Cubic critical

Percentage Accurate: 50.9% → 84.9%

Time: 23.6s

Alternatives: 11

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: 50.9% 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: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e+55)
   (/ 1.0 (* (/ a b) -1.5))
   (if (<= b 5.6e-104)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e+55) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.6e-104) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.75d+55)) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else if (b <= 5.6d-104) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e+55) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.6e-104) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.75e+55:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= 5.6e-104:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.75e+55)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= 5.6e-104)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.75e+55)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= 5.6e-104)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.75e+55], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-104], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.75000000000000005e55

   1. Initial program 56.2%

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

      1. sqr-neg 56.2%

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

      2. sqr-neg 56.2%

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

      3. associate-*l* 56.2%

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

   3. Simplified 56.2%

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

      1. frac-2neg 56.2%

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

      2. div-inv 56.2%

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

   6. Applied egg-rr 56.2%

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

      1. un-div-inv 56.2%

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

      2. clear-num 56.1%

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

      3. fma-undefine 56.1%

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

      4. add-sqr-sqrt 36.7%

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

      5. hypot-define 48.7%

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

   8. Applied egg-rr 48.7%

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

   9. Taylor expanded in b around -inf 95.7%

      \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. *-commutative 95.7%

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

   11. Simplified 95.7%

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

   if -1.75000000000000005e55 < b < 5.6e-104

   1. Initial program 81.8%

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

   if 5.6e-104 < b

   1. Initial program 20.0%

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

      1. sqr-neg 20.0%

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

      2. sqr-neg 20.0%

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

      3. associate-*l* 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around inf 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e+55)
   (/ 1.0 (* (/ a b) -1.5))
   (if (<= b 5.6e-104)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e+55) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.6e-104) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.75d+55)) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else if (b <= 5.6d-104) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e+55) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.6e-104) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.75e+55:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= 5.6e-104:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.75e+55)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= 5.6e-104)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.75e+55)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= 5.6e-104)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.75e+55], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-104], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.75000000000000005e55

   1. Initial program 56.2%

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

      1. sqr-neg 56.2%

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

      2. sqr-neg 56.2%

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

      3. associate-*l* 56.2%

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

   3. Simplified 56.2%

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

      1. frac-2neg 56.2%

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

      2. div-inv 56.2%

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

   6. Applied egg-rr 56.2%

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

      1. un-div-inv 56.2%

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

      2. clear-num 56.1%

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

      3. fma-undefine 56.1%

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

      4. add-sqr-sqrt 36.7%

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

      5. hypot-define 48.7%

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

   8. Applied egg-rr 48.7%

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

   9. Taylor expanded in b around -inf 95.7%

      \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. *-commutative 95.7%

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

   11. Simplified 95.7%

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

   if -1.75000000000000005e55 < b < 5.6e-104

   1. Initial program 81.8%

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

      1. sqr-neg 81.8%

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

      2. sqr-neg 81.8%

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

      3. associate-*l* 81.6%

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

   3. Simplified 81.6%

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

   if 5.6e-104 < b

   1. Initial program 20.0%

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

      1. sqr-neg 20.0%

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

      2. sqr-neg 20.0%

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

      3. associate-*l* 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around inf 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-104}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-35)
   (/ 1.0 (* (/ a b) -1.5))
   (if (<= b 5.6e-104)
     (* (- b (sqrt (* c (* a -3.0)))) (/ 1.0 (* a -3.0)))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-35) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.6e-104) {
		tmp = (b - sqrt((c * (a * -3.0)))) * (1.0 / (a * -3.0));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.3d-35)) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else if (b <= 5.6d-104) then
        tmp = (b - sqrt((c * (a * (-3.0d0))))) * (1.0d0 / (a * (-3.0d0)))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-35) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.6e-104) {
		tmp = (b - Math.sqrt((c * (a * -3.0)))) * (1.0 / (a * -3.0));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-35:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= 5.6e-104:
		tmp = (b - math.sqrt((c * (a * -3.0)))) * (1.0 / (a * -3.0))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-35)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= 5.6e-104)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) * Float64(1.0 / Float64(a * -3.0)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e-35)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= 5.6e-104)
		tmp = (b - sqrt((c * (a * -3.0)))) * (1.0 / (a * -3.0));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-35], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-104], N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999999e-35

   1. Initial program 63.1%

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

      1. sqr-neg 63.1%

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

      2. sqr-neg 63.1%

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

      3. associate-*l* 63.1%

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

   3. Simplified 63.1%

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

      1. frac-2neg 63.1%

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

      2. div-inv 63.1%

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

   6. Applied egg-rr 63.1%

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

      1. un-div-inv 63.1%

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

      2. clear-num 63.1%

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

      3. fma-undefine 63.1%

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

      4. add-sqr-sqrt 40.3%

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

      5. hypot-define 50.1%

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

   8. Applied egg-rr 50.1%

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

   9. Taylor expanded in b around -inf 88.9%

      \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. *-commutative 88.9%

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

   11. Simplified 88.9%

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

   if -2.2999999999999999e-35 < b < 5.6e-104

   1. Initial program 79.0%

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

      1. sqr-neg 79.0%

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

      2. sqr-neg 79.0%

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

      3. associate-*l* 78.8%

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

   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. Step-by-step derivation

      1. frac-2neg 78.8%

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

      2. div-inv 78.8%

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in b around 0 72.5%

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

      1. *-commutative 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-*r* 72.7%

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

   9. Simplified 72.7%

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

   if 5.6e-104 < b

   1. Initial program 20.0%

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

      1. sqr-neg 20.0%

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

      2. sqr-neg 20.0%

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

      3. associate-*l* 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around inf 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-104}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-105}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-36)
   (/ 1.0 (* (/ a b) -1.5))
   (if (<= b 2.45e-105)
     (* 0.3333333333333333 (/ (+ b (sqrt (* (* a c) -3.0))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-36) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 2.45e-105) {
		tmp = 0.3333333333333333 * ((b + sqrt(((a * c) * -3.0))) / 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 <= (-8d-36)) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else if (b <= 2.45d-105) then
        tmp = 0.3333333333333333d0 * ((b + sqrt(((a * c) * (-3.0d0)))) / 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 <= -8e-36) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 2.45e-105) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt(((a * c) * -3.0))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8e-36:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= 2.45e-105:
		tmp = 0.3333333333333333 * ((b + math.sqrt(((a * c) * -3.0))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e-36)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= 2.45e-105)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(Float64(a * c) * -3.0))) / 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 <= -8e-36)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= 2.45e-105)
		tmp = 0.3333333333333333 * ((b + sqrt(((a * c) * -3.0))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e-36], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.45e-105], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.9999999999999995e-36

   1. Initial program 63.1%

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

      1. sqr-neg 63.1%

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

      2. sqr-neg 63.1%

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

      3. associate-*l* 63.1%

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

   3. Simplified 63.1%

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

      1. frac-2neg 63.1%

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

      2. div-inv 63.1%

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

   6. Applied egg-rr 63.1%

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

      1. un-div-inv 63.1%

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

      2. clear-num 63.1%

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

      3. fma-undefine 63.1%

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

      4. add-sqr-sqrt 40.3%

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

      5. hypot-define 50.1%

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

   8. Applied egg-rr 50.1%

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

   9. Taylor expanded in b around -inf 88.9%

      \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. *-commutative 88.9%

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

   11. Simplified 88.9%

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

   if -7.9999999999999995e-36 < b < 2.45e-105

   1. Initial program 79.0%

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

      1. sqr-neg 79.0%

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

      2. sqr-neg 79.0%

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

      3. associate-*l* 78.8%

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

   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. Step-by-step derivation

      1. *-un-lft-identity 78.8%

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

      2. times-frac 78.8%

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

      3. metadata-eval 78.8%

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

      4. add-sqr-sqrt 46.9%

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

      5. sqrt-unprod 76.0%

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

      6. sqr-neg 76.0%

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

      7. sqrt-prod 29.3%

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

      8. add-sqr-sqrt 72.0%

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

      9. fma-neg 72.0%

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

      10. associate-*r* 71.9%

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

      11. *-commutative 71.9%

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

      12. distribute-rgt-neg-in 71.9%

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

      13. *-commutative 71.9%

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

      14. distribute-rgt-neg-in 71.9%

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

      15. metadata-eval 71.9%

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

   6. Applied egg-rr 71.9%

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

   7. Taylor expanded in b around 0 71.9%

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

   if 2.45e-105 < b

   1. Initial program 20.0%

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

      1. sqr-neg 20.0%

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

      2. sqr-neg 20.0%

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

      3. associate-*l* 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around inf 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-105}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e-24)
   (/ 1.0 (* (/ a b) -1.5))
   (if (<= b 5.6e-104)
     (/ (- (sqrt (* (* a c) -3.0)) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-24) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.6e-104) {
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.6d-24)) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else if (b <= 5.6d-104) then
        tmp = (sqrt(((a * c) * (-3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-24) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.6e-104) {
		tmp = (Math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.6e-24:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= 5.6e-104:
		tmp = (math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e-24)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= 5.6e-104)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.6e-24)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= 5.6e-104)
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e-24], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-104], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.60000000000000052e-24

   1. Initial program 63.1%

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

      1. sqr-neg 63.1%

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

      2. sqr-neg 63.1%

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

      3. associate-*l* 63.1%

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

   3. Simplified 63.1%

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

      1. frac-2neg 63.1%

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

      2. div-inv 63.1%

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

   6. Applied egg-rr 63.1%

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

      1. un-div-inv 63.1%

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

      2. clear-num 63.1%

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

      3. fma-undefine 63.1%

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

      4. add-sqr-sqrt 40.3%

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

      5. hypot-define 50.1%

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

   8. Applied egg-rr 50.1%

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

   9. Taylor expanded in b around -inf 88.9%

      \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. *-commutative 88.9%

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

   11. Simplified 88.9%

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

   if -7.60000000000000052e-24 < b < 5.6e-104

   1. Initial program 79.0%

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

      1. sqr-neg 79.0%

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

      2. sqr-neg 79.0%

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

      3. associate-*l* 78.8%

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

   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 0 72.5%

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

   if 5.6e-104 < b

   1. Initial program 20.0%

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

      1. sqr-neg 20.0%

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

      2. sqr-neg 20.0%

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

      3. associate-*l* 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around inf 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.8e-38)
   (/ 1.0 (* (/ a b) -1.5))
   (if (<= b 2.2e-104)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.8e-38) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 2.2e-104) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.8d-38)) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else if (b <= 2.2d-104) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.8e-38) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 2.2e-104) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.8e-38:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= 2.2e-104:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.8e-38)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= 2.2e-104)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.8e-38)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= 2.2e-104)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.8e-38], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e-104], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.80000000000000044e-38

   1. Initial program 63.1%

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

      1. sqr-neg 63.1%

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

      2. sqr-neg 63.1%

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

      3. associate-*l* 63.1%

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

   3. Simplified 63.1%

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

      1. frac-2neg 63.1%

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

      2. div-inv 63.1%

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

   6. Applied egg-rr 63.1%

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

      1. un-div-inv 63.1%

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

      2. clear-num 63.1%

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

      3. fma-undefine 63.1%

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

      4. add-sqr-sqrt 40.3%

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

      5. hypot-define 50.1%

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

   8. Applied egg-rr 50.1%

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

   9. Taylor expanded in b around -inf 88.9%

      \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. *-commutative 88.9%

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

   11. Simplified 88.9%

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

   if -4.80000000000000044e-38 < b < 2.20000000000000012e-104

   1. Initial program 79.0%

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

      1. sqr-neg 79.0%

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

      2. sqr-neg 79.0%

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

      3. associate-*l* 78.8%

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

   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 0 72.5%

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

      1. *-commutative 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-*r* 72.7%

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

   7. Simplified 72.7%

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

   if 2.20000000000000012e-104 < b

   1. Initial program 20.0%

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

      1. sqr-neg 20.0%

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

      2. sqr-neg 20.0%

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

      3. associate-*l* 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around inf 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.4e-292) (/ 1.0 (* (/ a b) -1.5)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4e-292) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.4d-292) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4e-292) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.4e-292:
		tmp = 1.0 / ((a / b) * -1.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.4e-292)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.4e-292)
		tmp = 1.0 / ((a / b) * -1.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.4e-292], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4000000000000001e-292

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. sqr-neg 70.4%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

      1. frac-2neg 70.3%

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

      2. div-inv 70.2%

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

   6. Applied egg-rr 70.3%

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

      1. un-div-inv 70.4%

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

      2. clear-num 70.3%

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

      3. fma-undefine 70.3%

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

      4. add-sqr-sqrt 52.9%

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

      5. hypot-define 59.7%

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

   8. Applied egg-rr 59.7%

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

   9. Taylor expanded in b around -inf 65.4%

      \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. *-commutative 65.4%

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

   11. Simplified 65.4%

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

   if 1.4000000000000001e-292 < b

   1. Initial program 31.2%

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

      1. sqr-neg 31.2%

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

      2. sqr-neg 31.2%

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

      3. associate-*l* 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in b around inf 71.9%

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

      1. *-commutative 71.9%

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

   7. Simplified 71.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-291}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4e-291) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-291) {
		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 <= 4d-291) 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 <= 4e-291) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4e-291:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4e-291)
		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 <= 4e-291)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4e-291], 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 < 3.99999999999999985e-291

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. sqr-neg 70.4%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in b around -inf 65.3%

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

      1. *-commutative 65.3%

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

   7. Simplified 65.3%

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

      1. *-commutative 65.3%

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

      2. clear-num 65.2%

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

      3. un-div-inv 65.2%

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

   9. Applied egg-rr 65.2%

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

       1. associate-/r/ 65.3%

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

   11. Simplified 65.3%

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

   if 3.99999999999999985e-291 < b

   1. Initial program 31.2%

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

      1. sqr-neg 31.2%

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

      2. sqr-neg 31.2%

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

      3. associate-*l* 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in b around inf 54.4%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

   7. Simplified 54.5%

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

      1. associate-/l* 63.0%

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

      2. *-commutative 63.0%

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

      3. times-frac 71.6%

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

   9. Applied egg-rr 71.6%

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

       1. *-inverses 71.6%

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

       2. *-lft-identity 71.6%

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

       3. associate-/l/ 71.7%

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

       4. *-commutative 71.7%

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

   11. Simplified 71.7%

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

       1. associate-/l* 71.6%

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

       2. *-commutative 71.6%

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

       3. *-commutative 71.6%

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

       4. associate-/r* 71.7%

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

       5. metadata-eval 71.7%

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

   13. Applied egg-rr 71.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-291}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-294}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4e-294) (* (/ b a) -0.6666666666666666) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-294) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4d-294) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-294) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4e-294:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4e-294)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	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 <= 4e-294)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4e-294], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000007e-294

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. sqr-neg 70.4%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in b around -inf 65.3%

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

      1. *-commutative 65.3%

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

   7. Simplified 65.3%

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

   if 4.00000000000000007e-294 < b

   1. Initial program 31.2%

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

      1. sqr-neg 31.2%

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

      2. sqr-neg 31.2%

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

      3. associate-*l* 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in b around inf 54.4%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

   7. Simplified 54.5%

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

      1. associate-/l* 63.0%

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

      2. *-commutative 63.0%

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

      3. times-frac 71.6%

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

   9. Applied egg-rr 71.6%

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

       1. *-inverses 71.6%

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

       2. *-lft-identity 71.6%

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

       3. associate-/l/ 71.7%

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

       4. *-commutative 71.7%

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

   11. Simplified 71.7%

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

       1. associate-/l* 71.6%

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

       2. *-commutative 71.6%

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

       3. *-commutative 71.6%

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

       4. associate-/r* 71.7%

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

       5. metadata-eval 71.7%

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

   13. Applied egg-rr 71.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-294}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-294}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.1e-294) (* (/ b a) -0.6666666666666666) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.1e-294) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 <= 3.1d-294) then
        tmp = (b / a) * (-0.6666666666666666d0)
    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 <= 3.1e-294) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.1e-294:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.1e-294)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	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 <= 3.1e-294)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.1e-294], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.10000000000000003e-294

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. sqr-neg 70.4%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in b around -inf 65.3%

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

      1. *-commutative 65.3%

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

   7. Simplified 65.3%

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

   if 3.10000000000000003e-294 < b

   1. Initial program 31.2%

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

      1. sqr-neg 31.2%

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

      2. sqr-neg 31.2%

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

      3. associate-*l* 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in b around inf 71.9%

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

      1. *-commutative 71.9%

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

   7. Simplified 71.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-294}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

   1. sqr-neg 50.2%

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

   2. sqr-neg 50.2%

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

   3. associate-*l* 50.1%

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

3. Simplified 50.1%

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

5. Taylor expanded in b around -inf 33.2%

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

   1. *-commutative 33.2%

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

7. Simplified 33.2%

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

   1. *-commutative 33.2%

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

   2. clear-num 33.2%

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

   3. un-div-inv 33.2%

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

9. Applied egg-rr 33.2%

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

    1. associate-/r/ 33.2%

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

11. Simplified 33.2%

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

12. Final simplification 33.2%

    \[\leadsto b \cdot \frac{-0.6666666666666666}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))