Cubic critical

Percentage Accurate: 53.1% → 85.3%

Time: 12.0s

Alternatives: 9

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: 53.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+153)
   (* (/ b a) -0.6666666666666666)
   (if (<= b 4.2e-57)
     (/ (- (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 <= -2.3e+153) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 4.2e-57) {
		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 <= (-2.3d+153)) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else if (b <= 4.2d-57) 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 <= -2.3e+153) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 4.2e-57) {
		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 <= -2.3e+153:
		tmp = (b / a) * -0.6666666666666666
	elif b <= 4.2e-57:
		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 <= -2.3e+153)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	elseif (b <= 4.2e-57)
		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 <= -2.3e+153)
		tmp = (b / a) * -0.6666666666666666;
	elseif (b <= 4.2e-57)
		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, -2.3e+153], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], If[LessEqual[b, 4.2e-57], 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 < -2.3000000000000001e153

   1. Initial program 46.5%

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

      1. sqr-neg 46.5%

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

      2. sqr-neg 46.5%

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

      3. associate-*l* 46.5%

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

   3. Simplified 46.5%

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

      1. sub-neg 46.5%

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

      2. sub-neg 46.5%

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

      3. *-commutative 46.5%

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

      4. cancel-sign-sub-inv 46.5%

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

      5. distribute-lft-neg-in 46.5%

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

      6. distribute-rgt-neg-in 46.5%

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

      7. metadata-eval 46.5%

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

      8. associate-*r* 46.5%

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

      9. fma-undefine 46.5%

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

      10. add-sqr-sqrt 46.5%

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

      11. pow2 46.5%

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

      12. fma-undefine 46.5%

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

      13. add-sqr-sqrt 33.3%

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

      14. hypot-define 56.7%

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

   6. Applied egg-rr 56.7%

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

   7. Taylor expanded in b around -inf 91.4%

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

      1. *-commutative 91.4%

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

   9. Simplified 91.4%

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

   if -2.3000000000000001e153 < b < 4.1999999999999999e-57

   1. Initial program 86.1%

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

   if 4.1999999999999999e-57 < b

   1. Initial program 8.7%

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

      1. sqr-neg 8.7%

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

      2. sqr-neg 8.7%

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

      3. associate-*l* 8.7%

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

   3. Simplified 8.7%

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

   5. Taylor expanded in b around inf 78.0%

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

      1. associate-/l* 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in a around 0 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*l/ 93.9%

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

   10. Simplified 93.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e+153)
   (* (/ b a) -0.6666666666666666)
   (if (<= b 3.7e-47)
     (/ (- (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+153) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 3.7e-47) {
		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+153)) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else if (b <= 3.7d-47) 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+153) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 3.7e-47) {
		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+153:
		tmp = (b / a) * -0.6666666666666666
	elif b <= 3.7e-47:
		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+153)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	elseif (b <= 3.7e-47)
		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+153)
		tmp = (b / a) * -0.6666666666666666;
	elseif (b <= 3.7e-47)
		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+153], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], If[LessEqual[b, 3.7e-47], 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 < -9.0000000000000002e153

   1. Initial program 46.5%

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

      1. sqr-neg 46.5%

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

      2. sqr-neg 46.5%

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

      3. associate-*l* 46.5%

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

   3. Simplified 46.5%

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

      1. sub-neg 46.5%

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

      2. sub-neg 46.5%

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

      3. *-commutative 46.5%

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

      4. cancel-sign-sub-inv 46.5%

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

      5. distribute-lft-neg-in 46.5%

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

      6. distribute-rgt-neg-in 46.5%

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

      7. metadata-eval 46.5%

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

      8. associate-*r* 46.5%

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

      9. fma-undefine 46.5%

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

      10. add-sqr-sqrt 46.5%

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

      11. pow2 46.5%

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

      12. fma-undefine 46.5%

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

      13. add-sqr-sqrt 33.3%

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

      14. hypot-define 56.7%

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

   6. Applied egg-rr 56.7%

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

   7. Taylor expanded in b around -inf 91.4%

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

      1. *-commutative 91.4%

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

   9. Simplified 91.4%

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

   if -9.0000000000000002e153 < b < 3.7e-47

   1. Initial program 86.1%

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

      1. sqr-neg 86.1%

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

      2. sqr-neg 86.1%

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

      3. associate-*l* 86.1%

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

   3. Simplified 86.1%

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

   if 3.7e-47 < b

   1. Initial program 8.7%

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

      1. sqr-neg 8.7%

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

      2. sqr-neg 8.7%

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

      3. associate-*l* 8.7%

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

   3. Simplified 8.7%

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

   5. Taylor expanded in b around inf 78.0%

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

      1. associate-/l* 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in a around 0 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*l/ 93.9%

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

   10. Simplified 93.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-128}:\\ \;\;\;\;b \cdot \left(\frac{c \cdot -0.5}{-{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-128)
   (* b (- (/ (* c -0.5) (- (pow b 2.0))) (/ 0.6666666666666666 a)))
   (if (<= b 4.9e-52)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-128) {
		tmp = b * (((c * -0.5) / -pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 4.9e-52) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.5d-128)) then
        tmp = b * (((c * (-0.5d0)) / -(b ** 2.0d0)) - (0.6666666666666666d0 / a))
    else if (b <= 4.9d-52) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-128) {
		tmp = b * (((c * -0.5) / -Math.pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 4.9e-52) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.5e-128:
		tmp = b * (((c * -0.5) / -math.pow(b, 2.0)) - (0.6666666666666666 / a))
	elif b <= 4.9e-52:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-128)
		tmp = Float64(b * Float64(Float64(Float64(c * -0.5) / Float64(-(b ^ 2.0))) - Float64(0.6666666666666666 / a)));
	elseif (b <= 4.9e-52)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.5e-128)
		tmp = b * (((c * -0.5) / -(b ^ 2.0)) - (0.6666666666666666 / a));
	elseif (b <= 4.9e-52)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-128], N[(b * N[(N[(N[(c * -0.5), $MachinePrecision] / (-N[Power[b, 2.0], $MachinePrecision])), $MachinePrecision] - N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.9e-52], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5000000000000004e-128

   1. Initial program 80.0%

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

      1. sqr-neg 80.0%

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

      2. sqr-neg 80.0%

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

      3. associate-*l* 80.0%

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

   3. Simplified 80.0%

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

      1. sub-neg 80.0%

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

      2. sub-neg 80.0%

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

      3. *-commutative 80.0%

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

      4. cancel-sign-sub-inv 80.0%

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

      5. distribute-lft-neg-in 80.0%

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

      6. distribute-rgt-neg-in 80.0%

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

      7. metadata-eval 80.0%

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

      8. associate-*r* 80.0%

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

      9. fma-undefine 80.0%

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

      10. add-sqr-sqrt 79.7%

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

      11. pow2 79.7%

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

      12. fma-undefine 79.7%

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

      13. add-sqr-sqrt 61.8%

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

      14. hypot-define 69.6%

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

   6. Applied egg-rr 69.6%

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

   7. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto \color{blue}{-b \cdot \left(0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto -\color{blue}{\left(0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 0.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. associate-*r/ 0.0%

         \[\leadsto \left(\color{blue}{\frac{0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{{b}^{2}}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      5. *-commutative 0.0%

         \[\leadsto \left(\frac{0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      6. associate-*r* 0.0%

         \[\leadsto \left(\frac{\color{blue}{\left(0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      7. unpow2 0.0%

         \[\leadsto \left(\frac{\left(0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      8. rem-square-sqrt 81.3%

         \[\leadsto \left(\frac{\left(0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      9. metadata-eval 81.3%

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

      10. *-commutative 81.3%

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

      11. associate-*r/ 81.3%

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

      12. metadata-eval 81.3%

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

   9. Simplified 81.3%

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

   if -5.5000000000000004e-128 < b < 4.90000000000000019e-52

   1. Initial program 75.5%

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

      1. sqr-neg 75.5%

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

      2. sqr-neg 75.5%

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

      3. associate-*l* 75.4%

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

   3. Simplified 75.4%

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

      1. add-cube-cbrt 74.9%

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

      2. pow3 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. add-cube-cbrt 74.4%

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

      2. pow3 74.4%

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

      3. unpow3 74.4%

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

      4. add-cube-cbrt 74.8%

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

   8. Applied egg-rr 74.8%

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

   9. Taylor expanded in a around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 71.2%

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

       4. associate-*r* 71.2%

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

       5. metadata-eval 71.2%

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

       6. rem-cube-cbrt 71.4%

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

   11. Simplified 71.4%

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

   if 4.90000000000000019e-52 < b

   1. Initial program 8.7%

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

      1. sqr-neg 8.7%

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

      2. sqr-neg 8.7%

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

      3. associate-*l* 8.7%

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

   3. Simplified 8.7%

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

   5. Taylor expanded in b around inf 78.0%

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

      1. associate-/l* 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in a around 0 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*l/ 93.9%

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

   10. Simplified 93.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-128}:\\ \;\;\;\;b \cdot \left(\frac{c \cdot -0.5}{-{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-147}:\\ \;\;\;\;b \cdot \left(\frac{c \cdot -0.5}{-{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e-147)
   (* b (- (/ (* c -0.5) (- (pow b 2.0))) (/ 0.6666666666666666 a)))
   (if (<= b 4.8e-105)
     (* (sqrt (* c (/ -3.0 a))) 0.3333333333333333)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.12e-147) {
		tmp = b * (((c * -0.5) / -pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 4.8e-105) {
		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 <= (-1.12d-147)) then
        tmp = b * (((c * (-0.5d0)) / -(b ** 2.0d0)) - (0.6666666666666666d0 / a))
    else if (b <= 4.8d-105) 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 <= -1.12e-147) {
		tmp = b * (((c * -0.5) / -Math.pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 4.8e-105) {
		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 <= -1.12e-147:
		tmp = b * (((c * -0.5) / -math.pow(b, 2.0)) - (0.6666666666666666 / a))
	elif b <= 4.8e-105:
		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 <= -1.12e-147)
		tmp = Float64(b * Float64(Float64(Float64(c * -0.5) / Float64(-(b ^ 2.0))) - Float64(0.6666666666666666 / a)));
	elseif (b <= 4.8e-105)
		tmp = Float64(sqrt(Float64(c * Float64(-3.0 / a))) * 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 <= -1.12e-147)
		tmp = b * (((c * -0.5) / -(b ^ 2.0)) - (0.6666666666666666 / a));
	elseif (b <= 4.8e-105)
		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, -1.12e-147], N[(b * N[(N[(N[(c * -0.5), $MachinePrecision] / (-N[Power[b, 2.0], $MachinePrecision])), $MachinePrecision] - N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.8e-105], 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 < -1.12e-147

   1. Initial program 80.4%

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

      1. sqr-neg 80.4%

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

      2. sqr-neg 80.4%

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

      3. associate-*l* 80.3%

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

   3. Simplified 80.3%

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

      1. sub-neg 80.3%

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

      2. sub-neg 80.3%

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

      3. *-commutative 80.3%

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

      4. cancel-sign-sub-inv 80.3%

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

      5. distribute-lft-neg-in 80.3%

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

      6. distribute-rgt-neg-in 80.3%

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

      7. metadata-eval 80.3%

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

      8. associate-*r* 80.3%

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

      9. fma-undefine 80.3%

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

      10. add-sqr-sqrt 80.1%

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

      11. pow2 80.1%

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

      12. fma-undefine 80.1%

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

      13. add-sqr-sqrt 62.6%

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

      14. hypot-define 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto \color{blue}{-b \cdot \left(0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto -\color{blue}{\left(0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 0.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. associate-*r/ 0.0%

         \[\leadsto \left(\color{blue}{\frac{0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{{b}^{2}}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      5. *-commutative 0.0%

         \[\leadsto \left(\frac{0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      6. associate-*r* 0.0%

         \[\leadsto \left(\frac{\color{blue}{\left(0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      7. unpow2 0.0%

         \[\leadsto \left(\frac{\left(0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      8. rem-square-sqrt 80.7%

         \[\leadsto \left(\frac{\left(0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right) \]

      9. metadata-eval 80.7%

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

      10. *-commutative 80.7%

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

      11. associate-*r/ 80.7%

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

      12. metadata-eval 80.7%

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

   9. Simplified 80.7%

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

   if -1.12e-147 < b < 4.8000000000000003e-105

   1. Initial program 76.8%

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

      1. sqr-neg 76.8%

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

      2. sqr-neg 76.8%

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

      3. associate-*l* 76.8%

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

   3. Simplified 76.8%

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

      1. add-cube-cbrt 76.1%

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

      2. pow3 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. add-cube-cbrt 75.6%

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

      2. pow3 75.6%

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

      3. unpow3 75.7%

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

      4. add-cube-cbrt 76.1%

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

   8. Applied egg-rr 76.1%

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

   9. Taylor expanded in a around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 30.1%

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

       4. associate-*l* 30.1%

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

       5. associate-/l* 30.1%

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

       6. rem-cube-cbrt 30.3%

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

       7. metadata-eval 30.3%

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

   11. Simplified 30.3%

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

   if 4.8000000000000003e-105 < b

   1. Initial program 13.3%

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

      1. sqr-neg 13.3%

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

      2. sqr-neg 13.3%

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

      3. associate-*l* 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in b around inf 73.4%

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

      1. associate-/l* 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in a around 0 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

   10. Simplified 89.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-147}:\\ \;\;\;\;b \cdot \left(\frac{c \cdot -0.5}{-{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.35 \cdot 10^{-186}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.35e-186)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 4.8e-105)
     (* (sqrt (* c (/ -3.0 a))) 0.3333333333333333)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.35e-186) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 4.8e-105) {
		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 <= (-3.35d-186)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 4.8d-105) 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 <= -3.35e-186) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 4.8e-105) {
		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 <= -3.35e-186:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 4.8e-105:
		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 <= -3.35e-186)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 4.8e-105)
		tmp = Float64(sqrt(Float64(c * Float64(-3.0 / a))) * 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 <= -3.35e-186)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 4.8e-105)
		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, -3.35e-186], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.8e-105], 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 < -3.35000000000000017e-186

   1. Initial program 79.2%

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

      1. sqr-neg 79.2%

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

      2. sqr-neg 79.2%

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

      3. associate-*l* 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in b around -inf 77.2%

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

      1. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   if -3.35000000000000017e-186 < b < 4.8000000000000003e-105

   1. Initial program 79.0%

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

      1. sqr-neg 79.0%

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

      2. sqr-neg 79.0%

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

      3. associate-*l* 78.9%

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

   3. Simplified 78.9%

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

      1. add-cube-cbrt 78.3%

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

      2. pow3 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. add-cube-cbrt 77.7%

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

      2. pow3 77.7%

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

      3. unpow3 77.7%

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

      4. add-cube-cbrt 78.1%

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

   8. Applied egg-rr 78.1%

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

   9. Taylor expanded in a around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 30.1%

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

       4. associate-*l* 30.1%

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

       5. associate-/l* 30.2%

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

       6. rem-cube-cbrt 30.4%

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

       7. metadata-eval 30.4%

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

   11. Simplified 30.4%

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

   if 4.8000000000000003e-105 < b

   1. Initial program 13.3%

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

      1. sqr-neg 13.3%

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

      2. sqr-neg 13.3%

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

      3. associate-*l* 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in b around inf 73.4%

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

      1. associate-/l* 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in a around 0 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

   10. Simplified 89.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.35 \cdot 10^{-186}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-290}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.85e-290) (/ (* b -2.0) (* a 3.0)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-290) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.85d-290) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-290) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.85e-290:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e-290)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.85e-290)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.85e-290], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.84999999999999989e-290

   1. Initial program 79.7%

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

      1. sqr-neg 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-*l* 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. *-commutative 67.1%

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

   7. Simplified 67.1%

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

   if 1.84999999999999989e-290 < b

   1. Initial program 27.0%

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

      1. sqr-neg 27.0%

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

      2. sqr-neg 27.0%

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

      3. associate-*l* 27.0%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in b around inf 58.6%

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

      1. associate-/l* 60.9%

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

   7. Simplified 60.9%

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

   8. Taylor expanded in a around 0 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-*l/ 73.4%

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

   10. Simplified 73.4%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-290}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-290}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.85e-290) (* (/ b a) -0.6666666666666666) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-290) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.85d-290) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-290) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.85e-290:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e-290)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.85e-290)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.85e-290], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.84999999999999989e-290

   1. Initial program 79.7%

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

      1. sqr-neg 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-*l* 79.7%

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

   3. Simplified 79.7%

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

      1. sub-neg 79.7%

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

      2. sub-neg 79.7%

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

      3. *-commutative 79.7%

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

      4. cancel-sign-sub-inv 79.7%

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

      5. distribute-lft-neg-in 79.7%

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

      6. distribute-rgt-neg-in 79.7%

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

      7. metadata-eval 79.7%

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

      8. associate-*r* 79.7%

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

      9. fma-undefine 79.7%

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

      10. add-sqr-sqrt 79.4%

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

      11. pow2 79.4%

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

      12. fma-undefine 79.4%

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

      13. add-sqr-sqrt 65.4%

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

      14. hypot-define 72.8%

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

   6. Applied egg-rr 72.8%

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

   7. Taylor expanded in b around -inf 67.1%

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

      1. *-commutative 67.1%

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

   9. Simplified 67.1%

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

   if 1.84999999999999989e-290 < b

   1. Initial program 27.0%

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

      1. sqr-neg 27.0%

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

      2. sqr-neg 27.0%

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

      3. associate-*l* 27.0%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in b around inf 58.6%

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

      1. associate-/l* 60.9%

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

   7. Simplified 60.9%

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

   8. Taylor expanded in a around 0 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-*l/ 73.4%

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

   10. Simplified 73.4%

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

Alternative 8: 67.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-290}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.85e-290) (* (/ b a) -0.6666666666666666) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-290) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 <= 1.85d-290) then
        tmp = (b / a) * (-0.6666666666666666d0)
    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 <= 1.85e-290) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.85e-290:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e-290)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	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 <= 1.85e-290)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.85e-290], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.84999999999999989e-290

   1. Initial program 79.7%

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

      1. sqr-neg 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-*l* 79.7%

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

   3. Simplified 79.7%

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

      1. sub-neg 79.7%

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

      2. sub-neg 79.7%

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

      3. *-commutative 79.7%

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

      4. cancel-sign-sub-inv 79.7%

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

      5. distribute-lft-neg-in 79.7%

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

      6. distribute-rgt-neg-in 79.7%

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

      7. metadata-eval 79.7%

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

      8. associate-*r* 79.7%

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

      9. fma-undefine 79.7%

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

      10. add-sqr-sqrt 79.4%

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

      11. pow2 79.4%

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

      12. fma-undefine 79.4%

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

      13. add-sqr-sqrt 65.4%

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

      14. hypot-define 72.8%

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

   6. Applied egg-rr 72.8%

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

   7. Taylor expanded in b around -inf 67.1%

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

      1. *-commutative 67.1%

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

   9. Simplified 67.1%

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

   if 1.84999999999999989e-290 < b

   1. Initial program 27.0%

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

      1. sqr-neg 27.0%

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

      2. sqr-neg 27.0%

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

      3. associate-*l* 27.0%

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

   3. Simplified 27.0%

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

      1. sub-neg 27.0%

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

      2. sub-neg 27.0%

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

      3. *-commutative 27.0%

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

      4. cancel-sign-sub-inv 27.0%

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

      5. distribute-lft-neg-in 27.0%

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

      6. distribute-rgt-neg-in 27.0%

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

      7. metadata-eval 27.0%

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

      8. associate-*r* 27.0%

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

      9. fma-undefine 27.0%

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

      10. add-sqr-sqrt 25.0%

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

      11. pow2 25.0%

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

      12. fma-undefine 25.0%

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

      13. add-sqr-sqrt 23.6%

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

      14. hypot-define 27.4%

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

   6. Applied egg-rr 27.4%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} \]

      3. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} \]

      4. unpow2 0.0%

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

      5. rem-square-sqrt 73.4%

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

      6. metadata-eval 73.4%

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

      7. associate-*r/ 73.4%

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

      8. *-commutative 73.4%

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

   9. Simplified 73.4%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-290}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. sqr-neg 52.1%

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

   2. sqr-neg 52.1%

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

   3. associate-*l* 52.1%

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

3. Simplified 52.1%

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

   1. sub-neg 52.1%

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

   2. sub-neg 52.1%

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

   3. *-commutative 52.1%

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

   4. cancel-sign-sub-inv 52.1%

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

   5. distribute-lft-neg-in 52.1%

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

   6. distribute-rgt-neg-in 52.1%

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

   7. metadata-eval 52.1%

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

   8. associate-*r* 52.1%

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

   9. fma-undefine 52.1%

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

   10. add-sqr-sqrt 51.0%

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

   11. pow2 51.0%

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

   12. fma-undefine 51.0%

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

   13. add-sqr-sqrt 43.5%

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

   14. hypot-define 49.0%

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

6. Applied egg-rr 49.0%

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

7. Taylor expanded in b around -inf 33.3%

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

   1. *-commutative 33.3%

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

9. Simplified 33.3%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))