Cubic critical

Percentage Accurate: 60.8% → 82.8%

Time: 16.8s

Alternatives: 9

Speedup: 1.0×

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: 60.8% 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: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -24000000:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -24000000.0)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 6.5e+86)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (* (/ (* c (/ a b)) a) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -24000000.0) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 6.5e+86) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	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 <= (-24000000.0d0)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 6.5d+86) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = ((c * (a / b)) / a) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -24000000.0) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 6.5e+86) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -24000000.0:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 6.5e+86:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = ((c * (a / b)) / a) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -24000000.0)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 6.5e+86)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * Float64(a / b)) / a) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -24000000.0)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 6.5e+86)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = ((c * (a / b)) / a) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -24000000.0], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+86], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4e7

   1. Initial program 64.7%

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

   3. Taylor expanded in b around -inf 94.2%

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

      1. *-commutative 94.2%

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

   5. Simplified 94.2%

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

   if -2.4e7 < b < 6.49999999999999996e86

   1. Initial program 74.3%

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

   if 6.49999999999999996e86 < b

   1. Initial program 24.8%

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

   3. Taylor expanded in b around inf 67.5%

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

      1. associate-/l* 73.6%

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

      2. associate-/r/ 76.1%

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

   5. Simplified 76.1%

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

      1. *-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. times-frac 76.2%

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

      4. *-commutative 76.2%

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

      5. metadata-eval 76.2%

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

   7. Applied egg-rr 76.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -24000000:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-55}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-18}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-55)
   (* b (/ -0.6666666666666666 a))
   (if (<= b 8e-18)
     (* 0.3333333333333333 (/ (+ b (sqrt (* c (* a -3.0)))) a))
     (* (/ (* c (/ a b)) a) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-55) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 8e-18) {
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	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 <= (-3d-55)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else if (b <= 8d-18) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((c * (a * (-3.0d0))))) / a)
    else
        tmp = ((c * (a / b)) / a) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-55) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 8e-18) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-55:
		tmp = b * (-0.6666666666666666 / a)
	elif b <= 8e-18:
		tmp = 0.3333333333333333 * ((b + math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = ((c * (a / b)) / a) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-55)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 8e-18)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / a));
	else
		tmp = Float64(Float64(Float64(c * Float64(a / b)) / a) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-55)
		tmp = b * (-0.6666666666666666 / a);
	elseif (b <= 8e-18)
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	else
		tmp = ((c * (a / b)) / a) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-55], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-18], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000016e-55

   1. Initial program 70.5%

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

   4. Applied egg-rr 57.6%

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

      1. sub-neg 57.6%

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

      2. distribute-rgt-out-- 57.6%

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

   6. Simplified 57.6%

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

      1. expm1-log1p-u 38.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   8. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   9. Taylor expanded in b around -inf 91.9%

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

       1. metadata-eval 91.9%

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

       2. distribute-lft-neg-in 91.9%

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

       3. *-lft-identity 91.9%

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

       4. associate-*l/ 91.8%

          \[\leadsto -0.6666666666666666 \cdot \color{blue}{\left(\frac{1}{a} \cdot b\right)} \]

       5. associate-*r* 91.8%

          \[\leadsto -\color{blue}{\left(0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

       6. *-commutative 91.8%

          \[\leadsto -\color{blue}{b \cdot \left(0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       7. distribute-rgt-neg-in 91.8%

          \[\leadsto \color{blue}{b \cdot \left(-0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       8. associate-*r/ 91.9%

          \[\leadsto b \cdot \left(-\color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \]

       9. metadata-eval 91.9%

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

       10. distribute-neg-frac 91.9%

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

       11. metadata-eval 91.9%

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

   11. Simplified 91.9%

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

   if -3.00000000000000016e-55 < b < 8.0000000000000006e-18

   1. Initial program 71.8%

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

   3. Taylor expanded in b around 0 63.2%

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

      1. associate-*r* 63.2%

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

      2. *-commutative 63.2%

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

      3. *-commutative 63.2%

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

   5. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. times-frac 63.2%

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

      3. metadata-eval 63.2%

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

      4. +-commutative 63.2%

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

      5. add-sqr-sqrt 35.5%

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

      6. sqrt-unprod 62.9%

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

      7. sqr-neg 62.9%

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

      8. sqrt-unprod 28.1%

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

      9. add-sqr-sqrt 62.1%

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

   7. Applied egg-rr 62.1%

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

   if 8.0000000000000006e-18 < b

   1. Initial program 39.4%

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

   3. Taylor expanded in b around inf 61.6%

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

      1. associate-/l* 65.7%

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

      2. associate-/r/ 67.4%

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

   5. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. *-commutative 67.4%

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

      3. times-frac 67.5%

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

      4. *-commutative 67.5%

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

      5. metadata-eval 67.5%

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

   7. Applied egg-rr 67.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-55}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-18}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-57}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e-57)
   (* b (/ -0.6666666666666666 a))
   (if (<= b 7.6e-20)
     (/ (- (sqrt (* a (* c -3.0))) b) (* 3.0 a))
     (* (/ (* c (/ a b)) a) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-57) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 7.6e-20) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	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 <= (-2.1d-57)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else if (b <= 7.6d-20) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = ((c * (a / b)) / a) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-57) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 7.6e-20) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e-57:
		tmp = b * (-0.6666666666666666 / a)
	elif b <= 7.6e-20:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (3.0 * a)
	else:
		tmp = ((c * (a / b)) / a) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e-57)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 7.6e-20)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * Float64(a / b)) / a) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e-57)
		tmp = b * (-0.6666666666666666 / a);
	elseif (b <= 7.6e-20)
		tmp = (sqrt((a * (c * -3.0))) - b) / (3.0 * a);
	else
		tmp = ((c * (a / b)) / a) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e-57], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.6e-20], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0999999999999999e-57

   1. Initial program 70.5%

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

   4. Applied egg-rr 57.6%

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

      1. sub-neg 57.6%

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

      2. distribute-rgt-out-- 57.6%

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

   6. Simplified 57.6%

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

      1. expm1-log1p-u 38.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   8. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   9. Taylor expanded in b around -inf 91.9%

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

       1. metadata-eval 91.9%

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

       2. distribute-lft-neg-in 91.9%

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

       3. *-lft-identity 91.9%

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

       4. associate-*l/ 91.8%

          \[\leadsto -0.6666666666666666 \cdot \color{blue}{\left(\frac{1}{a} \cdot b\right)} \]

       5. associate-*r* 91.8%

          \[\leadsto -\color{blue}{\left(0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

       6. *-commutative 91.8%

          \[\leadsto -\color{blue}{b \cdot \left(0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       7. distribute-rgt-neg-in 91.8%

          \[\leadsto \color{blue}{b \cdot \left(-0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       8. associate-*r/ 91.9%

          \[\leadsto b \cdot \left(-\color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \]

       9. metadata-eval 91.9%

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

       10. distribute-neg-frac 91.9%

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

       11. metadata-eval 91.9%

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

   11. Simplified 91.9%

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

   if -2.0999999999999999e-57 < b < 7.5999999999999995e-20

   1. Initial program 71.8%

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

   3. Taylor expanded in b around 0 63.2%

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

      1. associate-*r* 63.2%

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

      2. *-commutative 63.2%

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

      3. *-commutative 63.2%

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

   5. Simplified 63.2%

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

      1. +-commutative 63.2%

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

      2. unsub-neg 63.2%

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

   7. Applied egg-rr 63.2%

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

      1. associate-*r* 63.2%

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

      2. *-commutative 63.2%

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

      3. rem-square-sqrt 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*r* 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 63.1%

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

   9. Simplified 63.1%

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

   if 7.5999999999999995e-20 < b

   1. Initial program 39.4%

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

   3. Taylor expanded in b around inf 61.6%

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

      1. associate-/l* 65.7%

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

      2. associate-/r/ 67.4%

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

   5. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. *-commutative 67.4%

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

      3. times-frac 67.5%

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

      4. *-commutative 67.5%

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

      5. metadata-eval 67.5%

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

   7. Applied egg-rr 67.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-57}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-54}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e-54)
   (* b (/ -0.6666666666666666 a))
   (if (<= b 5.3e-23)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (* (/ (* c (/ a b)) a) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-54) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 5.3e-23) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	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 <= (-6.5d-54)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else if (b <= 5.3d-23) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = ((c * (a / b)) / a) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-54) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 5.3e-23) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-54:
		tmp = b * (-0.6666666666666666 / a)
	elif b <= 5.3e-23:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (3.0 * a)
	else:
		tmp = ((c * (a / b)) / a) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-54)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 5.3e-23)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * Float64(a / b)) / a) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-54)
		tmp = b * (-0.6666666666666666 / a);
	elseif (b <= 5.3e-23)
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	else
		tmp = ((c * (a / b)) / a) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-54], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.3e-23], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.49999999999999991e-54

   1. Initial program 70.5%

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

   4. Applied egg-rr 57.6%

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

      1. sub-neg 57.6%

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

      2. distribute-rgt-out-- 57.6%

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

   6. Simplified 57.6%

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

      1. expm1-log1p-u 38.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   8. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   9. Taylor expanded in b around -inf 91.9%

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

       1. metadata-eval 91.9%

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

       2. distribute-lft-neg-in 91.9%

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

       3. *-lft-identity 91.9%

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

       4. associate-*l/ 91.8%

          \[\leadsto -0.6666666666666666 \cdot \color{blue}{\left(\frac{1}{a} \cdot b\right)} \]

       5. associate-*r* 91.8%

          \[\leadsto -\color{blue}{\left(0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

       6. *-commutative 91.8%

          \[\leadsto -\color{blue}{b \cdot \left(0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       7. distribute-rgt-neg-in 91.8%

          \[\leadsto \color{blue}{b \cdot \left(-0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       8. associate-*r/ 91.9%

          \[\leadsto b \cdot \left(-\color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \]

       9. metadata-eval 91.9%

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

       10. distribute-neg-frac 91.9%

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

       11. metadata-eval 91.9%

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

   11. Simplified 91.9%

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

   if -6.49999999999999991e-54 < b < 5.30000000000000042e-23

   1. Initial program 71.8%

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

   3. Taylor expanded in b around 0 63.2%

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

      1. associate-*r* 63.2%

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

      2. *-commutative 63.2%

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

      3. *-commutative 63.2%

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

   5. Simplified 63.2%

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

      1. +-commutative 63.2%

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

      2. unsub-neg 63.2%

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

   7. Applied egg-rr 63.2%

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

   if 5.30000000000000042e-23 < b

   1. Initial program 39.4%

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

   3. Taylor expanded in b around inf 61.6%

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

      1. associate-/l* 65.7%

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

      2. associate-/r/ 67.4%

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

   5. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. *-commutative 67.4%

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

      3. times-frac 67.5%

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

      4. *-commutative 67.5%

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

      5. metadata-eval 67.5%

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

   7. Applied egg-rr 67.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-54}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right) - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (* b (/ -0.6666666666666666 a))
   (if (<= b 1.8e+86)
     (/ (- (+ b (* -1.5 (/ (* a c) b))) b) (* 3.0 a))
     (* (/ (* c (/ a b)) a) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 1.8e+86) {
		tmp = ((b + (-1.5 * ((a * c) / b))) - b) / (3.0 * a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else if (b <= 1.8d+86) then
        tmp = ((b + ((-1.5d0) * ((a * c) / b))) - b) / (3.0d0 * a)
    else
        tmp = ((c * (a / b)) / a) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 1.8e+86) {
		tmp = ((b + (-1.5 * ((a * c) / b))) - b) / (3.0 * a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b * (-0.6666666666666666 / a)
	elif b <= 1.8e+86:
		tmp = ((b + (-1.5 * ((a * c) / b))) - b) / (3.0 * a)
	else:
		tmp = ((c * (a / b)) / a) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 1.8e+86)
		tmp = Float64(Float64(Float64(b + Float64(-1.5 * Float64(Float64(a * c) / b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * Float64(a / b)) / a) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b * (-0.6666666666666666 / a);
	elseif (b <= 1.8e+86)
		tmp = ((b + (-1.5 * ((a * c) / b))) - b) / (3.0 * a);
	else
		tmp = ((c * (a / b)) / a) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e+86], N[(N[(N[(b + N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 72.6%

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

   4. Applied egg-rr 65.3%

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

      1. sub-neg 65.3%

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

      2. distribute-rgt-out-- 65.3%

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

   6. Simplified 65.3%

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

      1. expm1-log1p-u 46.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   8. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   9. Taylor expanded in b around -inf 66.9%

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

       1. metadata-eval 66.9%

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

       2. distribute-lft-neg-in 66.9%

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

       3. *-lft-identity 66.9%

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

       4. associate-*l/ 66.8%

          \[\leadsto -0.6666666666666666 \cdot \color{blue}{\left(\frac{1}{a} \cdot b\right)} \]

       5. associate-*r* 66.8%

          \[\leadsto -\color{blue}{\left(0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

       6. *-commutative 66.8%

          \[\leadsto -\color{blue}{b \cdot \left(0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       7. distribute-rgt-neg-in 66.8%

          \[\leadsto \color{blue}{b \cdot \left(-0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       8. associate-*r/ 66.9%

          \[\leadsto b \cdot \left(-\color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \]

       9. metadata-eval 66.9%

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

       10. distribute-neg-frac 66.9%

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

       11. metadata-eval 66.9%

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

   11. Simplified 66.9%

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

   if -4.999999999999985e-310 < b < 1.80000000000000003e86

   1. Initial program 68.1%

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

   3. Taylor expanded in b around inf 33.1%

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

   if 1.80000000000000003e86 < b

   1. Initial program 24.8%

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

   3. Taylor expanded in b around inf 67.5%

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

      1. associate-/l* 73.6%

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

      2. associate-/r/ 76.1%

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

   5. Simplified 76.1%

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

      1. *-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. times-frac 76.2%

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

      4. *-commutative 76.2%

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

      5. metadata-eval 76.2%

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

   7. Applied egg-rr 76.2%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right) - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-309}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2e-309)
   (* b (/ -0.6666666666666666 a))
   (* (/ (* c (/ a b)) a) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-309) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	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-309) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = ((c * (a / b)) / a) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-309) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = ((c * (a / b)) / a) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2e-309:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = ((c * (a / b)) / a) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2e-309)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(Float64(c * Float64(a / b)) / a) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2e-309)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = ((c * (a / b)) / a) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2e-309], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999988e-309

   1. Initial program 72.6%

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

   4. Applied egg-rr 65.3%

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

      1. sub-neg 65.3%

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

      2. distribute-rgt-out-- 65.3%

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

   6. Simplified 65.3%

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

      1. expm1-log1p-u 46.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   8. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   9. Taylor expanded in b around -inf 66.9%

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

       1. metadata-eval 66.9%

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

       2. distribute-lft-neg-in 66.9%

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

       3. *-lft-identity 66.9%

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

       4. associate-*l/ 66.8%

          \[\leadsto -0.6666666666666666 \cdot \color{blue}{\left(\frac{1}{a} \cdot b\right)} \]

       5. associate-*r* 66.8%

          \[\leadsto -\color{blue}{\left(0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

       6. *-commutative 66.8%

          \[\leadsto -\color{blue}{b \cdot \left(0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       7. distribute-rgt-neg-in 66.8%

          \[\leadsto \color{blue}{b \cdot \left(-0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       8. associate-*r/ 66.9%

          \[\leadsto b \cdot \left(-\color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \]

       9. metadata-eval 66.9%

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

       10. distribute-neg-frac 66.9%

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

       11. metadata-eval 66.9%

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

   11. Simplified 66.9%

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

   if 1.9999999999999988e-309 < b

   1. Initial program 48.3%

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

   3. Taylor expanded in b around inf 45.5%

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

      1. associate-/l* 48.5%

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

      2. associate-/r/ 49.6%

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

   5. Simplified 49.6%

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

      1. *-commutative 49.6%

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

      2. *-commutative 49.6%

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

      3. times-frac 49.6%

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

      4. *-commutative 49.6%

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

      5. metadata-eval 49.6%

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

   7. Applied egg-rr 49.6%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-309}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{a} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 25.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.85 \cdot 10^{+25}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= a 2.85e+25) 0.0 (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (a <= 2.85e+25) {
		tmp = 0.0;
	} 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 (a <= 2.85d+25) then
        tmp = 0.0d0
    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 (a <= 2.85e+25) {
		tmp = 0.0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if a <= 2.85e+25:
		tmp = 0.0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (a <= 2.85e+25)
		tmp = 0.0;
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (a <= 2.85e+25)
		tmp = 0.0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[a, 2.85e+25], 0.0, N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.8499999999999998e25

   1. Initial program 66.8%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in b around inf 2.3%

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

      1. expm1-log1p-u 1.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{b}{a}\right)\right)} \]

      2. expm1-udef 3.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{b}{a}\right)} - 1} \]

   7. Applied egg-rr 3.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{b}{a}\right)} - 1} \]

   8. Taylor expanded in b around 0 30.2%

      \[\leadsto \color{blue}{1} - 1 \]

   if 2.8499999999999998e25 < a

   1. Initial program 39.8%

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

   3. Taylor expanded in b around inf 24.0%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.85 \cdot 10^{+25}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* b (/ -0.6666666666666666 a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 72.6%

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

   4. Applied egg-rr 65.3%

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

      1. sub-neg 65.3%

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

      2. distribute-rgt-out-- 65.3%

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

   6. Simplified 65.3%

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

      1. expm1-log1p-u 46.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   8. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right)} \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) - b\right) \]

   9. Taylor expanded in b around -inf 66.9%

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

       1. metadata-eval 66.9%

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

       2. distribute-lft-neg-in 66.9%

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

       3. *-lft-identity 66.9%

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

       4. associate-*l/ 66.8%

          \[\leadsto -0.6666666666666666 \cdot \color{blue}{\left(\frac{1}{a} \cdot b\right)} \]

       5. associate-*r* 66.8%

          \[\leadsto -\color{blue}{\left(0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

       6. *-commutative 66.8%

          \[\leadsto -\color{blue}{b \cdot \left(0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       7. distribute-rgt-neg-in 66.8%

          \[\leadsto \color{blue}{b \cdot \left(-0.6666666666666666 \cdot \frac{1}{a}\right)} \]

       8. associate-*r/ 66.9%

          \[\leadsto b \cdot \left(-\color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \]

       9. metadata-eval 66.9%

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

       10. distribute-neg-frac 66.9%

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

       11. metadata-eval 66.9%

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

   11. Simplified 66.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 48.3%

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

   4. Applied egg-rr 19.8%

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

   5. Taylor expanded in b around inf 3.5%

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

      1. expm1-log1p-u 2.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{b}{a}\right)\right)} \]

      2. expm1-udef 6.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{b}{a}\right)} - 1} \]

   7. Applied egg-rr 6.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{b}{a}\right)} - 1} \]

   8. Taylor expanded in b around 0 47.0%

      \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.7% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 60.3%

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

4. Applied egg-rr 17.7%

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

5. Taylor expanded in b around inf 2.6%

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

   1. expm1-log1p-u 2.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{b}{a}\right)\right)} \]

   2. expm1-udef 4.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{b}{a}\right)} - 1} \]

7. Applied egg-rr 4.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{b}{a}\right)} - 1} \]

8. Taylor expanded in b around 0 25.1%

   \[\leadsto \color{blue}{1} - 1 \]

9. Final simplification 25.1%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))