Cubic critical

Percentage Accurate: 52.2% → 85.3%

Time: 14.8s

Alternatives: 10

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+118}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+118)
   (/ b (* a -1.5))
   (if (<= b 4.1e-132)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+118) {
		tmp = b / (a * -1.5);
	} else if (b <= 4.1e-132) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -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+118)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 4.1d-132) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+118) {
		tmp = b / (a * -1.5);
	} else if (b <= 4.1e-132) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+118:
		tmp = b / (a * -1.5)
	elif b <= 4.1e-132:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+118)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 4.1e-132)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+118)
		tmp = b / (a * -1.5);
	elseif (b <= 4.1e-132)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+118], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.1e-132], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.99999999999999972e118

   1. Initial program 49.3%

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

      1. sqr-neg 49.3%

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

      2. sqr-neg 49.3%

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

      3. associate-*l* 49.3%

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

   3. Simplified 49.3%

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

   6. Applied egg-rr 46.7%

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

   7. Taylor expanded in b around -inf 96.6%

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

      1. *-commutative 96.6%

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

      2. associate-*l/ 96.6%

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

      3. associate-/l* 96.5%

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

   9. Simplified 96.5%

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

       1. clear-num 96.6%

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

       2. un-div-inv 96.7%

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

       3. div-inv 96.7%

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

       4. metadata-eval 96.7%

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

   11. Applied egg-rr 96.7%

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

   if -4.99999999999999972e118 < b < 4.10000000000000007e-132

   1. Initial program 87.1%

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

   if 4.10000000000000007e-132 < b

   1. Initial program 19.9%

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

      1. sqr-neg 19.9%

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

      2. sqr-neg 19.9%

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

      3. associate-*l* 19.9%

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

   3. Simplified 19.9%

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

   5. Taylor expanded in b around inf 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+118}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+118}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+118)
   (/ b (* a -1.5))
   (if (<= b 3.9e-132)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+118) {
		tmp = b / (a * -1.5);
	} else if (b <= 3.9e-132) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -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+118)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 3.9d-132) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+118) {
		tmp = b / (a * -1.5);
	} else if (b <= 3.9e-132) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+118:
		tmp = b / (a * -1.5)
	elif b <= 3.9e-132:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+118)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 3.9e-132)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+118)
		tmp = b / (a * -1.5);
	elseif (b <= 3.9e-132)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+118], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-132], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.99999999999999972e118

   1. Initial program 49.3%

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

      1. sqr-neg 49.3%

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

      2. sqr-neg 49.3%

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

      3. associate-*l* 49.3%

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

   3. Simplified 49.3%

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

   6. Applied egg-rr 46.7%

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

   7. Taylor expanded in b around -inf 96.6%

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

      1. *-commutative 96.6%

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

      2. associate-*l/ 96.6%

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

      3. associate-/l* 96.5%

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

   9. Simplified 96.5%

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

       1. clear-num 96.6%

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

       2. un-div-inv 96.7%

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

       3. div-inv 96.7%

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

       4. metadata-eval 96.7%

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

   11. Applied egg-rr 96.7%

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

   if -4.99999999999999972e118 < b < 3.89999999999999982e-132

   1. Initial program 87.1%

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

      1. sqr-neg 87.1%

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

      2. sqr-neg 87.1%

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

      3. associate-*l* 87.0%

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

   3. Simplified 87.0%

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

   if 3.89999999999999982e-132 < b

   1. Initial program 19.9%

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

      1. sqr-neg 19.9%

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

      2. sqr-neg 19.9%

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

      3. associate-*l* 19.9%

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

   3. Simplified 19.9%

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

   5. Taylor expanded in b around inf 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+118}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.7e-38)
   (/ b (* a -1.5))
   (if (<= b 7.5e-145)
     (* (/ -0.3333333333333333 a) (- b (sqrt (* c (* a -3.0)))))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.7e-38) {
		tmp = b / (a * -1.5);
	} else if (b <= 7.5e-145) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt((c * (a * -3.0))));
	} else {
		tmp = (c / b) * -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 <= (-4.7d-38)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 7.5d-145) then
        tmp = ((-0.3333333333333333d0) / a) * (b - sqrt((c * (a * (-3.0d0)))))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.7e-38) {
		tmp = b / (a * -1.5);
	} else if (b <= 7.5e-145) {
		tmp = (-0.3333333333333333 / a) * (b - Math.sqrt((c * (a * -3.0))));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.7e-38:
		tmp = b / (a * -1.5)
	elif b <= 7.5e-145:
		tmp = (-0.3333333333333333 / a) * (b - math.sqrt((c * (a * -3.0))))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.7e-38)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 7.5e-145)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(Float64(c * Float64(a * -3.0)))));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.7e-38)
		tmp = b / (a * -1.5);
	elseif (b <= 7.5e-145)
		tmp = (-0.3333333333333333 / a) * (b - sqrt((c * (a * -3.0))));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.7e-38], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-145], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.69999999999999998e-38

   1. Initial program 69.9%

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

      1. sqr-neg 69.9%

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

      2. sqr-neg 69.9%

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

      3. associate-*l* 69.9%

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

   3. Simplified 69.9%

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

   6. Applied egg-rr 59.3%

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

   7. Taylor expanded in b around -inf 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l/ 87.1%

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

      3. associate-/l* 87.1%

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

   9. Simplified 87.1%

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

       1. clear-num 87.0%

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

       2. un-div-inv 87.1%

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

       3. div-inv 87.2%

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

       4. metadata-eval 87.2%

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

   11. Applied egg-rr 87.2%

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

   if -4.69999999999999998e-38 < b < 7.4999999999999996e-145

   1. Initial program 83.9%

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

   3. Taylor expanded in b around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*r* 72.9%

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

   5. Simplified 72.9%

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

      1. frac-2neg 72.9%

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

      2. div-inv 72.7%

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

      3. distribute-neg-in 72.7%

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

      4. add-sqr-sqrt 39.8%

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

      5. sqrt-unprod 72.6%

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

      6. sqr-neg 72.6%

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

      7. sqrt-unprod 33.0%

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

      8. add-sqr-sqrt 70.5%

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

      9. sub-neg 70.5%

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

      10. add-sqr-sqrt 37.5%

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

      11. sqrt-unprod 70.4%

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

      12. sqr-neg 70.4%

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

      13. sqrt-unprod 32.9%

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

      14. add-sqr-sqrt 72.7%

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

      15. *-commutative 72.7%

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

      16. associate-*r* 72.8%

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

      17. distribute-lft-neg-in 72.8%

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

      18. metadata-eval 72.8%

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

   7. Applied egg-rr 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-/r* 72.6%

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

      3. metadata-eval 72.6%

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

      4. *-commutative 72.6%

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

   9. Simplified 72.6%

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

   if 7.4999999999999996e-145 < b

   1. Initial program 20.6%

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

      1. sqr-neg 20.6%

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

      2. sqr-neg 20.6%

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

      3. associate-*l* 20.6%

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

   3. Simplified 20.6%

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

   5. Taylor expanded in b around inf 83.4%

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

      1. *-commutative 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -14800000000:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -14800000000.0)
   (/ b (* a -1.5))
   (if (<= b 1.35e-144)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -14800000000.0) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.35e-144) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -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 <= (-14800000000.0d0)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.35d-144) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -14800000000.0) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.35e-144) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -14800000000.0:
		tmp = b / (a * -1.5)
	elif b <= 1.35e-144:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -14800000000.0)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.35e-144)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -14800000000.0)
		tmp = b / (a * -1.5);
	elseif (b <= 1.35e-144)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -14800000000.0], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-144], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.48e10

   1. Initial program 65.9%

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

      1. sqr-neg 65.9%

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

      2. sqr-neg 65.9%

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

      3. associate-*l* 65.9%

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

   3. Simplified 65.9%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in b around -inf 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*l/ 93.8%

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

      3. associate-/l* 93.7%

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

   9. Simplified 93.7%

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

       1. clear-num 93.6%

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

       2. un-div-inv 93.7%

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

       3. div-inv 93.8%

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

       4. metadata-eval 93.8%

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

   11. Applied egg-rr 93.8%

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

   if -1.48e10 < b < 1.34999999999999988e-144

   1. Initial program 85.3%

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

   3. Taylor expanded in b around 0 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

   5. Simplified 69.2%

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

      1. +-commutative 69.2%

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

      2. unsub-neg 69.2%

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

      3. *-commutative 69.2%

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

      4. associate-*r* 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. associate-*r* 69.2%

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

      2. rem-square-sqrt 0.0%

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

      3. unpow2 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 69.2%

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

   9. Simplified 69.2%

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

   if 1.34999999999999988e-144 < b

   1. Initial program 20.6%

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

      1. sqr-neg 20.6%

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

      2. sqr-neg 20.6%

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

      3. associate-*l* 20.6%

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

   3. Simplified 20.6%

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

   5. Taylor expanded in b around inf 83.4%

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

      1. *-commutative 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -14800000000:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -14800000000:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.18 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -14800000000.0)
   (/ b (* a -1.5))
   (if (<= b 1.18e-144)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -14800000000.0) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.18e-144) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -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 <= (-14800000000.0d0)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.18d-144) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -14800000000.0) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.18e-144) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -14800000000.0:
		tmp = b / (a * -1.5)
	elif b <= 1.18e-144:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -14800000000.0)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.18e-144)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -14800000000.0)
		tmp = b / (a * -1.5);
	elseif (b <= 1.18e-144)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -14800000000.0], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.18e-144], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.48e10

   1. Initial program 65.9%

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

      1. sqr-neg 65.9%

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

      2. sqr-neg 65.9%

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

      3. associate-*l* 65.9%

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

   3. Simplified 65.9%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in b around -inf 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*l/ 93.8%

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

      3. associate-/l* 93.7%

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

   9. Simplified 93.7%

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

       1. clear-num 93.6%

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

       2. un-div-inv 93.7%

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

       3. div-inv 93.8%

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

       4. metadata-eval 93.8%

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

   11. Applied egg-rr 93.8%

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

   if -1.48e10 < b < 1.18e-144

   1. Initial program 85.3%

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

   3. Taylor expanded in b around 0 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

   5. Simplified 69.2%

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

      1. +-commutative 69.2%

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

      2. unsub-neg 69.2%

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

      3. *-commutative 69.2%

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

      4. associate-*r* 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. *-commutative 69.3%

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

   9. Simplified 69.3%

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

   if 1.18e-144 < b

   1. Initial program 20.6%

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

      1. sqr-neg 20.6%

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

      2. sqr-neg 20.6%

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

      3. associate-*l* 20.6%

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

   3. Simplified 20.6%

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

   5. Taylor expanded in b around inf 83.4%

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

      1. *-commutative 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -14800000000:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.18 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.8e-5) (* b (/ -0.6666666666666666 a)) (* (/ c b) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8e-5) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 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 <= 4.8d-5) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8e-5) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.8e-5:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8e-5)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.8e-5)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8e-5], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8000000000000001e-5

   1. Initial program 68.5%

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

      1. sqr-neg 68.5%

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

      2. sqr-neg 68.5%

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

      3. associate-*l* 68.4%

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

   3. Simplified 68.4%

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

   6. Applied egg-rr 61.7%

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

   7. Taylor expanded in b around -inf 47.7%

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

      1. *-commutative 47.7%

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

      2. associate-*l/ 47.7%

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

      3. associate-/l* 47.7%

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

   9. Simplified 47.7%

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

   if 4.8000000000000001e-5 < b

   1. Initial program 18.0%

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

      1. sqr-neg 18.0%

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

      2. sqr-neg 18.0%

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

      3. associate-*l* 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in b around inf 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-/l* 72.2%

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

      3. associate-*r* 72.3%

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

      4. *-commutative 72.3%

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

      5. associate-*r/ 72.2%

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

   7. Simplified 72.2%

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

      1. expm1-log1p-u 62.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \frac{-1.5 \cdot c}{b}}{3 \cdot a}\right)\right)} \]

      2. expm1-undefine 35.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot \frac{-1.5 \cdot c}{b}}{3 \cdot a}\right)} - 1} \]

      3. associate-/l* 34.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{a \cdot \frac{\frac{-1.5 \cdot c}{b}}{3 \cdot a}}\right)} - 1 \]

      4. associate-/l* 34.5%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \frac{\color{blue}{-1.5 \cdot \frac{c}{b}}}{3 \cdot a}\right)} - 1 \]

      5. *-commutative 34.5%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \frac{-1.5 \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}}\right)} - 1 \]

   9. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \frac{-1.5 \cdot \frac{c}{b}}{a \cdot 3}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-define 76.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{-1.5 \cdot \frac{c}{b}}{a \cdot 3}\right)\right)} \]

       2. associate-*r/ 62.7%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a \cdot \left(-1.5 \cdot \frac{c}{b}\right)}{a \cdot 3}}\right)\right) \]

       3. times-frac 80.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a}{a} \cdot \frac{-1.5 \cdot \frac{c}{b}}{3}}\right)\right) \]

       4. *-inverses 80.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \frac{-1.5 \cdot \frac{c}{b}}{3}\right)\right) \]

       5. *-commutative 80.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 \cdot \frac{\color{blue}{\frac{c}{b} \cdot -1.5}}{3}\right)\right) \]

   11. Simplified 80.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 \cdot \frac{\frac{c}{b} \cdot -1.5}{3}\right)\right)} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 90.5%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{c}{b} \cdot -1.5}{3}} \]

       2. *-un-lft-identity 90.5%

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

       3. frac-2neg 90.5%

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

       4. associate-*l/ 90.3%

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

       5. distribute-neg-frac2 90.3%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 30.7%

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

       8. sqr-neg 30.7%

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

       9. sqrt-unprod 30.2%

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

       10. add-sqr-sqrt 30.2%

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

       11. metadata-eval 30.2%

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

   13. Applied egg-rr 30.2%

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

       1. associate-/l/ 30.2%

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

       2. *-commutative 30.2%

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

       3. times-frac 30.2%

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

       4. metadata-eval 30.2%

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

   15. Simplified 30.2%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 75.4%

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

      1. sqr-neg 75.4%

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

      2. sqr-neg 75.4%

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

      3. associate-*l* 75.4%

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

   3. Simplified 75.4%

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

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in b around -inf 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-*l/ 67.7%

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

      3. associate-/l* 67.7%

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

   9. Simplified 67.7%

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

   if -4.999999999999985e-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. Step-by-step derivation

      1. sqr-neg 33.0%

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

      2. sqr-neg 33.0%

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

      3. associate-*l* 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in b around inf 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-/l* 53.7%

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

      3. associate-*r* 53.8%

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

      4. *-commutative 53.8%

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

      5. associate-*r/ 53.7%

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

   7. Simplified 53.7%

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

      1. expm1-log1p-u 46.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \frac{-1.5 \cdot c}{b}}{3 \cdot a}\right)\right)} \]

      2. expm1-undefine 23.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot \frac{-1.5 \cdot c}{b}}{3 \cdot a}\right)} - 1} \]

      3. associate-/l* 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{a \cdot \frac{\frac{-1.5 \cdot c}{b}}{3 \cdot a}}\right)} - 1 \]

      4. associate-/l* 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \frac{\color{blue}{-1.5 \cdot \frac{c}{b}}}{3 \cdot a}\right)} - 1 \]

      5. *-commutative 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \frac{-1.5 \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}}\right)} - 1 \]

   9. Applied egg-rr 22.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \frac{-1.5 \cdot \frac{c}{b}}{a \cdot 3}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-define 56.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{-1.5 \cdot \frac{c}{b}}{a \cdot 3}\right)\right)} \]

       2. associate-*r/ 47.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a \cdot \left(-1.5 \cdot \frac{c}{b}\right)}{a \cdot 3}}\right)\right) \]

       3. times-frac 60.3%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a}{a} \cdot \frac{-1.5 \cdot \frac{c}{b}}{3}}\right)\right) \]

       4. *-inverses 60.3%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \frac{-1.5 \cdot \frac{c}{b}}{3}\right)\right) \]

       5. *-commutative 60.3%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 \cdot \frac{\color{blue}{\frac{c}{b} \cdot -1.5}}{3}\right)\right) \]

   11. Simplified 60.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 \cdot \frac{\frac{c}{b} \cdot -1.5}{3}\right)\right)} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 67.2%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{c}{b} \cdot -1.5}{3}} \]

       2. *-un-lft-identity 67.2%

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

       3. associate-/l* 67.3%

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

       4. div-inv 67.1%

          \[\leadsto \color{blue}{\left(c \cdot \frac{1}{b}\right)} \cdot \frac{-1.5}{3} \]

       5. associate-*l* 67.1%

          \[\leadsto \color{blue}{c \cdot \left(\frac{1}{b} \cdot \frac{-1.5}{3}\right)} \]

       6. metadata-eval 67.1%

          \[\leadsto c \cdot \left(\frac{1}{b} \cdot \color{blue}{-0.5}\right) \]

   13. Applied egg-rr 67.1%

       \[\leadsto \color{blue}{c \cdot \left(\frac{1}{b} \cdot -0.5\right)} \]
   14. Step-by-step derivation

       1. associate-*l/ 67.1%

          \[\leadsto c \cdot \color{blue}{\frac{1 \cdot -0.5}{b}} \]

       2. metadata-eval 67.1%

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

   15. Simplified 67.1%

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

4. Final simplification 67.4%

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

Alternative 8: 67.7% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * -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);
	else
		tmp = (c / b) * -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], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 75.4%

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

      1. sqr-neg 75.4%

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

      2. sqr-neg 75.4%

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

      3. associate-*l* 75.4%

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

   3. Simplified 75.4%

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

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in b around -inf 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-*l/ 67.7%

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

      3. associate-/l* 67.7%

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

   9. Simplified 67.7%

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

   if -4.999999999999985e-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. Step-by-step derivation

      1. sqr-neg 33.0%

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

      2. sqr-neg 33.0%

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

      3. associate-*l* 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in b around inf 67.3%

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

      1. *-commutative 67.3%

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

   7. Simplified 67.3%

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

4. Final simplification 67.5%

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

Alternative 9: 67.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -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 / (a * (-1.5d0))
    else
        tmp = (c / b) * (-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 / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 75.4%

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

      1. sqr-neg 75.4%

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

      2. sqr-neg 75.4%

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

      3. associate-*l* 75.4%

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

   3. Simplified 75.4%

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

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in b around -inf 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-*l/ 67.7%

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

      3. associate-/l* 67.7%

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

   9. Simplified 67.7%

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

       1. clear-num 67.6%

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

       2. un-div-inv 67.6%

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

       3. div-inv 67.8%

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

       4. metadata-eval 67.8%

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

   11. Applied egg-rr 67.8%

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

   if -4.999999999999985e-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. Step-by-step derivation

      1. sqr-neg 33.0%

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

      2. sqr-neg 33.0%

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

      3. associate-*l* 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in b around inf 67.3%

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

      1. *-commutative 67.3%

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

   7. Simplified 67.3%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

   1. sqr-neg 53.7%

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

   2. sqr-neg 53.7%

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

   3. associate-*l* 53.7%

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

3. Simplified 53.7%

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

5. Taylor expanded in b around inf 25.8%

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

   1. *-commutative 25.8%

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

   2. associate-/l* 28.4%

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

   3. associate-*r* 28.5%

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

   4. *-commutative 28.5%

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

   5. associate-*r/ 28.4%

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

7. Simplified 28.4%

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

   1. expm1-log1p-u 24.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \frac{-1.5 \cdot c}{b}}{3 \cdot a}\right)\right)} \]

   2. expm1-undefine 13.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot \frac{-1.5 \cdot c}{b}}{3 \cdot a}\right)} - 1} \]

   3. associate-/l* 12.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{a \cdot \frac{\frac{-1.5 \cdot c}{b}}{3 \cdot a}}\right)} - 1 \]

   4. associate-/l* 12.2%

      \[\leadsto e^{\mathsf{log1p}\left(a \cdot \frac{\color{blue}{-1.5 \cdot \frac{c}{b}}}{3 \cdot a}\right)} - 1 \]

   5. *-commutative 12.2%

      \[\leadsto e^{\mathsf{log1p}\left(a \cdot \frac{-1.5 \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}}\right)} - 1 \]

9. Applied egg-rr 12.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \frac{-1.5 \cdot \frac{c}{b}}{a \cdot 3}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-define 29.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{-1.5 \cdot \frac{c}{b}}{a \cdot 3}\right)\right)} \]

    2. associate-*r/ 24.9%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a \cdot \left(-1.5 \cdot \frac{c}{b}\right)}{a \cdot 3}}\right)\right) \]

    3. times-frac 31.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a}{a} \cdot \frac{-1.5 \cdot \frac{c}{b}}{3}}\right)\right) \]

    4. *-inverses 31.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \frac{-1.5 \cdot \frac{c}{b}}{3}\right)\right) \]

    5. *-commutative 31.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 \cdot \frac{\color{blue}{\frac{c}{b} \cdot -1.5}}{3}\right)\right) \]

11. Simplified 31.8%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 \cdot \frac{\frac{c}{b} \cdot -1.5}{3}\right)\right)} \]
12. Step-by-step derivation

    1. expm1-log1p-u 35.3%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{c}{b} \cdot -1.5}{3}} \]

    2. *-un-lft-identity 35.3%

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

    3. frac-2neg 35.3%

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

    4. associate-*l/ 35.3%

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

    5. distribute-neg-frac2 35.3%

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

    6. add-sqr-sqrt 1.0%

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

    7. sqrt-unprod 10.2%

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

    8. sqr-neg 10.2%

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

    9. sqrt-unprod 9.2%

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

    10. add-sqr-sqrt 11.2%

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

    11. metadata-eval 11.2%

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

13. Applied egg-rr 11.2%

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

    1. associate-/l/ 11.2%

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

    2. *-commutative 11.2%

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

    3. times-frac 11.2%

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

    4. metadata-eval 11.2%

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

15. Simplified 11.2%

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

16. Final simplification 11.2%

    \[\leadsto \frac{c}{b} \cdot 0.5 \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))