Cubic critical

Percentage Accurate: 51.0% → 85.8%

Time: 12.7s

Alternatives: 12

Speedup: 16.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e+79)
   (/ (/ (* b -2.0) a) 3.0)
   (if (<= b 4.3e-48)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e+79) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else if (b <= 4.3e-48) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} 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 <= (-9d+79)) then
        tmp = ((b * (-2.0d0)) / a) / 3.0d0
    else if (b <= 4.3d-48) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    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 <= -9e+79) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else if (b <= 4.3e-48) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e+79:
		tmp = ((b * -2.0) / a) / 3.0
	elif b <= 4.3e-48:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e+79)
		tmp = Float64(Float64(Float64(b * -2.0) / a) / 3.0);
	elseif (b <= 4.3e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	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 <= -9e+79)
		tmp = ((b * -2.0) / a) / 3.0;
	elseif (b <= 4.3e-48)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e+79], N[(N[(N[(b * -2.0), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 4.3e-48], 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[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.99999999999999987e79

   1. Initial program 44.7%

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

   3. Applied egg-rr 39.2%

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

      1. associate-/r* 39.2%

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

   5. Simplified 39.2%

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

      1. associate-*r/ 39.2%

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

      2. frac-2neg 39.2%

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

      3. un-div-inv 39.2%

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

      4. associate-*r* 39.2%

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

      5. metadata-eval 39.2%

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

   7. Applied egg-rr 39.2%

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

      1. distribute-neg-frac 39.2%

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

      2. sub-neg 39.2%

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

      3. +-commutative 39.2%

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

      4. distribute-neg-in 39.2%

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

      5. remove-double-neg 39.2%

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

      6. sub-neg 39.2%

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

      7. associate-*l* 39.2%

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

   9. Simplified 39.2%

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

   10. Taylor expanded in b around -inf 95.0%

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

       1. *-commutative 95.0%

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

   12. Simplified 95.0%

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

   if -8.99999999999999987e79 < b < 4.3e-48

   1. Initial program 78.1%

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

   if 4.3e-48 < b

   1. Initial program 17.0%

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

   2. Taylor expanded in b around inf 89.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-61}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e-63)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 3e-61)
     (* -0.3333333333333333 (/ (- b (sqrt (* c (* a -3.0)))) a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-63) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 3e-61) {
		tmp = -0.3333333333333333 * ((b - sqrt((c * (a * -3.0)))) / 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 <= (-1.95d-63)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 3d-61) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt((c * (a * (-3.0d0))))) / 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 <= -1.95e-63) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 3e-61) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e-63:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 3e-61:
		tmp = -0.3333333333333333 * ((b - math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e-63)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 3e-61)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) / 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 <= -1.95e-63)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 3e-61)
		tmp = -0.3333333333333333 * ((b - sqrt((c * (a * -3.0)))) / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e-63], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e-61], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.95000000000000011e-63

   1. Initial program 62.9%

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

   2. Taylor expanded in b around -inf 88.3%

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

   if -1.95000000000000011e-63 < b < 3.00000000000000012e-61

   1. Initial program 71.6%

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

   2. Taylor expanded in b around 0 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l* 66.9%

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

   4. Simplified 66.9%

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

      1. expm1-log1p-u 42.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -3\right)}}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 19.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -3\right)}}{3 \cdot a}\right)} - 1} \]

   6. Applied egg-rr 19.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\left(c \cdot a\right) \cdot -3}}{a \cdot -3}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 42.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\left(c \cdot a\right) \cdot -3}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 67.0%

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

      3. *-lft-identity 67.0%

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

      4. *-commutative 67.0%

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

      5. times-frac 66.9%

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

      6. metadata-eval 66.9%

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

      7. associate-*l* 66.9%

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

   8. Simplified 66.9%

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

   if 3.00000000000000012e-61 < b

   1. Initial program 17.0%

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

   2. Taylor expanded in b around inf 89.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-61}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-54}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-63)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 8e-54)
     (* (- b (sqrt (* c (* a -3.0)))) (/ -0.3333333333333333 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-63) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 8e-54) {
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / 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 <= (-3.6d-63)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 8d-54) then
        tmp = (b - sqrt((c * (a * (-3.0d0))))) * ((-0.3333333333333333d0) / 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 <= -3.6e-63) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 8e-54) {
		tmp = (b - Math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e-63:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 8e-54:
		tmp = (b - math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-63)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 8e-54)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) * Float64(-0.3333333333333333 / 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 <= -3.6e-63)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 8e-54)
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-63], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-54], N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.60000000000000008e-63

   1. Initial program 62.9%

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

   2. Taylor expanded in b around -inf 88.3%

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

   if -3.60000000000000008e-63 < b < 8.0000000000000002e-54

   1. Initial program 71.6%

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

   2. Taylor expanded in b around 0 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l* 66.9%

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

   4. Simplified 66.9%

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

      1. frac-2neg 66.9%

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

      2. div-inv 66.8%

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

      3. neg-sub0 66.8%

         \[\leadsto \color{blue}{\left(0 - \left(\left(-b\right) + \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)} \cdot \frac{1}{-3 \cdot a} \]

      4. add-sqr-sqrt 34.6%

         \[\leadsto \left(0 - \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      5. sqrt-unprod 65.7%

         \[\leadsto \left(0 - \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      6. sqr-neg 65.7%

         \[\leadsto \left(0 - \left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      7. sqrt-unprod 32.0%

         \[\leadsto \left(0 - \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      8. add-sqr-sqrt 65.0%

         \[\leadsto \left(0 - \left(\color{blue}{b} + \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      9. associate--l- 65.0%

         \[\leadsto \color{blue}{\left(\left(0 - b\right) - \sqrt{a \cdot \left(c \cdot -3\right)}\right)} \cdot \frac{1}{-3 \cdot a} \]

      10. neg-sub0 65.0%

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

      11. add-sqr-sqrt 33.0%

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

      12. sqrt-unprod 65.3%

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

      13. sqr-neg 65.3%

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

      14. sqrt-unprod 32.2%

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

      15. add-sqr-sqrt 66.8%

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

      16. *-commutative 66.8%

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

      17. associate-*r* 67.0%

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

      18. *-commutative 67.0%

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

      19. associate-*r* 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-*l/ 66.9%

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

      3. metadata-eval 66.9%

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

      4. associate-*l* 66.9%

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

   8. Simplified 66.9%

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

   if 8.0000000000000002e-54 < b

   1. Initial program 17.0%

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

   2. Taylor expanded in b around inf 89.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-54}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-63}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.45e-63)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 2.9e-62)
     (/ (* -0.3333333333333333 (- b (sqrt (* c (* a -3.0))))) a)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-63) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 2.9e-62) {
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / 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 <= (-2.45d-63)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 2.9d-62) then
        tmp = ((-0.3333333333333333d0) * (b - sqrt((c * (a * (-3.0d0)))))) / 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 <= -2.45e-63) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 2.9e-62) {
		tmp = (-0.3333333333333333 * (b - Math.sqrt((c * (a * -3.0))))) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.45e-63:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 2.9e-62:
		tmp = (-0.3333333333333333 * (b - math.sqrt((c * (a * -3.0))))) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e-63)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 2.9e-62)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(Float64(c * Float64(a * -3.0))))) / 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 <= -2.45e-63)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 2.9e-62)
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e-63], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e-62], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.45000000000000008e-63

   1. Initial program 62.9%

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

   2. Taylor expanded in b around -inf 88.3%

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

   if -2.45000000000000008e-63 < b < 2.89999999999999986e-62

   1. Initial program 71.6%

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

   2. Taylor expanded in b around 0 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l* 66.9%

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

   4. Simplified 66.9%

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

      1. expm1-log1p-u 42.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -3\right)}}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 19.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -3\right)}}{3 \cdot a}\right)} - 1} \]

   6. Applied egg-rr 19.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\left(c \cdot a\right) \cdot -3}}{a \cdot -3}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 42.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\left(c \cdot a\right) \cdot -3}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 67.0%

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

      3. *-lft-identity 67.0%

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

      4. *-commutative 67.0%

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

      5. times-frac 66.9%

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

      6. metadata-eval 66.9%

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

      7. associate-*l* 66.9%

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

   8. Simplified 66.9%

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

      1. associate-*r/ 67.0%

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

   10. Applied egg-rr 67.0%

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

   if 2.89999999999999986e-62 < b

   1. Initial program 17.0%

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

   2. Taylor expanded in b around inf 89.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-63}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-63}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.8e-63)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 1.1e-51)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.8e-63) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.1e-51) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} 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 <= (-8.8d-63)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 1.1d-51) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    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 <= -8.8e-63) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.1e-51) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.8e-63:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 1.1e-51:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.8e-63)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1.1e-51)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	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 <= -8.8e-63)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 1.1e-51)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.8e-63], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-51], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.7999999999999998e-63

   1. Initial program 62.9%

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

   2. Taylor expanded in b around -inf 88.3%

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

   if -8.7999999999999998e-63 < b < 1.1e-51

   1. Initial program 71.6%

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

   2. Taylor expanded in b around 0 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l* 66.9%

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

   4. Simplified 66.9%

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

      1. +-commutative 66.9%

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

      2. unsub-neg 66.9%

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

      3. *-commutative 66.9%

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

      4. associate-*r* 67.1%

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

      5. *-commutative 67.1%

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

      6. associate-*r* 67.0%

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

   6. Applied egg-rr 67.0%

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

      1. associate-*l* 67.1%

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

   8. Simplified 67.1%

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

   if 1.1e-51 < b

   1. Initial program 17.0%

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

   2. Taylor expanded in b around inf 89.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-63}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6: 68.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.5e-309) (/ (/ (* b -2.0) a) 3.0) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-309) {
		tmp = ((b * -2.0) / a) / 3.0;
	} 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 <= 1.5d-309) then
        tmp = ((b * (-2.0d0)) / a) / 3.0d0
    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 <= 1.5e-309) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.5e-309:
		tmp = ((b * -2.0) / a) / 3.0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.5e-309)
		tmp = Float64(Float64(Float64(b * -2.0) / a) / 3.0);
	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 <= 1.5e-309)
		tmp = ((b * -2.0) / a) / 3.0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.5e-309], N[(N[(N[(b * -2.0), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e-309

   1. Initial program 66.3%

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

   3. Applied egg-rr 56.1%

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

      1. associate-/r* 56.2%

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

   5. Simplified 56.2%

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

      1. associate-*r/ 56.2%

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

      2. frac-2neg 56.2%

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

      3. un-div-inv 56.2%

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

      4. associate-*r* 56.2%

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

      5. metadata-eval 56.2%

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

   7. Applied egg-rr 56.2%

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

      1. distribute-neg-frac 56.2%

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

      2. sub-neg 56.2%

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

      3. +-commutative 56.2%

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

      4. distribute-neg-in 56.2%

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

      5. remove-double-neg 56.2%

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

      6. sub-neg 56.2%

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

      7. associate-*l* 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in b around -inf 68.2%

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

       1. *-commutative 68.2%

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

   12. Simplified 68.2%

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

   if 1.5e-309 < b

   1. Initial program 33.0%

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

   2. Taylor expanded in b around inf 69.1%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 48.8% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{-307}:\\ \;\;\;\;b \cdot \frac{-1.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.12e-307) (* b (/ -1.3333333333333333 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.12e-307) {
		tmp = b * (-1.3333333333333333 / 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 <= 1.12d-307) then
        tmp = b * ((-1.3333333333333333d0) / 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 <= 1.12e-307) {
		tmp = b * (-1.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.12e-307:
		tmp = b * (-1.3333333333333333 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.12e-307)
		tmp = Float64(b * Float64(-1.3333333333333333 / 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 <= 1.12e-307)
		tmp = b * (-1.3333333333333333 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.12e-307], N[(b * N[(-1.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.11999999999999994e-307

   1. Initial program 66.3%

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

      1. +-commutative 66.3%

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

      2. sqr-neg 66.3%

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

      3. unsub-neg 66.3%

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

      4. div-sub 66.3%

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

      5. --rgt-identity 66.3%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 66.3%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 66.2%

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

   5. Applied egg-rr 23.0%

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

      1. +-commutative 23.0%

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

      2. associate-+l+ 23.0%

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

      3. associate-*r* 22.9%

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

      4. *-commutative 22.9%

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

      5. associate-*l* 22.9%

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

      6. fma-udef 22.9%

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

      7. *-rgt-identity 22.9%

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

   7. Simplified 22.9%

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

   8. Taylor expanded in b around inf 1.4%

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

      1. *-commutative 1.4%

         \[\leadsto \frac{\color{blue}{b \cdot 4}}{3 \cdot a} \]

   10. Simplified 1.4%

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

       1. frac-2neg 1.4%

          \[\leadsto \color{blue}{\frac{-b \cdot 4}{-3 \cdot a}} \]

       2. div-inv 1.4%

          \[\leadsto \color{blue}{\left(-b \cdot 4\right) \cdot \frac{1}{-3 \cdot a}} \]

       3. distribute-rgt-neg-in 1.4%

          \[\leadsto \color{blue}{\left(b \cdot \left(-4\right)\right)} \cdot \frac{1}{-3 \cdot a} \]

       4. metadata-eval 1.4%

          \[\leadsto \left(b \cdot \color{blue}{-4}\right) \cdot \frac{1}{-3 \cdot a} \]

       5. distribute-lft-neg-in 1.4%

          \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\color{blue}{\left(-3\right) \cdot a}} \]

       6. metadata-eval 1.4%

          \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\color{blue}{-3} \cdot a} \]

       7. *-commutative 1.4%

          \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\color{blue}{a \cdot -3}} \]

       8. add-sqr-sqrt 0.7%

          \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\color{blue}{\sqrt{a \cdot -3} \cdot \sqrt{a \cdot -3}}} \]

       9. sqrt-unprod 10.0%

          \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\color{blue}{\sqrt{\left(a \cdot -3\right) \cdot \left(a \cdot -3\right)}}} \]

       10. swap-sqr 10.0%

           \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\sqrt{\color{blue}{\left(a \cdot a\right) \cdot \left(-3 \cdot -3\right)}}} \]

       11. metadata-eval 10.0%

           \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\sqrt{\left(a \cdot a\right) \cdot \color{blue}{9}}} \]

       12. metadata-eval 10.0%

           \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\sqrt{\left(a \cdot a\right) \cdot \color{blue}{\left(3 \cdot 3\right)}}} \]

       13. swap-sqr 10.0%

           \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\sqrt{\color{blue}{\left(a \cdot 3\right) \cdot \left(a \cdot 3\right)}}} \]

       14. *-commutative 10.0%

           \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\sqrt{\color{blue}{\left(3 \cdot a\right)} \cdot \left(a \cdot 3\right)}} \]

       15. *-commutative 10.0%

           \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\sqrt{\left(3 \cdot a\right) \cdot \color{blue}{\left(3 \cdot a\right)}}} \]

       16. sqrt-unprod 11.0%

           \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\color{blue}{\sqrt{3 \cdot a} \cdot \sqrt{3 \cdot a}}} \]

       17. add-sqr-sqrt 25.7%

           \[\leadsto \left(b \cdot -4\right) \cdot \frac{1}{\color{blue}{3 \cdot a}} \]

       18. associate-/r* 25.7%

           \[\leadsto \left(b \cdot -4\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}} \]

       19. metadata-eval 25.7%

           \[\leadsto \left(b \cdot -4\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} \]

   12. Applied egg-rr 25.7%

       \[\leadsto \color{blue}{\left(b \cdot -4\right) \cdot \frac{0.3333333333333333}{a}} \]
   13. Step-by-step derivation

       1. associate-*l* 25.7%

          \[\leadsto \color{blue}{b \cdot \left(-4 \cdot \frac{0.3333333333333333}{a}\right)} \]

       2. associate-*r/ 25.7%

          \[\leadsto b \cdot \color{blue}{\frac{-4 \cdot 0.3333333333333333}{a}} \]

       3. metadata-eval 25.7%

          \[\leadsto b \cdot \frac{\color{blue}{-1.3333333333333333}}{a} \]

   14. Simplified 25.7%

       \[\leadsto \color{blue}{b \cdot \frac{-1.3333333333333333}{a}} \]

   if 1.11999999999999994e-307 < b

   1. Initial program 33.0%

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

   2. Taylor expanded in b around inf 69.1%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{-307}:\\ \;\;\;\;b \cdot \frac{-1.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8: 68.6% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 66.3%

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

   2. Taylor expanded in b around -inf 68.2%

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

      1. *-commutative 68.2%

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

   4. Simplified 68.2%

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

      1. *-commutative 68.2%

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

      2. clear-num 68.1%

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

      3. un-div-inv 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. associate-/r/ 68.2%

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

   8. Simplified 68.2%

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

   if -1.999999999999994e-310 < b

   1. Initial program 33.0%

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

   2. Taylor expanded in b around inf 69.1%

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

4. Final simplification 68.6%

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

Alternative 9: 68.6% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 66.3%

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

   2. Taylor expanded in b around -inf 68.2%

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

      1. *-commutative 68.2%

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

   4. Simplified 68.2%

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

   if -1.999999999999994e-310 < b

   1. Initial program 33.0%

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

   2. Taylor expanded in b around inf 69.1%

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

4. Final simplification 68.6%

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

Alternative 10: 68.6% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ b (* a -1.5)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 66.3%

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

   3. Applied egg-rr 56.1%

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

      1. associate-/r* 56.2%

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

   5. Simplified 56.2%

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

      1. associate-*r/ 56.2%

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

      2. frac-2neg 56.2%

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

      3. un-div-inv 56.2%

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

      4. associate-*r* 56.2%

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

      5. metadata-eval 56.2%

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

   7. Applied egg-rr 56.2%

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

      1. distribute-neg-frac 56.2%

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

      2. sub-neg 56.2%

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

      3. +-commutative 56.2%

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

      4. distribute-neg-in 56.2%

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

      5. remove-double-neg 56.2%

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

      6. sub-neg 56.2%

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

      7. associate-*l* 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in b around -inf 68.2%

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

       1. *-commutative 68.2%

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

   12. Simplified 68.2%

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

   13. Taylor expanded in b around 0 68.2%

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

       1. metadata-eval 68.2%

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

       2. times-frac 68.2%

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

       3. *-commutative 68.2%

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

       4. associate-/r* 68.3%

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

       5. associate-/l* 68.3%

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

       6. metadata-eval 68.3%

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

       7. associate-/l/ 68.2%

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

   15. Simplified 68.2%

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

   if -1.999999999999994e-310 < b

   1. Initial program 33.0%

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

   2. Taylor expanded in b around inf 69.1%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 68.6% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 66.3%

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

   2. Taylor expanded in b around -inf 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-*l/ 68.2%

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

   4. Simplified 68.2%

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

   if -1.999999999999994e-310 < b

   1. Initial program 33.0%

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

   2. Taylor expanded in b around inf 69.1%

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

4. Final simplification 68.6%

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

Alternative 12: 36.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.8%

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

2. Taylor expanded in b around inf 33.3%

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

3. Final simplification 33.3%

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

Reproduce

herbie shell --seed 2023319 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))