Cubic critical

Percentage Accurate: 52.9% → 83.8%

Time: 18.5s

Alternatives: 15

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{hypot}\left(b, \sqrt{a \cdot -3} \cdot \sqrt{c}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+159)
   (/ b (* a -1.5))
   (if (<= b 1.45e-102)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (if (or (<= b 1.25e-38) (not (<= b 5100000.0)))
       (/ (* c -0.5) b)
       (/ (+ b (hypot b (* (sqrt (* a -3.0)) (sqrt c)))) (* a 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+159) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.45e-102) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else if ((b <= 1.25e-38) || !(b <= 5100000.0)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = (b + hypot(b, (sqrt((a * -3.0)) * sqrt(c)))) / (a * 3.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+159) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.45e-102) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else if ((b <= 1.25e-38) || !(b <= 5100000.0)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = (b + Math.hypot(b, (Math.sqrt((a * -3.0)) * Math.sqrt(c)))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.2e+159:
		tmp = b / (a * -1.5)
	elif b <= 1.45e-102:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	elif (b <= 1.25e-38) or not (b <= 5100000.0):
		tmp = (c * -0.5) / b
	else:
		tmp = (b + math.hypot(b, (math.sqrt((a * -3.0)) * math.sqrt(c)))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e+159)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.45e-102)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	elseif ((b <= 1.25e-38) || !(b <= 5100000.0))
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(Float64(b + hypot(b, Float64(sqrt(Float64(a * -3.0)) * sqrt(c)))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.2e+159)
		tmp = b / (a * -1.5);
	elseif (b <= 1.45e-102)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	elseif ((b <= 1.25e-38) || ~((b <= 5100000.0)))
		tmp = (c * -0.5) / b;
	else
		tmp = (b + hypot(b, (sqrt((a * -3.0)) * sqrt(c)))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+159], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e-102], 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], If[Or[LessEqual[b, 1.25e-38], N[Not[LessEqual[b, 5100000.0]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(N[(b + N[Sqrt[b ^ 2 + N[(N[Sqrt[N[(a * -3.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.2e159

   1. Initial program 56.7%

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

   4. Applied egg-rr 51.7%

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

      1. sub-neg 51.7%

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

   6. Applied egg-rr 51.7%

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

      1. unsub-neg 51.7%

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

      2. distribute-rgt-out-- 51.7%

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

      3. associate-*r* 51.7%

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

      4. *-commutative 51.7%

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

      5. associate-*r* 51.7%

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

   8. Simplified 51.7%

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

   9. Taylor expanded in b around -inf 99.7%

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

       1. *-commutative 99.7%

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

       2. metadata-eval 99.7%

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

       3. times-frac 99.9%

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

       4. *-rgt-identity 99.9%

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

   11. Simplified 99.9%

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

   if -1.2e159 < b < 1.44999999999999993e-102

   1. Initial program 83.3%

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

   if 1.44999999999999993e-102 < b < 1.25000000000000008e-38 or 5.1e6 < b

   1. Initial program 13.7%

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

   3. Taylor expanded in b around inf 89.0%

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

      1. associate-*r/ 89.0%

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

   5. Simplified 89.0%

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

   if 1.25000000000000008e-38 < b < 5.1e6

   1. Initial program 39.1%

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

      1. +-commutative 39.1%

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

      2. sqr-neg 39.1%

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

      3. unsub-neg 39.1%

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

      4. div-sub 39.1%

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

      5. --rgt-identity 39.1%

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

      6. div-sub 39.1%

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

   3. Simplified 39.1%

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

      1. sub-neg 39.1%

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

      2. fma-udef 39.1%

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

      3. add-sqr-sqrt 39.1%

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

      4. hypot-def 39.1%

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

      5. associate-*r* 39.1%

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

      6. *-commutative 39.1%

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

      7. metadata-eval 39.1%

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

      8. distribute-lft-neg-in 39.1%

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

      9. associate-*l* 39.1%

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

      10. *-commutative 39.1%

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

      11. distribute-rgt-neg-in 39.1%

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

      12. *-commutative 39.1%

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

      13. distribute-rgt-neg-in 39.1%

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

      14. metadata-eval 39.1%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 39.2%

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

      17. sqr-neg 39.2%

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

      18. sqrt-prod 39.2%

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

      19. add-sqr-sqrt 39.2%

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

   6. Applied egg-rr 39.2%

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

      1. *-commutative 39.2%

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

      2. sqrt-prod 75.2%

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

   8. Applied egg-rr 75.2%

      \[\leadsto \frac{\mathsf{hypot}\left(b, \color{blue}{\sqrt{a \cdot -3} \cdot \sqrt{c}}\right) + b}{3 \cdot a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{hypot}\left(b, \sqrt{a \cdot -3} \cdot \sqrt{c}\right)}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+159)
   (/ b (* a -1.5))
   (if (<= b 1.95e-103)
     (/ (- (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 <= -1.2e+159) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.95e-103) {
		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 <= (-1.2d+159)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.95d-103) 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 <= -1.2e+159) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.95e-103) {
		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 <= -1.2e+159:
		tmp = b / (a * -1.5)
	elif b <= 1.95e-103:
		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 <= -1.2e+159)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.95e-103)
		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 <= -1.2e+159)
		tmp = b / (a * -1.5);
	elseif (b <= 1.95e-103)
		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, -1.2e+159], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-103], 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 < -1.2e159

   1. Initial program 56.7%

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

   4. Applied egg-rr 51.7%

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

      1. sub-neg 51.7%

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

   6. Applied egg-rr 51.7%

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

      1. unsub-neg 51.7%

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

      2. distribute-rgt-out-- 51.7%

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

      3. associate-*r* 51.7%

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

      4. *-commutative 51.7%

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

      5. associate-*r* 51.7%

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

   8. Simplified 51.7%

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

   9. Taylor expanded in b around -inf 99.7%

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

       1. *-commutative 99.7%

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

       2. metadata-eval 99.7%

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

       3. times-frac 99.9%

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

       4. *-rgt-identity 99.9%

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

   11. Simplified 99.9%

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

   if -1.2e159 < b < 1.9500000000000001e-103

   1. Initial program 83.3%

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

   if 1.9500000000000001e-103 < b

   1. Initial program 15.6%

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

   3. Taylor expanded in b around inf 83.7%

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

      1. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-102}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e-47)
   (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
   (if (<= b 3.4e-102)
     (* (/ 0.3333333333333333 a) (+ b (sqrt (* c (* a -3.0)))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e-47) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 3.4e-102) {
		tmp = (0.3333333333333333 / a) * (b + sqrt((c * (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 <= (-1.02d-47)) then
        tmp = ((((a * 1.5d0) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 3.4d-102) then
        tmp = (0.3333333333333333d0 / a) * (b + sqrt((c * (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 <= -1.02e-47) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 3.4e-102) {
		tmp = (0.3333333333333333 / a) * (b + Math.sqrt((c * (a * -3.0))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.02e-47:
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 3.4e-102:
		tmp = (0.3333333333333333 / a) * (b + math.sqrt((c * (a * -3.0))))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.02e-47)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 3.4e-102)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(b + sqrt(Float64(c * 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 <= -1.02e-47)
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 3.4e-102)
		tmp = (0.3333333333333333 / a) * (b + sqrt((c * (a * -3.0))));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e-47], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.4e-102], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.02000000000000002e-47

   1. Initial program 79.7%

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

   3. Taylor expanded in b around -inf 81.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. +-commutative 81.2%

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

      2. mul-1-neg 81.2%

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

      3. unsub-neg 81.2%

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

      4. associate-/l* 83.7%

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

      5. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

   if -1.02000000000000002e-47 < b < 3.40000000000000013e-102

   1. Initial program 75.5%

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

   3. Taylor expanded in b around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*l* 70.5%

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

   5. Simplified 70.5%

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

      1. *-un-lft-identity 70.5%

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

      2. *-commutative 70.5%

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

      3. times-frac 70.2%

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

      4. div-inv 70.3%

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

      5. +-commutative 70.3%

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

      6. sqrt-prod 47.5%

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

      7. fma-def 47.5%

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

      8. add-sqr-sqrt 21.0%

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

      9. sqrt-unprod 46.6%

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

      10. sqr-neg 46.6%

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

      11. sqrt-unprod 25.6%

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

      12. add-sqr-sqrt 46.3%

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

      13. metadata-eval 46.3%

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

   7. Applied egg-rr 46.3%

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

      1. expm1-log1p-u 45.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{a} \cdot \left(\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -3}, b\right) \cdot 0.3333333333333333\right)\right)\right)} \]

      2. expm1-udef 22.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{a} \cdot \left(\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -3}, b\right) \cdot 0.3333333333333333\right)\right)} - 1} \]

      3. associate-*r* 22.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{a} \cdot \mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -3}, b\right)\right) \cdot 0.3333333333333333}\right)} - 1 \]

      4. *-commutative 22.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -3}, b\right)\right)}\right)} - 1 \]

      5. associate-*l/ 22.4%

         \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -3}, b\right)}{a}}\right)} - 1 \]

      6. *-un-lft-identity 22.4%

         \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -3}, b\right)}}{a}\right)} - 1 \]

      7. fma-udef 22.4%

         \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{a} \cdot \sqrt{c \cdot -3} + b}}{a}\right)} - 1 \]

      8. +-commutative 22.4%

         \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \frac{\color{blue}{b + \sqrt{a} \cdot \sqrt{c \cdot -3}}}{a}\right)} - 1 \]

      9. sqrt-unprod 16.2%

         \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \frac{b + \color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)}}}{a}\right)} - 1 \]

   9. Applied egg-rr 16.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 43.8%

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

       2. expm1-log1p 68.9%

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

       3. associate-*r/ 68.9%

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

       4. associate-*l/ 69.0%

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

       5. associate-*r* 68.8%

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

       6. *-commutative 68.8%

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

       7. associate-*l* 69.0%

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

   11. Simplified 69.0%

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

   if 3.40000000000000013e-102 < b

   1. Initial program 15.6%

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

   3. Taylor expanded in b around inf 83.7%

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

      1. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-102}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.3 \cdot 10^{-103}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-48)
   (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
   (if (<= b 8.3e-103)
     (/ (+ b (sqrt (* a (* c -3.0)))) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-48) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 8.3e-103) {
		tmp = (b + sqrt((a * (c * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.5d-48)) then
        tmp = ((((a * 1.5d0) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 8.3d-103) then
        tmp = (b + sqrt((a * (c * (-3.0d0))))) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-48) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 8.3e-103) {
		tmp = (b + Math.sqrt((a * (c * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-48:
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 8.3e-103:
		tmp = (b + math.sqrt((a * (c * -3.0)))) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-48)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 8.3e-103)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) / 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 <= -4.5e-48)
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 8.3e-103)
		tmp = (b + sqrt((a * (c * -3.0)))) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-48], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.3e-103], N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 < -4.49999999999999988e-48

   1. Initial program 79.7%

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

   3. Taylor expanded in b around -inf 81.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. +-commutative 81.2%

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

      2. mul-1-neg 81.2%

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

      3. unsub-neg 81.2%

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

      4. associate-/l* 83.7%

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

      5. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

   if -4.49999999999999988e-48 < b < 8.30000000000000006e-103

   1. Initial program 75.5%

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

   3. Taylor expanded in b around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*l* 70.5%

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

   5. Simplified 70.5%

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

      1. *-un-lft-identity 70.5%

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

      2. *-commutative 70.5%

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

      3. times-frac 70.2%

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

      4. div-inv 70.3%

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

      5. +-commutative 70.3%

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

      6. sqrt-prod 47.5%

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

      7. fma-def 47.5%

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

      8. add-sqr-sqrt 21.0%

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

      9. sqrt-unprod 46.6%

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

      10. sqr-neg 46.6%

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

      11. sqrt-unprod 25.6%

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

      12. add-sqr-sqrt 46.3%

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

      13. metadata-eval 46.3%

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

   7. Applied egg-rr 46.3%

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

      1. associate-*l/ 46.4%

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

      2. *-un-lft-identity 46.4%

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

      3. associate-/l* 46.2%

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

      4. fma-udef 46.2%

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

      5. +-commutative 46.2%

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

      6. sqrt-unprod 68.9%

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

      7. div-inv 69.0%

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

      8. metadata-eval 69.0%

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

      9. *-commutative 69.0%

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

   9. Applied egg-rr 69.0%

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

   if 8.30000000000000006e-103 < b

   1. Initial program 15.6%

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

   3. Taylor expanded in b around inf 83.7%

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

      1. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.3 \cdot 10^{-103}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-47)
   (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
   (if (<= b 2.95e-102)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-47) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 2.95e-102) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6d-47)) then
        tmp = ((((a * 1.5d0) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 2.95d-102) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-47) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 2.95e-102) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-47:
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 2.95e-102:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-47)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 2.95e-102)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e-47)
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 2.95e-102)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-47], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.95e-102], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.00000000000000033e-47

   1. Initial program 79.7%

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

   3. Taylor expanded in b around -inf 81.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. +-commutative 81.2%

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

      2. mul-1-neg 81.2%

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

      3. unsub-neg 81.2%

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

      4. associate-/l* 83.7%

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

      5. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

   if -6.00000000000000033e-47 < b < 2.9500000000000001e-102

   1. Initial program 75.5%

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

   3. Taylor expanded in b around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*l* 70.5%

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

   5. Simplified 70.5%

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

   if 2.9500000000000001e-102 < b

   1. Initial program 15.6%

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

   3. Taylor expanded in b around inf 83.7%

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

      1. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.8e-48)
   (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
   (if (<= b 1.35e-103)
     (/ (- (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 <= -8.8e-48) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 1.35e-103) {
		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 <= (-8.8d-48)) then
        tmp = ((((a * 1.5d0) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 1.35d-103) 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 <= -8.8e-48) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 1.35e-103) {
		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 <= -8.8e-48:
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 1.35e-103:
		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 <= -8.8e-48)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 1.35e-103)
		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 <= -8.8e-48)
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 1.35e-103)
		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, -8.8e-48], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-103], 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 < -8.8000000000000005e-48

   1. Initial program 79.7%

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

   3. Taylor expanded in b around -inf 81.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. +-commutative 81.2%

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

      2. mul-1-neg 81.2%

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

      3. unsub-neg 81.2%

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

      4. associate-/l* 83.7%

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

      5. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

   if -8.8000000000000005e-48 < b < 1.35000000000000005e-103

   1. Initial program 75.5%

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

   3. Taylor expanded in b around 0 70.3%

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

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

      3. *-commutative 70.6%

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

   5. Simplified 70.6%

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

   if 1.35000000000000005e-103 < b

   1. Initial program 15.6%

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

   3. Taylor expanded in b around inf 83.7%

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

      1. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 5.8× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((((a * 1.5) / (b / c)) - b) - 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 <= (-5d-310)) then
        tmp = ((((a * 1.5d0) / (b / c)) - b) - 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 <= -5e-310) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - 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 <= -5e-310)
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 81.4%

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

   3. Taylor expanded in b around -inf 62.0%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. +-commutative 62.0%

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

      2. mul-1-neg 62.0%

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

      3. unsub-neg 62.0%

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

      4. associate-/l* 63.7%

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

      5. associate-*r/ 63.7%

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

   5. Simplified 63.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 27.2%

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

   3. Taylor expanded in b around inf 66.0%

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

      1. associate-*r/ 66.0%

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

   5. Simplified 66.0%

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

4. Final simplification 65.0%

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

Alternative 8: 67.4% accurate, 7.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-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 <= -5e-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 <= (-5d-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 <= -5e-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 <= -5e-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 <= -5e-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 <= -5e-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, -5e-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 < -4.999999999999985e-310

   1. Initial program 81.4%

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

   3. Taylor expanded in b around -inf 63.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 27.2%

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

   3. Taylor expanded in b around inf 66.0%

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

      1. associate-*r/ 66.0%

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

   5. Simplified 66.0%

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

4. Final simplification 64.9%

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

Alternative 9: 43.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+19}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.9e+19) (* -0.6666666666666666 (/ b a)) (* c (/ 0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e+19) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.9d+19) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = c * (0.5d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e+19) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.9e+19:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = c * (0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.9e+19)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(c * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.9e+19)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = c * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.9e+19], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9e19

   1. Initial program 72.4%

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

   3. Taylor expanded in b around -inf 42.5%

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

      1. *-commutative 42.5%

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

   5. Simplified 42.5%

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

   if 2.9e19 < b

   1. Initial program 6.5%

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

   3. Taylor expanded in b around -inf 2.4%

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

   4. Taylor expanded in b around 0 24.1%

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

      1. associate-*r/ 24.1%

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

      2. associate-/l* 24.1%

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

      3. associate-/r/ 24.1%

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

   6. Simplified 24.1%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+19}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+21}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e+21) (/ -0.6666666666666666 (/ a b)) (* c (/ 0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e+21) {
		tmp = -0.6666666666666666 / (a / 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 <= 8d+21) then
        tmp = (-0.6666666666666666d0) / (a / 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 <= 8e+21) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e+21:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = c * (0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e+21)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(c * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e+21)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = c * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e+21], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8e21

   1. Initial program 72.4%

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

   3. Taylor expanded in b around -inf 42.5%

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

      1. *-commutative 42.5%

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

   5. Simplified 42.5%

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

      1. *-commutative 42.5%

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

      2. clear-num 42.6%

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

      3. un-div-inv 42.6%

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

   7. Applied egg-rr 42.6%

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

   if 8e21 < b

   1. Initial program 6.5%

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

   3. Taylor expanded in b around -inf 2.4%

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

   4. Taylor expanded in b around 0 24.1%

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

      1. associate-*r/ 24.1%

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

      2. associate-/l* 24.1%

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

      3. associate-/r/ 24.1%

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

   6. Simplified 24.1%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+21}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-281) (/ -0.6666666666666666 (/ a b)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-281) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.2d-281) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-281) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-281:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-281)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.2e-281)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-281], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2000000000000001e-281

   1. Initial program 80.1%

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

   3. Taylor expanded in b around -inf 62.1%

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

      1. *-commutative 62.1%

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

   5. Simplified 62.1%

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

      1. *-commutative 62.1%

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

      2. clear-num 62.2%

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

      3. un-div-inv 62.2%

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

   7. Applied egg-rr 62.2%

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

   if 5.2000000000000001e-281 < b

   1. Initial program 27.5%

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

   3. Taylor expanded in b around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. associate-/l* 64.9%

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

   5. Simplified 64.9%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-281) (/ b (* a -1.5)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-281) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.2d-281) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-281) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-281:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-281)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.2e-281)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-281], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2000000000000001e-281

   1. Initial program 80.1%

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

   4. Applied egg-rr 66.6%

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

      1. sub-neg 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. unsub-neg 66.6%

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

      2. distribute-rgt-out-- 66.6%

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

      3. associate-*r* 66.6%

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

      4. *-commutative 66.6%

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

      5. associate-*r* 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in b around -inf 62.1%

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

       1. *-commutative 62.1%

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

       2. metadata-eval 62.1%

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

       3. times-frac 62.4%

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

       4. *-rgt-identity 62.4%

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

   11. Simplified 62.4%

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

   if 5.2000000000000001e-281 < b

   1. Initial program 27.5%

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

   3. Taylor expanded in b around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. associate-/l* 64.9%

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

   5. Simplified 64.9%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-281) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-281) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.2d-281) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-281) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-281:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-281)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.2e-281)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-281], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2000000000000001e-281

   1. Initial program 80.1%

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

   4. Applied egg-rr 66.6%

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

      1. sub-neg 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. unsub-neg 66.6%

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

      2. distribute-rgt-out-- 66.6%

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

      3. associate-*r* 66.6%

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

      4. *-commutative 66.6%

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

      5. associate-*r* 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in b around -inf 62.1%

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

       1. *-commutative 62.1%

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

       2. metadata-eval 62.1%

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

       3. times-frac 62.4%

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

       4. *-rgt-identity 62.4%

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

   11. Simplified 62.4%

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

   if 5.2000000000000001e-281 < b

   1. Initial program 27.5%

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

   3. Taylor expanded in b around inf 66.9%

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

      1. associate-*r/ 66.9%

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

   5. Simplified 66.9%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 11.0% 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 51.5%

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

3. Taylor expanded in b around -inf 29.7%

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

4. Taylor expanded in b around 0 9.8%

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

5. Final simplification 9.8%

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

Alternative 15: 11.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (0.5 / 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 = c * (0.5d0 / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in b around -inf 29.7%

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

4. Taylor expanded in b around 0 9.8%

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

   1. associate-*r/ 9.8%

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

   2. associate-/l* 9.8%

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

   3. associate-/r/ 9.8%

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

6. Simplified 9.8%

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

7. Final simplification 9.8%

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

Reproduce

herbie shell --seed 2024027 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))