10^{-42}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right) \cdot \frac{-1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-127)
   (/ (fma b -2.0 (/ (* 1.5 a) (/ b c))) (* a 3.0))
   (if (<= b 1e-42)
     (* (- (sqrt (* c (* a -3.0))) b) (/ -1.0 (* a -3.0)))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-127) {
		tmp = fma(b, -2.0, ((1.5 * a) / (b / c))) / (a * 3.0);
	} else if (b <= 1e-42) {
		tmp = (sqrt((c * (a * -3.0))) - b) * (-1.0 / (a * -3.0));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-127)
		tmp = Float64(fma(b, -2.0, Float64(Float64(1.5 * a) / Float64(b / c))) / Float64(a * 3.0));
	elseif (b <= 1e-42)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) * 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, -2.9e-127], N[(N[(b * -2.0 + N[(N[(1.5 * a), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-42], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $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 < -2.9e-127

   1. Initial program 74.3%

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

   3. Taylor expanded in b around -inf 82.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 82.3%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 82.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 87.4%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, 1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

      4. associate-*r/ 87.4%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

   5. Simplified 87.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}}{3 \cdot a} \]

   if -2.9e-127 < b < 1.00000000000000004e-42

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0 82.7%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. *-commutative 83.1%

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

   5. Simplified 83.1%

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

      1. +-commutative 83.1%

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

      2. unsub-neg 83.1%

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

   7. Applied egg-rr 83.1%

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

      1. frac-2neg 83.1%

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

      2. div-inv 83.2%

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

      3. distribute-lft-neg-in 83.2%

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

      4. metadata-eval 83.2%

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

      5. *-commutative 83.2%

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

   9. Applied egg-rr 83.2%

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

   if 1.00000000000000004e-42 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 10^{-42}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right) \cdot \frac{-1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-41}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-127)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 5.6e-41)
     (* 0.3333333333333333 (/ (+ b (sqrt (* a (* c -3.0)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-127) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.6e-41) {
		tmp = 0.3333333333333333 * ((b + 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 <= (-2.9d-127)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 5.6d-41) then
        tmp = 0.3333333333333333d0 * ((b + 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 <= -2.9e-127) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.6e-41) {
		tmp = 0.3333333333333333 * ((b + 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 <= -2.9e-127:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 5.6e-41:
		tmp = 0.3333333333333333 * ((b + 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 <= -2.9e-127)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 5.6e-41)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + 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 <= -2.9e-127)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 5.6e-41)
		tmp = 0.3333333333333333 * ((b + 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, -2.9e-127], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-41], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $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 < -2.9e-127

   1. Initial program 74.3%

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

   3. Taylor expanded in b around -inf 87.4%

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

   if -2.9e-127 < b < 5.6000000000000003e-41

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0 82.7%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. *-commutative 83.1%

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

   5. Simplified 83.1%

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

      1. *-un-lft-identity 83.1%

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

      2. *-commutative 83.1%

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

      3. times-frac 83.0%

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

      4. div-inv 82.9%

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

      5. +-commutative 82.9%

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

      6. add-sqr-sqrt 41.7%

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

      7. sqrt-unprod 82.7%

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

      8. sqr-neg 82.7%

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

      9. sqrt-unprod 41.2%

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

      10. add-sqr-sqrt 82.6%

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

      11. metadata-eval 82.6%

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

   7. Applied egg-rr 82.6%

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

      1. associate-*l/ 82.7%

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

      2. *-lft-identity 82.7%

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

      3. *-commutative 82.7%

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

      4. associate-*r/ 82.7%

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

      5. +-commutative 82.7%

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

      6. associate-*r* 82.6%

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

      7. *-commutative 82.6%

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

      8. rem-square-sqrt 0.0%

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

      9. unpow2 0.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\left(a \cdot c\right) \cdot \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}}{a} \]

      10. associate-*r* 0.0%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\color{blue}{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}}}{a} \]

      11. unpow2 0.0%

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

      12. rem-square-sqrt 82.8%

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

   9. Simplified 82.8%

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

   if 5.6000000000000003e-41 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-41}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.66 \cdot 10^{-35}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-127)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.66e-35)
     (* (- b (sqrt (* c (* a -3.0)))) (/ -0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-127) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.66e-35) {
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / 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 <= (-2.9d-127)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.66d-35) then
        tmp = (b - sqrt((c * (a * (-3.0d0))))) * ((-0.3333333333333333d0) / 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 <= -2.9e-127) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.66e-35) {
		tmp = (b - Math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-127:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.66e-35:
		tmp = (b - math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-127)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.66e-35)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) * Float64(-0.3333333333333333 / 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 <= -2.9e-127)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.66e-35)
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-127], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.66e-35], N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.9e-127

   1. Initial program 74.3%

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

   3. Taylor expanded in b around -inf 87.4%

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

   if -2.9e-127 < b < 1.65999999999999999e-35

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0 82.7%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. *-commutative 83.1%

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

   5. Simplified 83.1%

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

      1. frac-2neg 83.1%

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

      2. div-inv 83.2%

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

      3. distribute-neg-in 83.2%

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

      4. add-sqr-sqrt 41.8%

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

      5. sqrt-unprod 82.9%

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

      6. sqr-neg 82.9%

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

      7. sqrt-unprod 41.3%

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

      8. add-sqr-sqrt 82.9%

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

      9. sub-neg 82.9%

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

      10. add-sqr-sqrt 41.6%

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

      11. sqrt-unprod 82.9%

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

      12. sqr-neg 82.9%

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

      13. sqrt-unprod 41.3%

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

      14. add-sqr-sqrt 83.2%

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

      15. distribute-lft-neg-in 83.2%

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

      16. metadata-eval 83.2%

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

      17. associate-/l/ 83.0%

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

      18. div-inv 82.8%

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

      19. metadata-eval 82.8%

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

   7. Applied egg-rr 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l/ 82.9%

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

      3. metadata-eval 82.9%

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

   9. Simplified 82.9%

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

   if 1.65999999999999999e-35 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.66 \cdot 10^{-35}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-127)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 5.2e-35)
     (/ (- (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 <= -2.9e-127) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.2e-35) {
		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 <= (-2.9d-127)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 5.2d-35) 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 <= -2.9e-127) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.2e-35) {
		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 <= -2.9e-127:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 5.2e-35:
		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 <= -2.9e-127)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 5.2e-35)
		tmp = Float64(Float64(sqrt(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 <= -2.9e-127)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 5.2e-35)
		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, -2.9e-127], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-35], N[(N[(N[Sqrt[N[(c * N[(a * -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 < -2.9e-127

   1. Initial program 74.3%

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

   3. Taylor expanded in b around -inf 87.4%

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

   if -2.9e-127 < b < 5.20000000000000009e-35

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0 82.7%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. *-commutative 83.1%

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

   5. Simplified 83.1%

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

      1. +-commutative 83.1%

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

      2. unsub-neg 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 5.20000000000000009e-35 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-129}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 10^{-42}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-129)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1e-42)
     (/ (/ (- (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 <= -3.8e-129) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1e-42) {
		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 <= (-3.8d-129)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1d-42) 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 <= -3.8e-129) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1e-42) {
		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 <= -3.8e-129:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1e-42:
		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 <= -3.8e-129)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1e-42)
		tmp = Float64(Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -3.8e-129)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1e-42)
		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, -3.8e-129], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-42], N[(N[(N[(N[Sqrt[N[(a * N[(c * -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 < -3.79999999999999985e-129

   1. Initial program 74.3%

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

   3. Taylor expanded in b around -inf 87.4%

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

   if -3.79999999999999985e-129 < b < 1.00000000000000004e-42

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0 82.7%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. *-commutative 83.1%

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

   5. Simplified 83.1%

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

      1. +-commutative 83.1%

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

      2. unsub-neg 83.1%

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

   7. Applied egg-rr 83.1%

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

      1. add-cube-cbrt 81.8%

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

      2. *-commutative 81.8%

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

      3. times-frac 81.7%

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

      4. pow2 81.7%

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

   9. Applied egg-rr 81.7%

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

       1. associate-*r/ 81.8%

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

       2. associate-*l/ 81.8%

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

       3. unpow2 81.8%

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

       4. rem-3cbrt-lft 83.1%

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

       5. associate-*r* 83.0%

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

       6. *-commutative 83.0%

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

       7. associate-*l* 83.1%

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

   11. Simplified 83.1%

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

   if 1.00000000000000004e-42 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-129}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 10^{-42}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-127)
   (/ (fma b -2.0 (/ (* 1.5 a) (/ b c))) (* a 3.0))
   (if (<= b 4.4e-42)
     (/ (/ (- (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 <= -2.9e-127) {
		tmp = fma(b, -2.0, ((1.5 * a) / (b / c))) / (a * 3.0);
	} else if (b <= 4.4e-42) {
		tmp = ((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 <= -2.9e-127)
		tmp = Float64(fma(b, -2.0, Float64(Float64(1.5 * a) / Float64(b / c))) / Float64(a * 3.0));
	elseif (b <= 4.4e-42)
		tmp = Float64(Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / a) / 3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-127], N[(N[(b * -2.0 + N[(N[(1.5 * a), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-42], N[(N[(N[(N[Sqrt[N[(a * N[(c * -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 < -2.9e-127

   1. Initial program 74.3%

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

   3. Taylor expanded in b around -inf 82.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 82.3%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 82.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 87.4%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, 1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

      4. associate-*r/ 87.4%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

   5. Simplified 87.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}}{3 \cdot a} \]

   if -2.9e-127 < b < 4.4000000000000001e-42

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0 82.7%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. *-commutative 83.1%

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

   5. Simplified 83.1%

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

      1. +-commutative 83.1%

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

      2. unsub-neg 83.1%

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

   7. Applied egg-rr 83.1%

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

      1. add-cube-cbrt 81.8%

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

      2. *-commutative 81.8%

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

      3. times-frac 81.7%

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

      4. pow2 81.7%

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

   9. Applied egg-rr 81.7%

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

       1. associate-*r/ 81.8%

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

       2. associate-*l/ 81.8%

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

       3. unpow2 81.8%

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

       4. rem-3cbrt-lft 83.1%

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

       5. associate-*r* 83.0%

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

       6. *-commutative 83.0%

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

       7. associate-*l* 83.1%

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

   11. Simplified 83.1%

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

   if 4.4000000000000001e-42 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{c \cdot 0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310)
   (fma -0.6666666666666666 (/ b a) (/ (* c 0.5) b))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = fma(-0.6666666666666666, (b / a), ((c * 0.5) / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = fma(-0.6666666666666666, Float64(b / a), Float64(Float64(c * 0.5) / b));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 78.4%

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

   3. Taylor expanded in b around -inf 68.2%

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

      1. fma-def 68.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)} \]

      2. associate-*r/ 68.2%

         \[\leadsto \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \color{blue}{\frac{0.5 \cdot c}{b}}\right) \]

   5. Simplified 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{0.5 \cdot c}{b}\right)} \]

   if -9.999999999999969e-311 < b

   1. Initial program 33.5%

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

   3. Taylor expanded in b around inf 69.0%

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

      1. associate-*r/ 69.0%

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

   5. Simplified 69.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{c \cdot 0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / 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 <= (-1d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / 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 <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / 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 <= -1e-310)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $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 < -9.999999999999969e-311

   1. Initial program 78.4%

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

   3. Taylor expanded in b around -inf 68.2%

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

   if -9.999999999999969e-311 < b

   1. Initial program 33.5%

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

   3. Taylor expanded in b around inf 69.0%

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

      1. associate-*r/ 69.0%

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

   5. Simplified 69.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (/ (* b -2.0) (* a 3.0)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = (b * -2.0) / (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 <= 5.1d-245) then
        tmp = (b * (-2.0d0)) / (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 <= 5.1e-245) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(Float64(b * -2.0) / 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 <= 5.1e-245)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

   3. Taylor expanded in b around -inf 66.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   5. Simplified 66.3%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

   3. Taylor expanded in b around inf 70.6%

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

      1. associate-*r/ 70.6%

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

   5. Simplified 70.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (/ (/ (* b -2.0) a) 3.0) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = ((b * -2.0) / 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 <= 5.1d-245) then
        tmp = ((b * (-2.0d0)) / 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 <= 5.1e-245) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = ((b * -2.0) / a) / 3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(Float64(Float64(b * -2.0) / 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 <= 5.1e-245)
		tmp = ((b * -2.0) / a) / 3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(N[(N[(b * -2.0), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

   4. Applied egg-rr 66.6%

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

      1. associate-/r* 66.5%

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

   6. Simplified 66.5%

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

      1. associate-*r/ 66.5%

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

      2. frac-2neg 66.5%

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

      3. un-div-inv 66.5%

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

      4. metadata-eval 66.5%

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

   8. Applied egg-rr 66.5%

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

      1. distribute-neg-frac 66.5%

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

      2. sub-neg 66.5%

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

      3. +-commutative 66.5%

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

      4. distribute-neg-in 66.5%

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

      5. remove-double-neg 66.5%

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

      6. sub-neg 66.5%

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

      7. associate-*r* 66.4%

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

      8. *-commutative 66.4%

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

      9. rem-square-sqrt 0.0%

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

      10. unpow2 0.0%

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

      11. associate-*r* 0.0%

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

      12. unpow2 0.0%

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

      13. rem-square-sqrt 66.5%

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

   10. Simplified 66.5%

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

   11. Taylor expanded in b around -inf 66.3%

       \[\leadsto \frac{\frac{\color{blue}{-2 \cdot b}}{a}}{3} \]
   12. Step-by-step derivation

       1. *-commutative 66.3%

          \[\leadsto \frac{\frac{\color{blue}{b \cdot -2}}{a}}{3} \]

   13. Simplified 66.3%

       \[\leadsto \frac{\frac{\color{blue}{b \cdot -2}}{a}}{3} \]

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

   3. Taylor expanded in b around inf 70.6%

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

      1. associate-*r/ 70.6%

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

   5. Simplified 70.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		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 <= 5.1d-245) 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 <= 5.1e-245) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-245)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

   3. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   5. Simplified 66.3%

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

      1. *-commutative 66.3%

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

      2. clear-num 66.2%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 66.3%

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

   7. Applied egg-rr 66.3%

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

      1. associate-/r/ 66.3%

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

   9. Simplified 66.3%

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

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

   3. Taylor expanded in b around inf 20.0%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

   4. Taylor expanded in b around 0 70.6%

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

      1. associate-*r/ 70.6%

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

      2. *-rgt-identity 70.6%

         \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b \cdot 1}} \]

      3. times-frac 70.4%

         \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot \frac{c}{1}} \]

      4. /-rgt-identity 70.4%

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

   6. Simplified 70.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (* -0.6666666666666666 (/ b a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = -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 <= 5.1d-245) then
        tmp = (-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 <= 5.1e-245) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-245)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

   3. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   5. Simplified 66.3%

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

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

   3. Taylor expanded in b around inf 20.0%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

   4. Taylor expanded in b around 0 70.6%

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

      1. associate-*r/ 70.6%

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

      2. *-rgt-identity 70.6%

         \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b \cdot 1}} \]

      3. times-frac 70.4%

         \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot \frac{c}{1}} \]

      4. /-rgt-identity 70.4%

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

   6. Simplified 70.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (/ -0.6666666666666666 (/ a b)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = -0.6666666666666666 / (a / 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 <= 5.1d-245) then
        tmp = (-0.6666666666666666d0) / (a / 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 <= 5.1e-245) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-245)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

   3. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   5. Simplified 66.3%

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

      1. *-commutative 66.3%

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

      2. clear-num 66.2%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 66.3%

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

   7. Applied egg-rr 66.3%

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

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

   3. Taylor expanded in b around inf 20.0%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

   4. Taylor expanded in b around 0 70.6%

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

      1. associate-*r/ 70.6%

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

      2. *-rgt-identity 70.6%

         \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b \cdot 1}} \]

      3. times-frac 70.4%

         \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot \frac{c}{1}} \]

      4. /-rgt-identity 70.4%

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

   6. Simplified 70.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (/ -0.6666666666666666 (/ a b)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = -0.6666666666666666 / (a / 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 <= 5.1d-245) then
        tmp = (-0.6666666666666666d0) / (a / 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 <= 5.1e-245) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(-0.6666666666666666 / Float64(a / 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 <= 5.1e-245)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

   3. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   5. Simplified 66.3%

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

      1. *-commutative 66.3%

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

      2. clear-num 66.2%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 66.3%

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

   7. Applied egg-rr 66.3%

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

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

   3. Taylor expanded in b around inf 70.6%

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

      1. associate-*r/ 70.6%

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

   5. Simplified 70.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.0% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{-0.6666666666666666}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* b (/ -0.6666666666666666 a)))

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

public static double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / a);
}

Python

def code(a, b, c):
	return b * (-0.6666666666666666 / a)

Julia

function code(a, b, c)
	return Float64(b * Float64(-0.6666666666666666 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b * (-0.6666666666666666 / a);
end

Wolfram

code[a_, b_, c_] := N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in b around -inf 35.4%

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

   1. *-commutative 35.4%

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

5. Simplified 35.4%

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

   1. *-commutative 35.4%

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

   2. clear-num 35.4%

      \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

   3. un-div-inv 35.5%

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

7. Applied egg-rr 35.5%

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

   1. associate-/r/ 35.4%

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

9. Simplified 35.4%

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

10. Final simplification 35.4%

    \[\leadsto b \cdot \frac{-0.6666666666666666}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))