Cubic critical

Percentage Accurate: 51.0% → 85.2%

Time: 13.0s

Alternatives: 12

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: 51.0% 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: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+100)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 3.5e-29)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+100) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 3.5e-29) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * 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 <= (-5.2d+100)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 3.5d-29) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * 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 <= -5.2e+100) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 3.5e-29) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.2e+100:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 3.5e-29:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - 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+100)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 3.5e-29)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * 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 <= -5.2e+100)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 3.5e-29)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - 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+100], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.5e-29], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $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 < -5.2000000000000003e100

   1. Initial program 55.9%

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

      1. add-cube-cbrt 55.9%

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

      2. pow3 55.9%

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

   4. Applied egg-rr 55.9%

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

   5. Taylor expanded in b around -inf 96.0%

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

      1. associate-*r/ 96.1%

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

      2. *-commutative 96.1%

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

   7. Simplified 96.1%

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

   if -5.2000000000000003e100 < b < 3.4999999999999997e-29

   1. Initial program 83.4%

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

   if 3.4999999999999997e-29 < b

   1. Initial program 12.2%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in b around inf 93.4%

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

      1. associate-*r/ 93.4%

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

   7. Applied egg-rr 93.4%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+100)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.95e-28)
     (/ (- (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 <= -5.2e+100) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.95e-28) {
		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 <= (-5.2d+100)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.95d-28) 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 <= -5.2e+100) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.95e-28) {
		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 <= -5.2e+100:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.95e-28:
		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 <= -5.2e+100)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.95e-28)
		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 <= -5.2e+100)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.95e-28)
		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, -5.2e+100], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.95e-28], 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 < -5.2000000000000003e100

   1. Initial program 55.9%

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

      1. add-cube-cbrt 55.9%

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

      2. pow3 55.9%

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

   4. Applied egg-rr 55.9%

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

   5. Taylor expanded in b around -inf 96.0%

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

      1. associate-*r/ 96.1%

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

      2. *-commutative 96.1%

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

   7. Simplified 96.1%

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

   if -5.2000000000000003e100 < b < 1.94999999999999999e-28

   1. Initial program 83.4%

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

      1. sqr-neg 83.4%

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

      2. sqr-neg 83.4%

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

      3. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

   if 1.94999999999999999e-28 < b

   1. Initial program 12.2%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in b around inf 93.4%

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

      1. associate-*r/ 93.4%

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

   7. Applied egg-rr 93.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e-62)
   (/ (/ b a) -1.5)
   (if (<= b 5.8e-29)
     (/ (- (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 <= -1.75e-62) {
		tmp = (b / a) / -1.5;
	} else if (b <= 5.8e-29) {
		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 <= (-1.75d-62)) then
        tmp = (b / a) / (-1.5d0)
    else if (b <= 5.8d-29) 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 <= -1.75e-62) {
		tmp = (b / a) / -1.5;
	} else if (b <= 5.8e-29) {
		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 <= -1.75e-62:
		tmp = (b / a) / -1.5
	elif b <= 5.8e-29:
		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 <= -1.75e-62)
		tmp = Float64(Float64(b / a) / -1.5);
	elseif (b <= 5.8e-29)
		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 <= -1.75e-62)
		tmp = (b / a) / -1.5;
	elseif (b <= 5.8e-29)
		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, -1.75e-62], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], If[LessEqual[b, 5.8e-29], 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 < -1.7500000000000001e-62

   1. Initial program 72.0%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in b around -inf 87.6%

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

      1. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. add-cube-cbrt 86.6%

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

      2. pow3 86.7%

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

      3. *-commutative 86.7%

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

      4. times-frac 86.6%

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

      5. metadata-eval 86.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{-0.6666666666666666} \cdot \frac{b}{a}}\right)}^{3} \]

   9. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-0.6666666666666666 \cdot \frac{b}{a}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 87.5%

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

       2. clear-num 87.4%

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

       3. div-inv 87.6%

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

       4. metadata-eval 87.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{\frac{a}{b}} \]

       5. associate-/r* 87.5%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

       6. *-commutative 87.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       7. associate-/r* 87.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{a}{b}}}{-1.5}} \]

       8. clear-num 87.6%

          \[\leadsto \frac{\color{blue}{\frac{b}{a}}}{-1.5} \]

   11. Applied egg-rr 87.6%

       \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

   if -1.7500000000000001e-62 < b < 5.80000000000000048e-29

   1. Initial program 78.4%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in b around 0 73.0%

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

      1. associate-*r* 73.2%

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

      2. *-commutative 73.2%

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

   7. Simplified 73.2%

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

   if 5.80000000000000048e-29 < b

   1. Initial program 12.2%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in b around inf 93.4%

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

      1. associate-*r/ 93.4%

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

   7. Applied egg-rr 93.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-62)
   (/ (/ b a) -1.5)
   (if (<= b 6.5e-29)
     (/ (- (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 <= -1.15e-62) {
		tmp = (b / a) / -1.5;
	} else if (b <= 6.5e-29) {
		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 <= (-1.15d-62)) then
        tmp = (b / a) / (-1.5d0)
    else if (b <= 6.5d-29) 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 <= -1.15e-62) {
		tmp = (b / a) / -1.5;
	} else if (b <= 6.5e-29) {
		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 <= -1.15e-62:
		tmp = (b / a) / -1.5
	elif b <= 6.5e-29:
		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 <= -1.15e-62)
		tmp = Float64(Float64(b / a) / -1.5);
	elseif (b <= 6.5e-29)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * 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 <= -1.15e-62)
		tmp = (b / a) / -1.5;
	elseif (b <= 6.5e-29)
		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, -1.15e-62], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], If[LessEqual[b, 6.5e-29], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $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 < -1.15e-62

   1. Initial program 72.0%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in b around -inf 87.6%

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

      1. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. add-cube-cbrt 86.6%

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

      2. pow3 86.7%

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

      3. *-commutative 86.7%

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

      4. times-frac 86.6%

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

      5. metadata-eval 86.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{-0.6666666666666666} \cdot \frac{b}{a}}\right)}^{3} \]

   9. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-0.6666666666666666 \cdot \frac{b}{a}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 87.5%

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

       2. clear-num 87.4%

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

       3. div-inv 87.6%

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

       4. metadata-eval 87.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{\frac{a}{b}} \]

       5. associate-/r* 87.5%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

       6. *-commutative 87.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       7. associate-/r* 87.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{a}{b}}}{-1.5}} \]

       8. clear-num 87.6%

          \[\leadsto \frac{\color{blue}{\frac{b}{a}}}{-1.5} \]

   11. Applied egg-rr 87.6%

       \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

   if -1.15e-62 < b < 6.5e-29

   1. Initial program 78.4%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in b around 0 73.0%

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

   if 6.5e-29 < b

   1. Initial program 12.2%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in b around inf 93.4%

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

      1. associate-*r/ 93.4%

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

   7. Applied egg-rr 93.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.65e-62)
   (/ (/ b a) -1.5)
   (if (<= b 1.2e-29) (/ (sqrt (* a (* c -3.0))) (* a 3.0)) (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.65e-62) {
		tmp = (b / a) / -1.5;
	} else if (b <= 1.2e-29) {
		tmp = sqrt((a * (c * -3.0))) / (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 <= (-1.65d-62)) then
        tmp = (b / a) / (-1.5d0)
    else if (b <= 1.2d-29) then
        tmp = sqrt((a * (c * (-3.0d0)))) / (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 <= -1.65e-62) {
		tmp = (b / a) / -1.5;
	} else if (b <= 1.2e-29) {
		tmp = Math.sqrt((a * (c * -3.0))) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.65e-62:
		tmp = (b / a) / -1.5
	elif b <= 1.2e-29:
		tmp = math.sqrt((a * (c * -3.0))) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.65e-62)
		tmp = Float64(Float64(b / a) / -1.5);
	elseif (b <= 1.2e-29)
		tmp = Float64(sqrt(Float64(a * Float64(c * -3.0))) / 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 <= -1.65e-62)
		tmp = (b / a) / -1.5;
	elseif (b <= 1.2e-29)
		tmp = sqrt((a * (c * -3.0))) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.65e-62], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], If[LessEqual[b, 1.2e-29], N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $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 < -1.65000000000000002e-62

   1. Initial program 72.0%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in b around -inf 87.6%

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

      1. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. add-cube-cbrt 86.6%

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

      2. pow3 86.7%

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

      3. *-commutative 86.7%

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

      4. times-frac 86.6%

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

      5. metadata-eval 86.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{-0.6666666666666666} \cdot \frac{b}{a}}\right)}^{3} \]

   9. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-0.6666666666666666 \cdot \frac{b}{a}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 87.5%

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

       2. clear-num 87.4%

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

       3. div-inv 87.6%

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

       4. metadata-eval 87.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{\frac{a}{b}} \]

       5. associate-/r* 87.5%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

       6. *-commutative 87.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       7. associate-/r* 87.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{a}{b}}}{-1.5}} \]

       8. clear-num 87.6%

          \[\leadsto \frac{\color{blue}{\frac{b}{a}}}{-1.5} \]

   11. Applied egg-rr 87.6%

       \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

   if -1.65000000000000002e-62 < b < 1.19999999999999996e-29

   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. Step-by-step derivation

      1. add-cube-cbrt 77.7%

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

      2. pow3 77.7%

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

   4. Applied egg-rr 77.7%

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

   5. Taylor expanded in a around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 71.6%

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

      5. distribute-lft-neg-in 71.6%

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

      6. metadata-eval 71.6%

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

      7. rem-cube-cbrt 71.8%

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

   7. Simplified 71.8%

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

   if 1.19999999999999996e-29 < b

   1. Initial program 12.2%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in b around inf 93.4%

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

      1. associate-*r/ 93.4%

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

   7. Applied egg-rr 93.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.7e-99)
   (/ (/ b a) -1.5)
   (if (<= b 8.6e-68)
     (* (sqrt (* c (/ -3.0 a))) (- -0.3333333333333333))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.7e-99) {
		tmp = (b / a) / -1.5;
	} else if (b <= 8.6e-68) {
		tmp = sqrt((c * (-3.0 / a))) * -(-0.3333333333333333);
	} 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 <= (-1.7d-99)) then
        tmp = (b / a) / (-1.5d0)
    else if (b <= 8.6d-68) then
        tmp = sqrt((c * ((-3.0d0) / a))) * -(-0.3333333333333333d0)
    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 <= -1.7e-99) {
		tmp = (b / a) / -1.5;
	} else if (b <= 8.6e-68) {
		tmp = Math.sqrt((c * (-3.0 / a))) * -(-0.3333333333333333);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.7e-99:
		tmp = (b / a) / -1.5
	elif b <= 8.6e-68:
		tmp = math.sqrt((c * (-3.0 / a))) * -(-0.3333333333333333)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.7e-99)
		tmp = Float64(Float64(b / a) / -1.5);
	elseif (b <= 8.6e-68)
		tmp = Float64(sqrt(Float64(c * Float64(-3.0 / a))) * Float64(-(-0.3333333333333333)));
	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 <= -1.7e-99)
		tmp = (b / a) / -1.5;
	elseif (b <= 8.6e-68)
		tmp = sqrt((c * (-3.0 / a))) * -(-0.3333333333333333);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.7e-99], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], If[LessEqual[b, 8.6e-68], N[(N[Sqrt[N[(c * N[(-3.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (--0.3333333333333333)), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.70000000000000003e-99

   1. Initial program 74.7%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in b around -inf 82.7%

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

      1. *-commutative 82.7%

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

   7. Simplified 82.7%

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

      1. add-cube-cbrt 81.8%

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

      2. pow3 81.8%

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

      3. *-commutative 81.8%

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

      4. times-frac 81.8%

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

      5. metadata-eval 81.8%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{-0.6666666666666666} \cdot \frac{b}{a}}\right)}^{3} \]

   9. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-0.6666666666666666 \cdot \frac{b}{a}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 82.6%

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

       2. clear-num 82.6%

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

       3. div-inv 82.7%

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

       4. metadata-eval 82.7%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{\frac{a}{b}} \]

       5. associate-/r* 82.6%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

       6. *-commutative 82.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       7. associate-/r* 82.8%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{a}{b}}}{-1.5}} \]

       8. clear-num 82.8%

          \[\leadsto \frac{\color{blue}{\frac{b}{a}}}{-1.5} \]

   11. Applied egg-rr 82.8%

       \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

   if -1.70000000000000003e-99 < b < 8.6000000000000002e-68

   1. Initial program 76.6%

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

      1. add-cube-cbrt 76.0%

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

      2. pow3 76.0%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

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

      3. rem-square-sqrt 40.5%

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

      4. rem-cube-cbrt 40.7%

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

      5. associate-/l* 40.8%

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

   7. Simplified 40.8%

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

   if 8.6000000000000002e-68 < b

   1. Initial program 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in b around inf 89.5%

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

      1. associate-*r/ 89.5%

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

   7. Applied egg-rr 89.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-307}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.4e-307) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-307) {
		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 <= 4.4d-307) 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 <= 4.4e-307) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.4e-307:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4e-307)
		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 <= 4.4e-307)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4e-307], 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 < 4.4e-307

   1. Initial program 77.2%

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

      1. add-cube-cbrt 76.9%

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

      2. pow3 77.0%

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in b around -inf 64.6%

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

      1. associate-*r/ 64.7%

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

      2. *-commutative 64.7%

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

   7. Simplified 64.7%

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

   if 4.4e-307 < b

   1. Initial program 32.7%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in b around inf 67.0%

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

      1. associate-*r/ 67.0%

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

   7. Applied egg-rr 67.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-307}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-306}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.32e-306) (/ -0.6666666666666666 (/ a b)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.32e-306) {
		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 <= 1.32d-306) 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 <= 1.32e-306) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.32e-306:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.32e-306)
		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 <= 1.32e-306)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.32e-306], 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 < 1.32e-306

   1. Initial program 77.2%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in b around -inf 64.7%

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

      1. *-commutative 64.7%

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

   7. Simplified 64.7%

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

      1. add-cube-cbrt 64.0%

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

      2. pow3 64.0%

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

      3. *-commutative 64.0%

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

      4. times-frac 64.0%

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

      5. metadata-eval 64.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{-0.6666666666666666} \cdot \frac{b}{a}}\right)}^{3} \]

   9. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-0.6666666666666666 \cdot \frac{b}{a}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 64.6%

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

       2. clear-num 64.6%

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

       3. div-inv 64.7%

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

   11. Applied egg-rr 64.7%

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

   if 1.32e-306 < b

   1. Initial program 32.7%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in b around inf 67.0%

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

      1. associate-*r/ 67.0%

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

   7. Applied egg-rr 67.0%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-306}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.3e-308) (/ -0.6666666666666666 (/ a b)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.3e-308) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.3d-308) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.3e-308) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.3e-308:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.3e-308)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.3e-308)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.3e-308], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.3000000000000002e-308

   1. Initial program 77.2%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in b around -inf 64.7%

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

      1. *-commutative 64.7%

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

   7. Simplified 64.7%

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

      1. add-cube-cbrt 64.0%

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

      2. pow3 64.0%

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

      3. *-commutative 64.0%

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

      4. times-frac 64.0%

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

      5. metadata-eval 64.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{-0.6666666666666666} \cdot \frac{b}{a}}\right)}^{3} \]

   9. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-0.6666666666666666 \cdot \frac{b}{a}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 64.6%

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

       2. clear-num 64.6%

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

       3. div-inv 64.7%

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

   11. Applied egg-rr 64.7%

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

   if 4.3000000000000002e-308 < b

   1. Initial program 32.7%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in b around inf 67.0%

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

Alternative 10: 68.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 77.2%

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

      1. add-cube-cbrt 76.9%

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

      2. pow3 77.0%

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

   4. Applied egg-rr 77.0%

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

      1. div-inv 76.9%

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

      2. neg-mul-1 76.9%

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

      3. fma-define 76.9%

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

      4. rem-cube-cbrt 77.1%

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

      5. pow2 77.1%

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

   6. Applied egg-rr 77.1%

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

      1. associate-/r* 77.0%

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

      2. metadata-eval 77.0%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} \]

      3. metadata-eval 77.0%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{\color{blue}{0.3333333333333333 \cdot 1}}{a} \]

      4. associate-*r/ 77.0%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right)} \]

      5. *-commutative 77.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right)} \]

      6. associate-*r/ 77.0%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{a}} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \]

      7. metadata-eval 77.0%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \]

      8. associate-*r* 77.0%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}}\right) \]

      9. cancel-sign-sub-inv 77.0%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-3\right) \cdot \left(a \cdot c\right)}}\right) \]

      10. unpow2 77.0%

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

      11. metadata-eval 77.0%

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

      12. fma-undefine 77.0%

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

      13. *-commutative 77.0%

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

      14. associate-*r* 77.0%

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

   8. Simplified 77.0%

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

   9. Taylor expanded in b around -inf 64.6%

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

       1. *-commutative 64.6%

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

       2. associate-*l/ 64.7%

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

       3. associate-*r/ 64.6%

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

   11. Simplified 64.6%

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

   if -1.999999999999994e-310 < b

   1. Initial program 32.7%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in b around inf 67.0%

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

Alternative 11: 36.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

3. Simplified 55.2%

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

5. Taylor expanded in b around inf 34.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Alternative 12: 11.5% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 55.3%

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

   1. add-cube-cbrt 55.1%

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

   2. pow3 55.1%

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

4. Applied egg-rr 55.1%

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

   1. div-inv 55.0%

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

   2. neg-mul-1 55.0%

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

   3. fma-define 55.0%

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

   4. rem-cube-cbrt 55.2%

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

   5. pow2 55.2%

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

6. Applied egg-rr 55.2%

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

   1. associate-/r* 55.2%

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

   2. metadata-eval 55.2%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} \]

   3. metadata-eval 55.2%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{\color{blue}{0.3333333333333333 \cdot 1}}{a} \]

   4. associate-*r/ 55.1%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right)} \]

   5. *-commutative 55.1%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right)} \]

   6. associate-*r/ 55.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{a}} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \]

   7. metadata-eval 55.2%

      \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}\right) \]

   8. associate-*r* 55.2%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}}\right) \]

   9. cancel-sign-sub-inv 55.2%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-3\right) \cdot \left(a \cdot c\right)}}\right) \]

   10. unpow2 55.2%

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

   11. metadata-eval 55.2%

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

   12. fma-undefine 55.2%

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

   13. *-commutative 55.2%

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

   14. associate-*r* 55.2%

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

8. Simplified 55.2%

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

9. Taylor expanded in a around 0 10.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 10.5%

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

    2. distribute-rgt1-in 10.5%

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

    3. metadata-eval 10.5%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 10.5%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 10.5%

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

11. Simplified 10.5%

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

12. Taylor expanded in a around 0 10.5%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))