Cubic critical

Percentage Accurate: 52.5% → 85.5%

Time: 14.9s

Alternatives: 10

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.5% 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: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.1e+94)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 2.55e-128)
     (/ (- (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 <= -3.1e+94) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2.55e-128) {
		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 <= (-3.1d+94)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 2.55d-128) 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 <= -3.1e+94) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2.55e-128) {
		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 <= -3.1e+94:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 2.55e-128:
		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 <= -3.1e+94)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 2.55e-128)
		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 <= -3.1e+94)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 2.55e-128)
		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, -3.1e+94], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.55e-128], 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 < -3.09999999999999991e94

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. sqr-neg 52.9%

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

      3. associate-*l* 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in b around -inf 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*l/ 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -3.09999999999999991e94 < b < 2.5500000000000002e-128

   1. Initial program 84.7%

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

   if 2.5500000000000002e-128 < b

   1. Initial program 14.1%

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

      1. add-cube-cbrt 14.0%

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

      2. pow3 14.0%

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in b around inf 87.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 87.6%

         \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b} \cdot -0.16666666666666666} \]

      2. rem-cube-cbrt 88.3%

         \[\leadsto \frac{c \cdot \color{blue}{3}}{b} \cdot -0.16666666666666666 \]

      3. associate-/l* 88.3%

         \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right)} \cdot -0.16666666666666666 \]

   7. Simplified 88.3%

      \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right) \cdot -0.16666666666666666} \]
   8. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(c \cdot \frac{3}{b}\right)} \]

      2. associate-*r/ 88.3%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\frac{c \cdot 3}{b}} \]

      3. associate-*r/ 88.5%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]

   9. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]
   10. Step-by-step derivation

       1. *-commutative 88.5%

          \[\leadsto \frac{\color{blue}{\left(c \cdot 3\right) \cdot -0.16666666666666666}}{b} \]

       2. associate-*l* 88.8%

          \[\leadsto \frac{\color{blue}{c \cdot \left(3 \cdot -0.16666666666666666\right)}}{b} \]

       3. metadata-eval 88.8%

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

   11. Simplified 88.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+94}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e+94)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 3.8e-128)
     (/ (- (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 <= -9e+94) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 3.8e-128) {
		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 <= (-9d+94)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 3.8d-128) 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 <= -9e+94) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 3.8e-128) {
		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 <= -9e+94:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 3.8e-128:
		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 <= -9e+94)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 3.8e-128)
		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 <= -9e+94)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 3.8e-128)
		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, -9e+94], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.8e-128], 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 < -8.99999999999999944e94

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. sqr-neg 52.9%

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

      3. associate-*l* 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in b around -inf 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*l/ 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -8.99999999999999944e94 < b < 3.8000000000000002e-128

   1. Initial program 84.7%

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

      1. sqr-neg 84.7%

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

      2. sqr-neg 84.7%

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

      3. associate-*l* 84.6%

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

   3. Simplified 84.6%

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

   if 3.8000000000000002e-128 < b

   1. Initial program 14.1%

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

      1. add-cube-cbrt 14.0%

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

      2. pow3 14.0%

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in b around inf 87.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 87.6%

         \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b} \cdot -0.16666666666666666} \]

      2. rem-cube-cbrt 88.3%

         \[\leadsto \frac{c \cdot \color{blue}{3}}{b} \cdot -0.16666666666666666 \]

      3. associate-/l* 88.3%

         \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right)} \cdot -0.16666666666666666 \]

   7. Simplified 88.3%

      \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right) \cdot -0.16666666666666666} \]
   8. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(c \cdot \frac{3}{b}\right)} \]

      2. associate-*r/ 88.3%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\frac{c \cdot 3}{b}} \]

      3. associate-*r/ 88.5%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]

   9. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]
   10. Step-by-step derivation

       1. *-commutative 88.5%

          \[\leadsto \frac{\color{blue}{\left(c \cdot 3\right) \cdot -0.16666666666666666}}{b} \]

       2. associate-*l* 88.8%

          \[\leadsto \frac{\color{blue}{c \cdot \left(3 \cdot -0.16666666666666666\right)}}{b} \]

       3. metadata-eval 88.8%

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

   11. Simplified 88.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+94}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-80}:\\ \;\;\;\;b \cdot \left(0.6666666666666666 \cdot \frac{-1}{a} - -0.5 \cdot \frac{c}{{b}^{2}}\right)\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-80)
   (* b (- (* 0.6666666666666666 (/ -1.0 a)) (* -0.5 (/ c (pow b 2.0)))))
   (if (<= b 3.8e-128)
     (/ (- (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 <= -1.8e-80) {
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / pow(b, 2.0))));
	} else if (b <= 3.8e-128) {
		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 <= (-1.8d-80)) then
        tmp = b * ((0.6666666666666666d0 * ((-1.0d0) / a)) - ((-0.5d0) * (c / (b ** 2.0d0))))
    else if (b <= 3.8d-128) 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 <= -1.8e-80) {
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / Math.pow(b, 2.0))));
	} else if (b <= 3.8e-128) {
		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 <= -1.8e-80:
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / math.pow(b, 2.0))))
	elif b <= 3.8e-128:
		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 <= -1.8e-80)
		tmp = Float64(b * Float64(Float64(0.6666666666666666 * Float64(-1.0 / a)) - Float64(-0.5 * Float64(c / (b ^ 2.0)))));
	elseif (b <= 3.8e-128)
		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 <= -1.8e-80)
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / (b ^ 2.0))));
	elseif (b <= 3.8e-128)
		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, -1.8e-80], N[(b * N[(N[(0.6666666666666666 * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-128], 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 < -1.8e-80

   1. Initial program 68.8%

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

      1. sqr-neg 68.8%

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

      2. sqr-neg 68.8%

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

      3. associate-*l* 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in b around -inf 84.6%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   if -1.8e-80 < b < 3.8000000000000002e-128

   1. Initial program 79.4%

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

      1. sqr-neg 79.4%

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

      2. sqr-neg 79.4%

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

      3. associate-*l* 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in b around 0 74.8%

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

      1. associate-*r* 74.9%

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

      2. *-commutative 74.9%

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

   7. Simplified 74.9%

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

   if 3.8000000000000002e-128 < b

   1. Initial program 14.1%

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

      1. add-cube-cbrt 14.0%

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

      2. pow3 14.0%

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in b around inf 87.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 87.6%

         \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b} \cdot -0.16666666666666666} \]

      2. rem-cube-cbrt 88.3%

         \[\leadsto \frac{c \cdot \color{blue}{3}}{b} \cdot -0.16666666666666666 \]

      3. associate-/l* 88.3%

         \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right)} \cdot -0.16666666666666666 \]

   7. Simplified 88.3%

      \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right) \cdot -0.16666666666666666} \]
   8. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(c \cdot \frac{3}{b}\right)} \]

      2. associate-*r/ 88.3%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\frac{c \cdot 3}{b}} \]

      3. associate-*r/ 88.5%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]

   9. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]
   10. Step-by-step derivation

       1. *-commutative 88.5%

          \[\leadsto \frac{\color{blue}{\left(c \cdot 3\right) \cdot -0.16666666666666666}}{b} \]

       2. associate-*l* 88.8%

          \[\leadsto \frac{\color{blue}{c \cdot \left(3 \cdot -0.16666666666666666\right)}}{b} \]

       3. metadata-eval 88.8%

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

   11. Simplified 88.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-80}:\\ \;\;\;\;b \cdot \left(0.6666666666666666 \cdot \frac{-1}{a} - -0.5 \cdot \frac{c}{{b}^{2}}\right)\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.3e-80)
   (* -0.6666666666666666 (/ b a))
   (if (<= b 3.1e-128)
     (/ (- (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 <= -1.3e-80) {
		tmp = -0.6666666666666666 * (b / a);
	} else if (b <= 3.1e-128) {
		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 <= (-1.3d-80)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else if (b <= 3.1d-128) 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 <= -1.3e-80) {
		tmp = -0.6666666666666666 * (b / a);
	} else if (b <= 3.1e-128) {
		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 <= -1.3e-80:
		tmp = -0.6666666666666666 * (b / a)
	elif b <= 3.1e-128:
		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 <= -1.3e-80)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	elseif (b <= 3.1e-128)
		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 <= -1.3e-80)
		tmp = -0.6666666666666666 * (b / a);
	elseif (b <= 3.1e-128)
		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, -1.3e-80], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.1e-128], 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 < -1.3e-80

   1. Initial program 68.8%

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

      1. sqr-neg 68.8%

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

      2. sqr-neg 68.8%

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

      3. associate-*l* 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in b around -inf 84.5%

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

      1. *-commutative 84.5%

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

   7. Simplified 84.5%

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

   if -1.3e-80 < b < 3.10000000000000003e-128

   1. Initial program 79.4%

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

      1. sqr-neg 79.4%

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

      2. sqr-neg 79.4%

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

      3. associate-*l* 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in b around 0 74.8%

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

      1. associate-*r* 74.9%

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

      2. *-commutative 74.9%

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

   7. Simplified 74.9%

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

   if 3.10000000000000003e-128 < b

   1. Initial program 14.1%

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

      1. add-cube-cbrt 14.0%

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

      2. pow3 14.0%

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in b around inf 87.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 87.6%

         \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b} \cdot -0.16666666666666666} \]

      2. rem-cube-cbrt 88.3%

         \[\leadsto \frac{c \cdot \color{blue}{3}}{b} \cdot -0.16666666666666666 \]

      3. associate-/l* 88.3%

         \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right)} \cdot -0.16666666666666666 \]

   7. Simplified 88.3%

      \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right) \cdot -0.16666666666666666} \]
   8. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(c \cdot \frac{3}{b}\right)} \]

      2. associate-*r/ 88.3%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\frac{c \cdot 3}{b}} \]

      3. associate-*r/ 88.5%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]

   9. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]
   10. Step-by-step derivation

       1. *-commutative 88.5%

          \[\leadsto \frac{\color{blue}{\left(c \cdot 3\right) \cdot -0.16666666666666666}}{b} \]

       2. associate-*l* 88.8%

          \[\leadsto \frac{\color{blue}{c \cdot \left(3 \cdot -0.16666666666666666\right)}}{b} \]

       3. metadata-eval 88.8%

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

   11. Simplified 88.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-80}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.8e-80)
   (* -0.6666666666666666 (/ b a))
   (if (<= b 3.8e-128)
     (/ (- (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 <= -8.8e-80) {
		tmp = -0.6666666666666666 * (b / a);
	} else if (b <= 3.8e-128) {
		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 <= (-8.8d-80)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else if (b <= 3.8d-128) 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 <= -8.8e-80) {
		tmp = -0.6666666666666666 * (b / a);
	} else if (b <= 3.8e-128) {
		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 <= -8.8e-80:
		tmp = -0.6666666666666666 * (b / a)
	elif b <= 3.8e-128:
		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 <= -8.8e-80)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	elseif (b <= 3.8e-128)
		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 <= -8.8e-80)
		tmp = -0.6666666666666666 * (b / a);
	elseif (b <= 3.8e-128)
		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, -8.8e-80], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-128], 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 < -8.80000000000000041e-80

   1. Initial program 68.8%

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

      1. sqr-neg 68.8%

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

      2. sqr-neg 68.8%

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

      3. associate-*l* 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in b around -inf 84.5%

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

      1. *-commutative 84.5%

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

   7. Simplified 84.5%

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

   if -8.80000000000000041e-80 < b < 3.8000000000000002e-128

   1. Initial program 79.4%

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

      1. sqr-neg 79.4%

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

      2. sqr-neg 79.4%

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

      3. associate-*l* 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in b around 0 74.8%

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

   if 3.8000000000000002e-128 < b

   1. Initial program 14.1%

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

      1. add-cube-cbrt 14.0%

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

      2. pow3 14.0%

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in b around inf 87.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 87.6%

         \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b} \cdot -0.16666666666666666} \]

      2. rem-cube-cbrt 88.3%

         \[\leadsto \frac{c \cdot \color{blue}{3}}{b} \cdot -0.16666666666666666 \]

      3. associate-/l* 88.3%

         \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right)} \cdot -0.16666666666666666 \]

   7. Simplified 88.3%

      \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right) \cdot -0.16666666666666666} \]
   8. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(c \cdot \frac{3}{b}\right)} \]

      2. associate-*r/ 88.3%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\frac{c \cdot 3}{b}} \]

      3. associate-*r/ 88.5%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]

   9. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]
   10. Step-by-step derivation

       1. *-commutative 88.5%

          \[\leadsto \frac{\color{blue}{\left(c \cdot 3\right) \cdot -0.16666666666666666}}{b} \]

       2. associate-*l* 88.8%

          \[\leadsto \frac{\color{blue}{c \cdot \left(3 \cdot -0.16666666666666666\right)}}{b} \]

       3. metadata-eval 88.8%

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

   11. Simplified 88.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-80}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-142}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-141}:\\ \;\;\;\;-\sqrt{\left(-3 \cdot \frac{c}{a}\right) \cdot 0.1111111111111111}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-142)
   (* -0.6666666666666666 (/ b a))
   (if (<= b 9.8e-141)
     (- (sqrt (* (* -3.0 (/ c a)) 0.1111111111111111)))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-142) {
		tmp = -0.6666666666666666 * (b / a);
	} else if (b <= 9.8e-141) {
		tmp = -sqrt(((-3.0 * (c / a)) * 0.1111111111111111));
	} 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 <= (-3.6d-142)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else if (b <= 9.8d-141) then
        tmp = -sqrt((((-3.0d0) * (c / a)) * 0.1111111111111111d0))
    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 <= -3.6e-142) {
		tmp = -0.6666666666666666 * (b / a);
	} else if (b <= 9.8e-141) {
		tmp = -Math.sqrt(((-3.0 * (c / a)) * 0.1111111111111111));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e-142:
		tmp = -0.6666666666666666 * (b / a)
	elif b <= 9.8e-141:
		tmp = -math.sqrt(((-3.0 * (c / a)) * 0.1111111111111111))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-142)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	elseif (b <= 9.8e-141)
		tmp = Float64(-sqrt(Float64(Float64(-3.0 * Float64(c / a)) * 0.1111111111111111)));
	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 <= -3.6e-142)
		tmp = -0.6666666666666666 * (b / a);
	elseif (b <= 9.8e-141)
		tmp = -sqrt(((-3.0 * (c / a)) * 0.1111111111111111));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-142], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.8e-141], (-N[Sqrt[N[(N[(-3.0 * N[(c / a), $MachinePrecision]), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]], $MachinePrecision]), N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.6e-142

   1. Initial program 71.0%

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

      1. sqr-neg 71.0%

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

      2. sqr-neg 71.0%

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

      3. associate-*l* 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in b around -inf 79.3%

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

      1. *-commutative 79.3%

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

   7. Simplified 79.3%

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

   if -3.6e-142 < b < 9.80000000000000012e-141

   1. Initial program 75.8%

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

      1. add-cube-cbrt 75.2%

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

      2. pow3 75.2%

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

   4. Applied egg-rr 75.2%

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

   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. associate-*r* 0.0%

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

      2. rem-cube-cbrt 0.0%

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

      3. associate-/l* 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 28.6%

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

   7. Simplified 28.6%

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

      1. add-sqr-sqrt 0.8%

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

      2. sqrt-unprod 41.5%

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

      3. *-commutative 41.5%

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

      4. *-commutative 41.5%

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

      5. swap-sqr 41.4%

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

      6. add-sqr-sqrt 41.5%

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

      7. associate-*r/ 41.6%

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

      8. *-commutative 41.6%

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

      9. associate-/l* 41.5%

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

      10. metadata-eval 41.5%

          \[\leadsto \sqrt{\left(-3 \cdot \frac{c}{a}\right) \cdot \color{blue}{0.1111111111111111}} \cdot -1 \]

   9. Applied egg-rr 41.5%

      \[\leadsto \color{blue}{\sqrt{\left(-3 \cdot \frac{c}{a}\right) \cdot 0.1111111111111111}} \cdot -1 \]

   if 9.80000000000000012e-141 < b

   1. Initial program 16.3%

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

      1. add-cube-cbrt 16.3%

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

      2. pow3 16.3%

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

   4. Applied egg-rr 16.3%

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

   5. Taylor expanded in b around inf 85.4%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 85.4%

         \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b} \cdot -0.16666666666666666} \]

      2. rem-cube-cbrt 86.1%

         \[\leadsto \frac{c \cdot \color{blue}{3}}{b} \cdot -0.16666666666666666 \]

      3. associate-/l* 86.1%

         \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right)} \cdot -0.16666666666666666 \]

   7. Simplified 86.1%

      \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right) \cdot -0.16666666666666666} \]
   8. Step-by-step derivation

      1. *-commutative 86.1%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(c \cdot \frac{3}{b}\right)} \]

      2. associate-*r/ 86.1%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\frac{c \cdot 3}{b}} \]

      3. associate-*r/ 86.3%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]

   9. Applied egg-rr 86.3%

      \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]
   10. Step-by-step derivation

       1. *-commutative 86.3%

          \[\leadsto \frac{\color{blue}{\left(c \cdot 3\right) \cdot -0.16666666666666666}}{b} \]

       2. associate-*l* 86.6%

          \[\leadsto \frac{\color{blue}{c \cdot \left(3 \cdot -0.16666666666666666\right)}}{b} \]

       3. metadata-eval 86.6%

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

   11. Simplified 86.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-142}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-141}:\\ \;\;\;\;-\sqrt{\left(-3 \cdot \frac{c}{a}\right) \cdot 0.1111111111111111}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 11.6× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* -0.6666666666666666 (/ b a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.4%

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

      1. sqr-neg 74.4%

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

      2. sqr-neg 74.4%

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

      3. associate-*l* 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in b around -inf 66.0%

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

      1. *-commutative 66.0%

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

   7. Simplified 66.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 24.5%

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

      1. add-cube-cbrt 24.3%

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

      2. pow3 24.4%

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

   4. Applied egg-rr 24.4%

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

   5. Taylor expanded in b around inf 72.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 72.7%

         \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt[3]{3}\right)}^{3}}{b} \cdot -0.16666666666666666} \]

      2. rem-cube-cbrt 73.2%

         \[\leadsto \frac{c \cdot \color{blue}{3}}{b} \cdot -0.16666666666666666 \]

      3. associate-/l* 73.2%

         \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right)} \cdot -0.16666666666666666 \]

   7. Simplified 73.2%

      \[\leadsto \color{blue}{\left(c \cdot \frac{3}{b}\right) \cdot -0.16666666666666666} \]
   8. Step-by-step derivation

      1. *-commutative 73.2%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(c \cdot \frac{3}{b}\right)} \]

      2. associate-*r/ 73.2%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\frac{c \cdot 3}{b}} \]

      3. associate-*r/ 73.4%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]

   9. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot 3\right)}{b}} \]
   10. Step-by-step derivation

       1. *-commutative 73.4%

          \[\leadsto \frac{\color{blue}{\left(c \cdot 3\right) \cdot -0.16666666666666666}}{b} \]

       2. associate-*l* 73.7%

          \[\leadsto \frac{\color{blue}{c \cdot \left(3 \cdot -0.16666666666666666\right)}}{b} \]

       3. metadata-eval 73.7%

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

   11. Simplified 73.7%

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

4. Final simplification 70.1%

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

Alternative 8: 68.4% accurate, 11.6× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* -0.6666666666666666 (/ b a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.6666666666666666 * (b / 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 <= (-5d-310)) then
        tmp = (-0.6666666666666666d0) * (b / 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 <= -5e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / 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 <= -5e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.4%

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

      1. sqr-neg 74.4%

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

      2. sqr-neg 74.4%

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

      3. associate-*l* 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in b around -inf 66.0%

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

      1. *-commutative 66.0%

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

   7. Simplified 66.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 24.5%

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

      1. sqr-neg 24.5%

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

      2. sqr-neg 24.5%

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

      3. associate-*l* 24.5%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in b around inf 73.7%

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

      1. *-commutative 73.7%

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

   7. Simplified 73.7%

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

4. Final simplification 70.1%

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

Alternative 9: 36.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.6666666666666666d0) * (b / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.5%

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

   1. sqr-neg 47.5%

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

   2. sqr-neg 47.5%

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

   3. associate-*l* 47.5%

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

3. Simplified 47.5%

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

5. Taylor expanded in b around -inf 31.9%

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

   1. *-commutative 31.9%

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

7. Simplified 31.9%

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

8. Final simplification 31.9%

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

Alternative 10: 36.6% 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 47.5%

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

   1. add-cube-cbrt 47.3%

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

   2. pow3 47.3%

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

4. Applied egg-rr 47.3%

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

5. Taylor expanded in b around -inf 31.9%

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

   1. associate-*r/ 31.9%

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

   2. *-commutative 31.9%

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

   3. associate-/l* 31.9%

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

7. Simplified 31.9%

   \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))