Cubic critical

Percentage Accurate: 52.6% → 86.2%

Time: 14.2s

Alternatives: 15

Speedup: 16.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.6% 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: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.72 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{b} \cdot a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.72e+72)
   (/ 1.0 (* (/ -1.5 b) a))
   (if (<= b 1.3e-104)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.72e+72) {
		tmp = 1.0 / ((-1.5 / b) * a);
	} else if (b <= 1.3e-104) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - 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 <= (-1.72d+72)) then
        tmp = 1.0d0 / (((-1.5d0) / b) * a)
    else if (b <= 1.3d-104) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - 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 <= -1.72e+72) {
		tmp = 1.0 / ((-1.5 / b) * a);
	} else if (b <= 1.3e-104) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.72e+72:
		tmp = 1.0 / ((-1.5 / b) * a)
	elif b <= 1.3e-104:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.72e+72)
		tmp = Float64(1.0 / Float64(Float64(-1.5 / b) * a));
	elseif (b <= 1.3e-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 * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.72e+72)
		tmp = 1.0 / ((-1.5 / b) * a);
	elseif (b <= 1.3e-104)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.72e+72], N[(1.0 / N[(N[(-1.5 / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e-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 * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.71999999999999993e72

   1. Initial program 53.9%

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

      1. /-rgt-identity 53.9%

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

      2. metadata-eval 53.9%

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

      3. associate-/r/ 53.9%

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

      4. metadata-eval 53.9%

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

      5. metadata-eval 53.9%

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

      6. times-frac 53.9%

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

      7. *-commutative 53.9%

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

      8. times-frac 53.8%

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

      9. *-commutative 53.8%

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

      10. associate-/r* 53.7%

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

      11. associate-*l/ 53.8%

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

   3. Simplified 53.8%

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

      1. clear-num 53.8%

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

      2. inv-pow 53.8%

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

      3. *-commutative 53.8%

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

      4. fma-udef 53.8%

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

      5. associate-*r* 53.8%

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

      6. add-sqr-sqrt 33.4%

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

      7. hypot-def 50.4%

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

   5. Applied egg-rr 50.4%

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

      1. unpow-1 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-*r* 50.4%

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

      4. *-commutative 50.4%

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

      5. associate-*l* 50.4%

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

   7. Simplified 50.4%

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

   8. Taylor expanded in b around -inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. associate-/l* 90.6%

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

   10. Simplified 90.6%

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

       1. associate-/r/ 90.6%

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

   12. Applied egg-rr 90.6%

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

   if -1.71999999999999993e72 < b < 1.30000000000000001e-104

   1. Initial program 72.4%

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

   if 1.30000000000000001e-104 < b

   1. Initial program 14.4%

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

   2. Taylor expanded in b around inf 88.4%

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

      1. associate-*r/ 88.4%

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

   4. Applied egg-rr 88.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.72 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{b} \cdot a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e+40)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 3.6e-106)
     (* -0.3333333333333333 (/ (- b (sqrt (- (* b b) (* 3.0 (* a c))))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e+40) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 3.6e-106) {
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (3.0 * (a * c))))) / 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 <= (-3.7d+40)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 3.6d-106) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((b * b) - (3.0d0 * (a * c))))) / 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 <= -3.7e+40) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 3.6e-106) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt(((b * b) - (3.0 * (a * c))))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e+40:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 3.6e-106:
		tmp = -0.3333333333333333 * ((b - math.sqrt(((b * b) - (3.0 * (a * c))))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e+40)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 3.6e-106)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c))))) / 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 <= -3.7e+40)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 3.6e-106)
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (3.0 * (a * c))))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e+40], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e-106], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $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 < -3.7e40

   1. Initial program 60.6%

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

   2. Taylor expanded in b around -inf 90.6%

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

      1. *-commutative 90.6%

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

   4. Simplified 90.6%

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

   if -3.7e40 < b < 3.60000000000000013e-106

   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. /-rgt-identity 70.4%

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

      2. metadata-eval 70.4%

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

      3. associate-/l* 70.4%

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

      4. associate-*r/ 70.3%

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

      5. *-commutative 70.3%

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

      6. associate-*l/ 70.4%

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

      7. associate-*r/ 70.4%

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

      8. metadata-eval 70.4%

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

      9. metadata-eval 70.4%

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

      10. times-frac 70.4%

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

      11. neg-mul-1 70.4%

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

      12. distribute-rgt-neg-in 70.4%

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

      13. times-frac 70.2%

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

      14. metadata-eval 70.2%

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

      15. neg-mul-1 70.2%

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

   3. Simplified 70.1%

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

      1. fma-udef 70.1%

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

      2. associate-*r* 70.2%

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

      3. *-commutative 70.2%

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

      4. metadata-eval 70.2%

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

      5. cancel-sign-sub-inv 70.2%

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

   5. Applied egg-rr 70.2%

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

   if 3.60000000000000013e-106 < b

   1. Initial program 14.4%

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

   2. Taylor expanded in b around inf 88.4%

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

      1. associate-*r/ 88.4%

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

   4. Applied egg-rr 88.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{b} \cdot a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e+72)
   (/ 1.0 (* (/ -1.5 b) a))
   (if (<= b 1.2e-104)
     (/ (- (sqrt (+ (* b b) (* 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 <= -1.75e+72) {
		tmp = 1.0 / ((-1.5 / b) * a);
	} else if (b <= 1.2e-104) {
		tmp = (sqrt(((b * b) + (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 <= (-1.75d+72)) then
        tmp = 1.0d0 / (((-1.5d0) / b) * a)
    else if (b <= 1.2d-104) then
        tmp = (sqrt(((b * b) + (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 <= -1.75e+72) {
		tmp = 1.0 / ((-1.5 / b) * a);
	} else if (b <= 1.2e-104) {
		tmp = (Math.sqrt(((b * b) + (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 <= -1.75e+72:
		tmp = 1.0 / ((-1.5 / b) * a)
	elif b <= 1.2e-104:
		tmp = (math.sqrt(((b * b) + (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 <= -1.75e+72)
		tmp = Float64(1.0 / Float64(Float64(-1.5 / b) * a));
	elseif (b <= 1.2e-104)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + 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 <= -1.75e+72)
		tmp = 1.0 / ((-1.5 / b) * a);
	elseif (b <= 1.2e-104)
		tmp = (sqrt(((b * b) + (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, -1.75e+72], N[(1.0 / N[(N[(-1.5 / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.2e-104], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -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 < -1.75000000000000005e72

   1. Initial program 53.9%

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

      1. /-rgt-identity 53.9%

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

      2. metadata-eval 53.9%

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

      3. associate-/r/ 53.9%

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

      4. metadata-eval 53.9%

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

      5. metadata-eval 53.9%

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

      6. times-frac 53.9%

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

      7. *-commutative 53.9%

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

      8. times-frac 53.8%

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

      9. *-commutative 53.8%

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

      10. associate-/r* 53.7%

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

      11. associate-*l/ 53.8%

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

   3. Simplified 53.8%

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

      1. clear-num 53.8%

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

      2. inv-pow 53.8%

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

      3. *-commutative 53.8%

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

      4. fma-udef 53.8%

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

      5. associate-*r* 53.8%

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

      6. add-sqr-sqrt 33.4%

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

      7. hypot-def 50.4%

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

   5. Applied egg-rr 50.4%

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

      1. unpow-1 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-*r* 50.4%

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

      4. *-commutative 50.4%

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

      5. associate-*l* 50.4%

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

   7. Simplified 50.4%

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

   8. Taylor expanded in b around -inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. associate-/l* 90.6%

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

   10. Simplified 90.6%

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

       1. associate-/r/ 90.6%

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

   12. Applied egg-rr 90.6%

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

   if -1.75000000000000005e72 < b < 1.2e-104

   1. Initial program 72.4%

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

      1. associate-*r* 72.4%

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

      2. cancel-sign-sub-inv 72.4%

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

      3. metadata-eval 72.4%

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

      4. *-commutative 72.4%

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

      5. associate-*r* 72.4%

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

   3. Applied egg-rr 72.4%

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

   if 1.2e-104 < b

   1. Initial program 14.4%

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

   2. Taylor expanded in b around inf 88.4%

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

      1. associate-*r/ 88.4%

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

   4. Applied egg-rr 88.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{b} \cdot a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{b} \cdot a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e+72)
   (/ 1.0 (* (/ -1.5 b) a))
   (if (<= b 5.6e-107)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e+72) {
		tmp = 1.0 / ((-1.5 / b) * a);
	} else if (b <= 5.6e-107) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - 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 <= (-1.75d+72)) then
        tmp = 1.0d0 / (((-1.5d0) / b) * a)
    else if (b <= 5.6d-107) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - 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 <= -1.75e+72) {
		tmp = 1.0 / ((-1.5 / b) * a);
	} else if (b <= 5.6e-107) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.75e+72:
		tmp = 1.0 / ((-1.5 / b) * a)
	elif b <= 5.6e-107:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.75e+72)
		tmp = Float64(1.0 / Float64(Float64(-1.5 / b) * a));
	elseif (b <= 5.6e-107)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - 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 <= -1.75e+72)
		tmp = 1.0 / ((-1.5 / b) * a);
	elseif (b <= 5.6e-107)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.75e+72], N[(1.0 / N[(N[(-1.5 / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-107], 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 * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.75000000000000005e72

   1. Initial program 53.9%

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

      1. /-rgt-identity 53.9%

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

      2. metadata-eval 53.9%

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

      3. associate-/r/ 53.9%

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

      4. metadata-eval 53.9%

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

      5. metadata-eval 53.9%

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

      6. times-frac 53.9%

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

      7. *-commutative 53.9%

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

      8. times-frac 53.8%

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

      9. *-commutative 53.8%

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

      10. associate-/r* 53.7%

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

      11. associate-*l/ 53.8%

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

   3. Simplified 53.8%

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

      1. clear-num 53.8%

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

      2. inv-pow 53.8%

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

      3. *-commutative 53.8%

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

      4. fma-udef 53.8%

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

      5. associate-*r* 53.8%

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

      6. add-sqr-sqrt 33.4%

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

      7. hypot-def 50.4%

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

   5. Applied egg-rr 50.4%

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

      1. unpow-1 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-*r* 50.4%

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

      4. *-commutative 50.4%

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

      5. associate-*l* 50.4%

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

   7. Simplified 50.4%

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

   8. Taylor expanded in b around -inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. associate-/l* 90.6%

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

   10. Simplified 90.6%

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

       1. associate-/r/ 90.6%

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

   12. Applied egg-rr 90.6%

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

   if -1.75000000000000005e72 < b < 5.5999999999999998e-107

   1. Initial program 72.4%

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

      1. neg-sub0 72.4%

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

      2. associate-+l- 72.4%

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

      3. sub0-neg 72.4%

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

      4. neg-mul-1 72.4%

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

      5. associate-*r/ 72.4%

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

      6. metadata-eval 72.4%

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

      7. metadata-eval 72.4%

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

      8. times-frac 72.4%

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

      9. *-commutative 72.4%

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

      10. times-frac 72.3%

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

      11. associate-*l/ 72.4%

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

   3. Simplified 72.4%

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

   if 5.5999999999999998e-107 < b

   1. Initial program 14.4%

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

   2. Taylor expanded in b around inf 88.4%

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

      1. associate-*r/ 88.4%

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

   4. Applied egg-rr 88.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{b} \cdot a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-99}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-104}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-99)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.05e-104)
     (* -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 <= -2.3e-99) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.05e-104) {
		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 <= (-2.3d-99)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.05d-104) 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 <= -2.3e-99) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.05e-104) {
		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 <= -2.3e-99:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.05e-104:
		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 <= -2.3e-99)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.05e-104)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(a * 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 <= -2.3e-99)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.05e-104)
		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, -2.3e-99], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-104], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999998e-99

   1. Initial program 66.3%

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

   2. Taylor expanded in b around -inf 80.9%

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

   if -2.2999999999999998e-99 < b < 1.04999999999999999e-104

   1. Initial program 67.0%

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

      1. /-rgt-identity 67.0%

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

      2. metadata-eval 67.0%

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

      3. associate-/l* 67.0%

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

      4. associate-*r/ 66.9%

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

      5. *-commutative 66.9%

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

      6. associate-*l/ 67.0%

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

      7. associate-*r/ 67.0%

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

      8. metadata-eval 67.0%

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

      9. metadata-eval 67.0%

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

      10. times-frac 67.0%

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

      11. neg-mul-1 67.0%

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

      12. distribute-rgt-neg-in 67.0%

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

      13. times-frac 66.9%

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

      14. metadata-eval 66.9%

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

      15. neg-mul-1 66.9%

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

   3. Simplified 66.8%

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

      1. fma-udef 66.8%

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

      2. associate-*r* 66.9%

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

      3. *-commutative 66.9%

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

      4. metadata-eval 66.9%

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

      5. cancel-sign-sub-inv 66.9%

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

   5. Applied egg-rr 66.9%

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

   6. Taylor expanded in b around 0 64.9%

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

   if 1.04999999999999999e-104 < b

   1. Initial program 14.4%

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

   2. Taylor expanded in b around inf 88.4%

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

      1. associate-*r/ 88.4%

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

   4. Applied egg-rr 88.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-99}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-104}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-100}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.2e-100)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.75e-105)
     (/ (- (sqrt (* 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 <= -9.2e-100) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.75e-105) {
		tmp = (sqrt((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 <= (-9.2d-100)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.75d-105) then
        tmp = (sqrt((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 <= -9.2e-100) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.75e-105) {
		tmp = (Math.sqrt((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 <= -9.2e-100:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.75e-105:
		tmp = (math.sqrt((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 <= -9.2e-100)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.75e-105)
		tmp = Float64(Float64(sqrt(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 <= -9.2e-100)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.75e-105)
		tmp = (sqrt((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, -9.2e-100], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.75e-105], N[(N[(N[Sqrt[N[(c * N[(a * -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 < -9.19999999999999978e-100

   1. Initial program 66.3%

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

   2. Taylor expanded in b around -inf 80.9%

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

   if -9.19999999999999978e-100 < b < 1.75e-105

   1. Initial program 67.0%

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

   2. Taylor expanded in b around 0 64.9%

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

      1. expm1-log1p-u 62.8%

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

      2. expm1-udef 32.2%

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

      3. neg-mul-1 32.2%

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

      4. fma-def 32.2%

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

      5. *-commutative 32.2%

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

      6. associate-*r* 32.2%

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

   4. Applied egg-rr 32.2%

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

      1. expm1-def 62.9%

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

      2. expm1-log1p 65.0%

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

      3. fma-udef 65.0%

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

      4. neg-mul-1 65.0%

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

      5. +-commutative 65.0%

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

      6. unsub-neg 65.0%

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

   6. Simplified 65.0%

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

   if 1.75e-105 < b

   1. Initial program 14.4%

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

   2. Taylor expanded in b around inf 88.4%

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

      1. associate-*r/ 88.4%

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

   4. Applied egg-rr 88.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-100}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 68.5% accurate, 8.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} 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)) + (0.5d0 * (c / b))
    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)) + (0.5 * (c / b));
	} 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)) + (0.5 * (c / b))
	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(0.5 * Float64(c / b)));
	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)) + (0.5 * (c / b));
	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[(0.5 * N[(c / b), $MachinePrecision]), $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 67.9%

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

   2. Taylor expanded in b around -inf 66.2%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.0%

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

   2. Taylor expanded in b around inf 66.2%

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

      1. associate-*r/ 66.2%

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

   4. Applied egg-rr 66.2%

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

4. Final simplification 66.2%

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

Alternative 8: 68.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.42 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.42e-307) (/ 1.0 (* (/ -1.5 b) a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.42e-307) {
		tmp = 1.0 / ((-1.5 / 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 <= 1.42d-307) then
        tmp = 1.0d0 / (((-1.5d0) / 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 <= 1.42e-307) {
		tmp = 1.0 / ((-1.5 / b) * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.42e-307:
		tmp = 1.0 / ((-1.5 / b) * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.42e-307)
		tmp = Float64(1.0 / Float64(Float64(-1.5 / b) * 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 <= 1.42e-307)
		tmp = 1.0 / ((-1.5 / b) * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.42e-307], N[(1.0 / N[(N[(-1.5 / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.42000000000000001e-307

   1. Initial program 67.9%

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

      1. /-rgt-identity 67.9%

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

      2. metadata-eval 67.9%

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

      3. associate-/r/ 67.9%

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

      4. metadata-eval 67.9%

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

      5. metadata-eval 67.9%

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

      6. times-frac 67.9%

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

      7. *-commutative 67.9%

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

      8. times-frac 67.9%

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

      9. *-commutative 67.9%

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

      10. associate-/r* 67.7%

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

      11. associate-*l/ 67.8%

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

   3. Simplified 67.8%

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

      1. clear-num 67.8%

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

      2. inv-pow 67.8%

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

      3. *-commutative 67.8%

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

      4. fma-udef 67.8%

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

      5. associate-*r* 67.8%

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

      6. add-sqr-sqrt 51.1%

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

      7. hypot-def 59.5%

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

   5. Applied egg-rr 59.5%

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

      1. unpow-1 59.5%

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

      2. *-commutative 59.5%

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

      3. associate-*r* 59.5%

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

      4. *-commutative 59.5%

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

      5. associate-*l* 59.5%

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

   7. Simplified 59.5%

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

   8. Taylor expanded in b around -inf 65.8%

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

      1. associate-*r/ 65.8%

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

      2. associate-/l* 65.8%

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

   10. Simplified 65.8%

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

       1. associate-/r/ 65.8%

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

   12. Applied egg-rr 65.8%

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

   if 1.42000000000000001e-307 < b

   1. Initial program 30.0%

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

   2. Taylor expanded in b around inf 66.2%

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

      1. associate-*r/ 66.2%

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

   4. Applied egg-rr 66.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.42 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9: 68.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{\frac{b}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ 1.0 (/ -1.5 (/ b a))) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = 1.0 / (-1.5 / (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 <= (-1d-310)) then
        tmp = 1.0d0 / ((-1.5d0) / (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 <= -1e-310) {
		tmp = 1.0 / (-1.5 / (b / a));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = 1.0 / (-1.5 / (b / a))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(1.0 / Float64(-1.5 / Float64(b / 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 <= -1e-310)
		tmp = 1.0 / (-1.5 / (b / a));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(1.0 / N[(-1.5 / N[(b / a), $MachinePrecision]), $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 67.9%

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

      1. /-rgt-identity 67.9%

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

      2. metadata-eval 67.9%

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

      3. associate-/r/ 67.9%

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

      4. metadata-eval 67.9%

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

      5. metadata-eval 67.9%

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

      6. times-frac 67.9%

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

      7. *-commutative 67.9%

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

      8. times-frac 67.9%

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

      9. *-commutative 67.9%

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

      10. associate-/r* 67.7%

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

      11. associate-*l/ 67.8%

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

   3. Simplified 67.8%

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

      1. clear-num 67.8%

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

      2. inv-pow 67.8%

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

      3. *-commutative 67.8%

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

      4. fma-udef 67.8%

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

      5. associate-*r* 67.8%

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

      6. add-sqr-sqrt 51.1%

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

      7. hypot-def 59.5%

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

   5. Applied egg-rr 59.5%

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

      1. unpow-1 59.5%

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

      2. *-commutative 59.5%

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

      3. associate-*r* 59.5%

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

      4. *-commutative 59.5%

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

      5. associate-*l* 59.5%

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

   7. Simplified 59.5%

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

   8. Taylor expanded in b around -inf 65.8%

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

      1. associate-*r/ 65.8%

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

      2. associate-/l* 65.8%

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

   10. Simplified 65.8%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.0%

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

   2. Taylor expanded in b around inf 66.2%

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

      1. associate-*r/ 66.2%

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

   4. Applied egg-rr 66.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{-1.5}{\frac{b}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 68.3% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 67.9%

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

   2. Taylor expanded in b around -inf 65.7%

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

      1. *-commutative 65.7%

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

   4. Simplified 65.7%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.0%

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

   2. Taylor expanded in b around inf 66.2%

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

4. Final simplification 66.0%

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

Alternative 11: 68.3% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = b / (a / (-0.6666666666666666d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 67.9%

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

   2. Taylor expanded in b around -inf 65.7%

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

      1. *-commutative 65.7%

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

   4. Simplified 65.7%

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

   5. Taylor expanded in b around 0 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-*l/ 65.7%

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

      3. associate-/l* 65.7%

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

   7. Simplified 65.7%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.0%

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

   2. Taylor expanded in b around inf 66.2%

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

4. Final simplification 66.0%

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

Alternative 12: 68.3% accurate, 16.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1e-309) (/ b (/ a -0.6666666666666666)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-309) {
		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 <= 1d-309) 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 <= 1e-309) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1e-309:
		tmp = b / (a / -0.6666666666666666)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1e-309)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	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-309)
		tmp = b / (a / -0.6666666666666666);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1e-309], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.000000000000002e-309

   1. Initial program 67.9%

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

   2. Taylor expanded in b around -inf 65.7%

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

      1. *-commutative 65.7%

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

   4. Simplified 65.7%

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

   5. Taylor expanded in b around 0 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-*l/ 65.7%

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

      3. associate-/l* 65.7%

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

   7. Simplified 65.7%

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

   if 1.000000000000002e-309 < b

   1. Initial program 30.0%

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

   2. Taylor expanded in b around inf 66.2%

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

      1. associate-*r/ 66.2%

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

   4. Applied egg-rr 66.2%

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

4. Final simplification 66.0%

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

Alternative 13: 68.4% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(b * -0.6666666666666666) / 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 <= -1e-310)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $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 67.9%

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

   2. Taylor expanded in b around -inf 65.7%

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

      1. *-commutative 65.7%

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

   4. Simplified 65.7%

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

      1. associate-*l/ 65.7%

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

   6. Applied egg-rr 65.7%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.0%

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

   2. Taylor expanded in b around inf 66.2%

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

      1. associate-*r/ 66.2%

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

   4. Applied egg-rr 66.2%

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

4. Final simplification 66.0%

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

Alternative 14: 35.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.1%

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

2. Taylor expanded in b around inf 37.5%

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

3. Final simplification 37.5%

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

Alternative 15: 11.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 47.1%

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

2. Taylor expanded in b around inf 25.5%

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

   1. unpow2 25.5%

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

4. Simplified 25.5%

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

5. Taylor expanded in b around 0 13.4%

   \[\leadsto \color{blue}{0} \]

6. Final simplification 13.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))