Cubic critical

Percentage Accurate: 51.6% → 85.3%

Time: 18.2s

Alternatives: 14

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: 51.6% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+76}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+76)
   (/ (+ (* b -2.0) (* (/ c b) (* a 1.5))) (* a 3.0))
   (if (<= b 8e-43)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+76) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 8e-43) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+76)) then
        tmp = ((b * (-2.0d0)) + ((c / b) * (a * 1.5d0))) / (a * 3.0d0)
    else if (b <= 8d-43) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+76) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 8e-43) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+76:
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0)
	elif b <= 8e-43:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+76)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(c / b) * Float64(a * 1.5))) / Float64(a * 3.0));
	elseif (b <= 8e-43)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+76)
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	elseif (b <= 8e-43)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+76], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * N[(a * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-43], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e76

   1. Initial program 62.1%

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

      1. sqr-neg 62.1%

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

      2. sqr-neg 62.1%

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

      3. associate-*l* 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in b around -inf 88.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r/ 88.0%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]

   7. Applied egg-rr 88.0%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. associate-/l* 88.0%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

      2. associate-*r/ 95.6%

         \[\leadsto \frac{-2 \cdot b + 1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}}{3 \cdot a} \]

      3. *-commutative 95.6%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(a \cdot \frac{c}{b}\right) \cdot 1.5}}{3 \cdot a} \]

      4. *-commutative 95.6%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(\frac{c}{b} \cdot a\right)} \cdot 1.5}{3 \cdot a} \]

      5. associate-*l* 95.6%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   9. Simplified 95.6%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   if -1e76 < b < 8.00000000000000062e-43

   1. Initial program 88.6%

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

   if 8.00000000000000062e-43 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*l/ 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+76}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.1e+72)
   (/ (+ (* b -2.0) (* (/ c b) (* a 1.5))) (* a 3.0))
   (if (<= b 5.1e-34)
     (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e+72) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 5.1e-34) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.1d+72)) then
        tmp = ((b * (-2.0d0)) + ((c / b) * (a * 1.5d0))) / (a * 3.0d0)
    else if (b <= 5.1d-34) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e+72) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 5.1e-34) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.1e+72:
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0)
	elif b <= 5.1e-34:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.1e+72)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(c / b) * Float64(a * 1.5))) / Float64(a * 3.0));
	elseif (b <= 5.1e-34)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.1e+72)
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	elseif (b <= 5.1e-34)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.1e+72], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * N[(a * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.1e-34], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.09999999999999988e72

   1. Initial program 62.1%

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

      1. sqr-neg 62.1%

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

      2. sqr-neg 62.1%

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

      3. associate-*l* 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in b around -inf 88.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r/ 88.0%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]

   7. Applied egg-rr 88.0%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. associate-/l* 88.0%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

      2. associate-*r/ 95.6%

         \[\leadsto \frac{-2 \cdot b + 1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}}{3 \cdot a} \]

      3. *-commutative 95.6%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(a \cdot \frac{c}{b}\right) \cdot 1.5}}{3 \cdot a} \]

      4. *-commutative 95.6%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(\frac{c}{b} \cdot a\right)} \cdot 1.5}{3 \cdot a} \]

      5. associate-*l* 95.6%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   9. Simplified 95.6%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   if -3.09999999999999988e72 < b < 5.1000000000000001e-34

   1. Initial program 88.6%

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

      1. sqr-neg 88.6%

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

      2. sqr-neg 88.6%

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

      3. associate-*l* 88.4%

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

   3. Simplified 88.4%

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

   if 5.1000000000000001e-34 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*l/ 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-40}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-127)
   (/ (+ (* b -2.0) (* (/ c b) (* a 1.5))) (* a 3.0))
   (if (<= b 1.36e-40)
     (* 0.3333333333333333 (/ (+ b (sqrt (* a (* c -3.0)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-127) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 1.36e-40) {
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / 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 <= (-2.9d-127)) then
        tmp = ((b * (-2.0d0)) + ((c / b) * (a * 1.5d0))) / (a * 3.0d0)
    else if (b <= 1.36d-40) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((a * (c * (-3.0d0))))) / 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 <= -2.9e-127) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 1.36e-40) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-127:
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0)
	elif b <= 1.36e-40:
		tmp = 0.3333333333333333 * ((b + math.sqrt((a * (c * -3.0)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-127)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(c / b) * Float64(a * 1.5))) / Float64(a * 3.0));
	elseif (b <= 1.36e-40)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e-127)
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	elseif (b <= 1.36e-40)
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-127], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * N[(a * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.36e-40], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.9e-127

   1. Initial program 74.3%

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

      1. sqr-neg 74.3%

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

      2. sqr-neg 74.3%

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

      3. associate-*l* 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in b around -inf 82.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r/ 82.3%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]

   7. Applied egg-rr 82.3%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. associate-/l* 82.3%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

      2. associate-*r/ 87.4%

         \[\leadsto \frac{-2 \cdot b + 1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}}{3 \cdot a} \]

      3. *-commutative 87.4%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(a \cdot \frac{c}{b}\right) \cdot 1.5}}{3 \cdot a} \]

      4. *-commutative 87.4%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(\frac{c}{b} \cdot a\right)} \cdot 1.5}{3 \cdot a} \]

      5. associate-*l* 87.4%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   9. Simplified 87.4%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   if -2.9e-127 < b < 1.3599999999999999e-40

   1. Initial program 83.5%

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

      1. sqr-neg 83.5%

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

      2. sqr-neg 83.5%

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

      3. associate-*l* 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in b around 0 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*r* 82.9%

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

   7. Simplified 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. times-frac 83.0%

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

      3. metadata-eval 83.0%

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

      4. +-commutative 83.0%

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

      5. add-sqr-sqrt 41.8%

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

      6. sqrt-unprod 82.8%

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

      7. sqr-neg 82.8%

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

      8. sqrt-unprod 41.2%

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

      9. add-sqr-sqrt 82.8%

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

   9. Applied egg-rr 82.8%

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

   if 1.3599999999999999e-40 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*l/ 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-40}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-43}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.2e-128)
   (/ (+ (* b -2.0) (* (/ c b) (* a 1.5))) (* a 3.0))
   (if (<= b 8e-43)
     (* (- b (sqrt (* c (* a -3.0)))) (/ -0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.2e-128) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 8e-43) {
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / 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 <= (-9.2d-128)) then
        tmp = ((b * (-2.0d0)) + ((c / b) * (a * 1.5d0))) / (a * 3.0d0)
    else if (b <= 8d-43) then
        tmp = (b - sqrt((c * (a * (-3.0d0))))) * ((-0.3333333333333333d0) / 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 <= -9.2e-128) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 8e-43) {
		tmp = (b - Math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.2e-128:
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0)
	elif b <= 8e-43:
		tmp = (b - math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.2e-128)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(c / b) * Float64(a * 1.5))) / Float64(a * 3.0));
	elseif (b <= 8e-43)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.2e-128)
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	elseif (b <= 8e-43)
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.2e-128], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * N[(a * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-43], N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.2000000000000003e-128

   1. Initial program 74.3%

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

      1. sqr-neg 74.3%

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

      2. sqr-neg 74.3%

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

      3. associate-*l* 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in b around -inf 82.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r/ 82.3%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]

   7. Applied egg-rr 82.3%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. associate-/l* 82.3%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

      2. associate-*r/ 87.4%

         \[\leadsto \frac{-2 \cdot b + 1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}}{3 \cdot a} \]

      3. *-commutative 87.4%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(a \cdot \frac{c}{b}\right) \cdot 1.5}}{3 \cdot a} \]

      4. *-commutative 87.4%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(\frac{c}{b} \cdot a\right)} \cdot 1.5}{3 \cdot a} \]

      5. associate-*l* 87.4%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   9. Simplified 87.4%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   if -9.2000000000000003e-128 < b < 8.00000000000000062e-43

   1. Initial program 83.5%

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

      1. sqr-neg 83.5%

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

      2. sqr-neg 83.5%

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

      3. associate-*l* 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in b around 0 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. *-commutative 82.7%

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

   7. Applied egg-rr 83.1%

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

      1. associate-*l/ 83.1%

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

      2. associate-*r/ 83.2%

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

      3. *-commutative 83.2%

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

      4. associate-/r* 82.9%

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

      5. metadata-eval 82.9%

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

   9. Simplified 82.9%

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

   if 8.00000000000000062e-43 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*l/ 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-43}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-127)
   (/ (+ (* b -2.0) (* (/ c b) (* a 1.5))) (* a 3.0))
   (if (<= b 7.8e-43)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-127) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 7.8e-43) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.9d-127)) then
        tmp = ((b * (-2.0d0)) + ((c / b) * (a * 1.5d0))) / (a * 3.0d0)
    else if (b <= 7.8d-43) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-127) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else if (b <= 7.8e-43) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-127:
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0)
	elif b <= 7.8e-43:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-127)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(c / b) * Float64(a * 1.5))) / Float64(a * 3.0));
	elseif (b <= 7.8e-43)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e-127)
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	elseif (b <= 7.8e-43)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-127], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * N[(a * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.8e-43], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.9e-127

   1. Initial program 74.3%

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

      1. sqr-neg 74.3%

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

      2. sqr-neg 74.3%

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

      3. associate-*l* 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in b around -inf 82.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r/ 82.3%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]

   7. Applied egg-rr 82.3%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. associate-/l* 82.3%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

      2. associate-*r/ 87.4%

         \[\leadsto \frac{-2 \cdot b + 1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}}{3 \cdot a} \]

      3. *-commutative 87.4%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(a \cdot \frac{c}{b}\right) \cdot 1.5}}{3 \cdot a} \]

      4. *-commutative 87.4%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(\frac{c}{b} \cdot a\right)} \cdot 1.5}{3 \cdot a} \]

      5. associate-*l* 87.4%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   9. Simplified 87.4%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   if -2.9e-127 < b < 7.80000000000000001e-43

   1. Initial program 83.5%

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

      1. sqr-neg 83.5%

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

      2. sqr-neg 83.5%

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

      3. associate-*l* 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in b around 0 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*r* 82.9%

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

   7. Simplified 82.9%

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

   if 7.80000000000000001e-43 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*l/ 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310)
   (/ (+ (* b -2.0) (* (/ c b) (* a 1.5))) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = ((b * (-2.0d0)) + ((c / b) * (a * 1.5d0))) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(c / b) * Float64(a * 1.5))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = ((b * -2.0) + ((c / b) * (a * 1.5))) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * N[(a * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 78.4%

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

      1. sqr-neg 78.4%

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

      2. sqr-neg 78.4%

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

      3. associate-*l* 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in b around -inf 64.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r/ 64.1%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]

   7. Applied egg-rr 64.1%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. associate-/l* 64.1%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

      2. associate-*r/ 68.3%

         \[\leadsto \frac{-2 \cdot b + 1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}}{3 \cdot a} \]

      3. *-commutative 68.3%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(a \cdot \frac{c}{b}\right) \cdot 1.5}}{3 \cdot a} \]

      4. *-commutative 68.3%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left(\frac{c}{b} \cdot a\right)} \cdot 1.5}{3 \cdot a} \]

      5. associate-*l* 68.3%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   9. Simplified 68.3%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)}}{3 \cdot a} \]

   if -9.999999999999969e-311 < b

   1. Initial program 33.5%

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

      1. sqr-neg 33.5%

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

      2. sqr-neg 33.5%

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

      3. associate-*l* 33.5%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in b around inf 69.0%

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

      1. *-commutative 69.0%

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

      2. associate-*l/ 69.0%

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

   7. Simplified 69.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c}{b} \cdot \left(a \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} 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 <= (-1d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    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 <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 78.4%

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

      1. sqr-neg 78.4%

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

      2. sqr-neg 78.4%

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

      3. associate-*l* 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in b around -inf 68.2%

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

   if -9.999999999999969e-311 < b

   1. Initial program 33.5%

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

      1. sqr-neg 33.5%

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

      2. sqr-neg 33.5%

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

      3. associate-*l* 33.5%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in b around inf 69.0%

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

      1. *-commutative 69.0%

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

      2. associate-*l/ 69.0%

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

   7. Simplified 69.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (/ (/ (* b 2.0) a) -3.0) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-245) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = ((b * 2.0) / a) / -3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-245)
		tmp = ((b * 2.0) / a) / -3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

      1. sqr-neg 78.9%

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

      2. sqr-neg 78.9%

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

      3. associate-*l* 78.8%

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

   3. Simplified 78.8%

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

   6. Applied egg-rr 66.6%

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

      1. associate-/r* 66.6%

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

   8. Simplified 66.6%

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

      1. associate-*r/ 66.5%

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

   10. Applied egg-rr 66.5%

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

   11. Taylor expanded in b around -inf 66.3%

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

       1. associate-*r/ 66.3%

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

   13. Simplified 66.3%

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

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

      1. sqr-neg 31.9%

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

      2. sqr-neg 31.9%

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

      3. associate-*l* 31.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in b around inf 70.6%

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

      1. *-commutative 70.6%

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

      2. associate-*l/ 70.6%

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

   7. Simplified 70.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+18}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.3e+18) (* b (/ -0.6666666666666666 a)) (* (/ c b) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.3e+18) {
		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 <= 1.3d+18) 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 <= 1.3e+18) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.3e+18:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.3e+18)
		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 <= 1.3e+18)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.3e+18], 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 < 1.3e18

   1. Initial program 74.2%

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

      1. sqr-neg 74.2%

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

      2. sqr-neg 74.2%

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

      3. associate-*l* 74.1%

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

   3. Simplified 74.1%

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

   6. Applied egg-rr 65.0%

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

      1. associate-/r* 64.9%

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

   8. Simplified 64.9%

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

   9. Taylor expanded in b around -inf 48.6%

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

       1. associate-*r/ 48.6%

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

       2. *-commutative 48.6%

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

       3. associate-/l* 48.6%

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

   11. Simplified 48.6%

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

   if 1.3e18 < b

   1. Initial program 10.9%

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

      1. sqr-neg 10.9%

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

      2. sqr-neg 10.9%

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

      3. associate-*l* 10.9%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in b around -inf 2.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

   6. Taylor expanded in b around 0 28.2%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+18}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		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 <= 5.1d-245) 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 <= 5.1e-245) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		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 <= 5.1e-245)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], 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 < 5.1000000000000003e-245

   1. Initial program 78.9%

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

      1. sqr-neg 78.9%

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

      2. sqr-neg 78.9%

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

      3. associate-*l* 78.8%

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

   3. Simplified 78.8%

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

   6. Applied egg-rr 66.6%

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

      1. associate-/r* 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in b around -inf 66.3%

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

       1. associate-*r/ 66.3%

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

       2. *-commutative 66.3%

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

       3. associate-/l* 66.3%

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

   11. Simplified 66.3%

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

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

      1. sqr-neg 31.9%

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

      2. sqr-neg 31.9%

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

      3. associate-*l* 31.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in b around inf 52.3%

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

      1. associate-/l* 52.2%

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

      2. *-un-lft-identity 52.2%

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

      3. times-frac 52.3%

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

      4. metadata-eval 52.3%

         \[\leadsto -1.5 \cdot \left(\color{blue}{0.3333333333333333} \cdot \frac{\frac{a \cdot c}{b}}{a}\right) \]

      5. associate-/l* 56.8%

         \[\leadsto -1.5 \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{a}\right) \]

   7. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{-1.5 \cdot \left(0.3333333333333333 \cdot \frac{a \cdot \frac{c}{b}}{a}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 56.8%

         \[\leadsto \color{blue}{\left(-1.5 \cdot 0.3333333333333333\right) \cdot \frac{a \cdot \frac{c}{b}}{a}} \]

      2. clear-num 56.3%

         \[\leadsto \left(-1.5 \cdot 0.3333333333333333\right) \cdot \color{blue}{\frac{1}{\frac{a}{a \cdot \frac{c}{b}}}} \]

      3. un-div-inv 56.3%

         \[\leadsto \color{blue}{\frac{-1.5 \cdot 0.3333333333333333}{\frac{a}{a \cdot \frac{c}{b}}}} \]

      4. metadata-eval 56.3%

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

      5. *-un-lft-identity 56.3%

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

      6. *-commutative 56.3%

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

      7. times-frac 69.5%

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

      8. clear-num 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. *-inverses 69.3%

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

       2. associate-/r* 69.3%

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

       3. /-rgt-identity 69.3%

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

       4. associate-/r/ 70.4%

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

   11. Simplified 70.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (* -0.6666666666666666 (/ b a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-245) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(-0.6666666666666666 * Float64(b / 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 <= 5.1e-245)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

      1. sqr-neg 78.9%

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

      2. sqr-neg 78.9%

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

      3. associate-*l* 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   7. Simplified 66.3%

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

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

      1. sqr-neg 31.9%

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

      2. sqr-neg 31.9%

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

      3. associate-*l* 31.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in b around inf 52.3%

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

      1. associate-/l* 52.2%

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

      2. *-un-lft-identity 52.2%

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

      3. times-frac 52.3%

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

      4. metadata-eval 52.3%

         \[\leadsto -1.5 \cdot \left(\color{blue}{0.3333333333333333} \cdot \frac{\frac{a \cdot c}{b}}{a}\right) \]

      5. associate-/l* 56.8%

         \[\leadsto -1.5 \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{a}\right) \]

   7. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{-1.5 \cdot \left(0.3333333333333333 \cdot \frac{a \cdot \frac{c}{b}}{a}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 56.8%

         \[\leadsto \color{blue}{\left(-1.5 \cdot 0.3333333333333333\right) \cdot \frac{a \cdot \frac{c}{b}}{a}} \]

      2. clear-num 56.3%

         \[\leadsto \left(-1.5 \cdot 0.3333333333333333\right) \cdot \color{blue}{\frac{1}{\frac{a}{a \cdot \frac{c}{b}}}} \]

      3. un-div-inv 56.3%

         \[\leadsto \color{blue}{\frac{-1.5 \cdot 0.3333333333333333}{\frac{a}{a \cdot \frac{c}{b}}}} \]

      4. metadata-eval 56.3%

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

      5. *-un-lft-identity 56.3%

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

      6. *-commutative 56.3%

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

      7. times-frac 69.5%

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

      8. clear-num 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. *-inverses 69.3%

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

       2. associate-/r* 69.3%

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

       3. /-rgt-identity 69.3%

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

       4. associate-/r/ 70.4%

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

   11. Simplified 70.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (/ b (* a -1.5)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = b / (a * -1.5);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-245) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = b / (a * -1.5);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = b / (a * -1.5)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(b / Float64(a * -1.5));
	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 <= 5.1e-245)
		tmp = b / (a * -1.5);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

      1. sqr-neg 78.9%

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

      2. sqr-neg 78.9%

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

      3. associate-*l* 78.8%

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

   3. Simplified 78.8%

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

   6. Applied egg-rr 66.6%

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

      1. associate-/r* 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in b around -inf 66.3%

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

       1. associate-*r/ 66.3%

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

       2. *-commutative 66.3%

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

       3. associate-/l* 66.3%

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

   11. Simplified 66.3%

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

       1. add-cbrt-cube 42.6%

          \[\leadsto b \cdot \color{blue}{\sqrt[3]{\left(\frac{-0.6666666666666666}{a} \cdot \frac{-0.6666666666666666}{a}\right) \cdot \frac{-0.6666666666666666}{a}}} \]

       2. pow1/3 21.8%

          \[\leadsto b \cdot \color{blue}{{\left(\left(\frac{-0.6666666666666666}{a} \cdot \frac{-0.6666666666666666}{a}\right) \cdot \frac{-0.6666666666666666}{a}\right)}^{0.3333333333333333}} \]

       3. pow3 21.8%

          \[\leadsto b \cdot {\color{blue}{\left({\left(\frac{-0.6666666666666666}{a}\right)}^{3}\right)}}^{0.3333333333333333} \]

   13. Applied egg-rr 21.8%

       \[\leadsto b \cdot \color{blue}{{\left({\left(\frac{-0.6666666666666666}{a}\right)}^{3}\right)}^{0.3333333333333333}} \]
   14. Step-by-step derivation

       1. unpow1/3 42.6%

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

       2. rem-cbrt-cube 66.3%

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

       3. clear-num 66.3%

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

       4. un-div-inv 66.3%

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

       5. div-inv 66.3%

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

       6. metadata-eval 66.3%

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

   15. Applied egg-rr 66.3%

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

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

      1. sqr-neg 31.9%

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

      2. sqr-neg 31.9%

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

      3. associate-*l* 31.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in b around inf 52.3%

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

      1. associate-/l* 52.2%

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

      2. *-un-lft-identity 52.2%

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

      3. times-frac 52.3%

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

      4. metadata-eval 52.3%

         \[\leadsto -1.5 \cdot \left(\color{blue}{0.3333333333333333} \cdot \frac{\frac{a \cdot c}{b}}{a}\right) \]

      5. associate-/l* 56.8%

         \[\leadsto -1.5 \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{a}\right) \]

   7. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{-1.5 \cdot \left(0.3333333333333333 \cdot \frac{a \cdot \frac{c}{b}}{a}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 56.8%

         \[\leadsto \color{blue}{\left(-1.5 \cdot 0.3333333333333333\right) \cdot \frac{a \cdot \frac{c}{b}}{a}} \]

      2. clear-num 56.3%

         \[\leadsto \left(-1.5 \cdot 0.3333333333333333\right) \cdot \color{blue}{\frac{1}{\frac{a}{a \cdot \frac{c}{b}}}} \]

      3. un-div-inv 56.3%

         \[\leadsto \color{blue}{\frac{-1.5 \cdot 0.3333333333333333}{\frac{a}{a \cdot \frac{c}{b}}}} \]

      4. metadata-eval 56.3%

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

      5. *-un-lft-identity 56.3%

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

      6. *-commutative 56.3%

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

      7. times-frac 69.5%

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

      8. clear-num 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. *-inverses 69.3%

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

       2. associate-/r* 69.3%

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

       3. /-rgt-identity 69.3%

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

       4. associate-/r/ 70.4%

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

   11. Simplified 70.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-245) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-245) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-245) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-245:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-245)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-245)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-245], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000003e-245

   1. Initial program 78.9%

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

      1. sqr-neg 78.9%

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

      2. sqr-neg 78.9%

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

      3. associate-*l* 78.8%

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

   3. Simplified 78.8%

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

   6. Applied egg-rr 66.6%

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

      1. associate-/r* 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in b around -inf 66.3%

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

       1. associate-*r/ 66.3%

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

       2. *-commutative 66.3%

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

       3. associate-/l* 66.3%

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

   11. Simplified 66.3%

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

       1. add-cbrt-cube 42.6%

          \[\leadsto b \cdot \color{blue}{\sqrt[3]{\left(\frac{-0.6666666666666666}{a} \cdot \frac{-0.6666666666666666}{a}\right) \cdot \frac{-0.6666666666666666}{a}}} \]

       2. pow1/3 21.8%

          \[\leadsto b \cdot \color{blue}{{\left(\left(\frac{-0.6666666666666666}{a} \cdot \frac{-0.6666666666666666}{a}\right) \cdot \frac{-0.6666666666666666}{a}\right)}^{0.3333333333333333}} \]

       3. pow3 21.8%

          \[\leadsto b \cdot {\color{blue}{\left({\left(\frac{-0.6666666666666666}{a}\right)}^{3}\right)}}^{0.3333333333333333} \]

   13. Applied egg-rr 21.8%

       \[\leadsto b \cdot \color{blue}{{\left({\left(\frac{-0.6666666666666666}{a}\right)}^{3}\right)}^{0.3333333333333333}} \]
   14. Step-by-step derivation

       1. unpow1/3 42.6%

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

       2. rem-cbrt-cube 66.3%

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

       3. clear-num 66.3%

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

       4. un-div-inv 66.3%

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

       5. div-inv 66.3%

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

       6. metadata-eval 66.3%

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

   15. Applied egg-rr 66.3%

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

   if 5.1000000000000003e-245 < b

   1. Initial program 31.9%

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

      1. sqr-neg 31.9%

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

      2. sqr-neg 31.9%

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

      3. associate-*l* 31.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in b around inf 70.6%

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

      1. *-commutative 70.6%

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

      2. associate-*l/ 70.6%

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

   7. Simplified 70.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 11.2% 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 56.1%

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

   1. sqr-neg 56.1%

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

   2. sqr-neg 56.1%

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

   3. associate-*l* 56.0%

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

3. Simplified 56.0%

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

5. Taylor expanded in b around -inf 33.4%

   \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

6. Taylor expanded in b around 0 10.2%

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

7. Final simplification 10.2%

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

Reproduce

herbie shell --seed 2024041 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))