Cubic critical

Percentage Accurate: 51.9% → 84.3%

Time: 15.8s

Alternatives: 8

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, 2, \frac{a \cdot -1.5}{b} \cdot c\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-71}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot {c}^{2}\right) + \frac{c}{\frac{b}{-0.5}}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= b -2.1e+128)
     (/ (/ (fma b 2.0 (* (/ (* a -1.5) b) c)) a) -3.0)
     (if (<= b 9.8e-88)
       t_0
       (if (<= b 5.8e-71)
         (+ (* -0.375 (* (/ a (pow b 3.0)) (pow c 2.0))) (/ c (/ b -0.5)))
         (if (<= b 6.5e+27) t_0 (/ (* c -0.5) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -2.1e+128) {
		tmp = (fma(b, 2.0, (((a * -1.5) / b) * c)) / a) / -3.0;
	} else if (b <= 9.8e-88) {
		tmp = t_0;
	} else if (b <= 5.8e-71) {
		tmp = (-0.375 * ((a / pow(b, 3.0)) * pow(c, 2.0))) + (c / (b / -0.5));
	} else if (b <= 6.5e+27) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -2.1e+128)
		tmp = Float64(Float64(fma(b, 2.0, Float64(Float64(Float64(a * -1.5) / b) * c)) / a) / -3.0);
	elseif (b <= 9.8e-88)
		tmp = t_0;
	elseif (b <= 5.8e-71)
		tmp = Float64(Float64(-0.375 * Float64(Float64(a / (b ^ 3.0)) * (c ^ 2.0))) + Float64(c / Float64(b / -0.5)));
	elseif (b <= 6.5e+27)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.1e+128], N[(N[(N[(b * 2.0 + N[(N[(N[(a * -1.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 9.8e-88], t$95$0, If[LessEqual[b, 5.8e-71], N[(N[(-0.375 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / N[(b / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+27], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.1e128

   1. Initial program 44.2%

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

   4. Applied egg-rr 60.7%

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

      1. associate-/r* 60.8%

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

   6. Simplified 60.8%

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

      1. associate-*r/ 60.7%

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

   8. Applied egg-rr 60.7%

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

      1. un-div-inv 60.8%

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

      2. div-sub 60.8%

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

   10. Applied egg-rr 60.8%

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

       1. div-sub 60.8%

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

   12. Simplified 60.8%

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

   13. Taylor expanded in b around -inf 0.0%

       \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b} + 2 \cdot b}}{a}}{-3} \]
   14. Step-by-step derivation

       1. +-commutative 0.0%

          \[\leadsto \frac{\frac{\color{blue}{2 \cdot b + 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}}{a}}{-3} \]

       2. *-commutative 0.0%

          \[\leadsto \frac{\frac{\color{blue}{b \cdot 2} + 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}{a}}{-3} \]

       3. fma-def 0.0%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}\right)}}{a}}{-3} \]

       4. associate-*r/ 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \color{blue}{\frac{0.5 \cdot \left(a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)}{b}}\right)}{a}}{-3} \]

       5. *-commutative 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{0.5 \cdot \color{blue}{\left(\left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot a\right)}}{b}\right)}{a}}{-3} \]

       6. associate-*r* 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\color{blue}{\left(0.5 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right) \cdot a}}{b}\right)}{a}}{-3} \]

       7. *-commutative 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}\right) \cdot a}{b}\right)}{a}}{-3} \]

       8. unpow2 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \left(\color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)} \cdot c\right)\right) \cdot a}{b}\right)}{a}}{-3} \]

       9. rem-square-sqrt 96.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \left(\color{blue}{-3} \cdot c\right)\right) \cdot a}{b}\right)}{a}}{-3} \]

       10. associate-*r* 96.0%

           \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\color{blue}{\left(\left(0.5 \cdot -3\right) \cdot c\right)} \cdot a}{b}\right)}{a}}{-3} \]

       11. metadata-eval 96.0%

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

       12. associate-*r* 96.0%

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

       13. *-commutative 96.0%

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

       14. associate-*r* 96.0%

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

       15. associate-*l/ 96.2%

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

       16. *-commutative 96.2%

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

   15. Simplified 96.2%

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

   if -2.1e128 < b < 9.80000000000000055e-88 or 5.7999999999999997e-71 < b < 6.5000000000000005e27

   1. Initial program 86.6%

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

   if 9.80000000000000055e-88 < b < 5.7999999999999997e-71

   1. Initial program 7.9%

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

   3. Taylor expanded in b around inf 98.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   4. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}} \]

      2. fma-def 98.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)} \]

      3. associate-/l* 80.1%

         \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}}, -0.5 \cdot \frac{c}{b}\right) \]

      4. *-commutative 80.1%

         \[\leadsto \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{{c}^{2}}}, \color{blue}{\frac{c}{b} \cdot -0.5}\right) \]

      5. associate-*l/ 80.1%

         \[\leadsto \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{{c}^{2}}}, \color{blue}{\frac{c \cdot -0.5}{b}}\right) \]

   5. Simplified 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{{c}^{2}}}, \frac{c \cdot -0.5}{b}\right)} \]
   6. Step-by-step derivation

      1. fma-udef 80.1%

         \[\leadsto \color{blue}{-0.375 \cdot \frac{a}{\frac{{b}^{3}}{{c}^{2}}} + \frac{c \cdot -0.5}{b}} \]

      2. associate-/r/ 100.0%

         \[\leadsto -0.375 \cdot \color{blue}{\left(\frac{a}{{b}^{3}} \cdot {c}^{2}\right)} + \frac{c \cdot -0.5}{b} \]

      3. associate-/l* 100.0%

         \[\leadsto -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot {c}^{2}\right) + \color{blue}{\frac{c}{\frac{b}{-0.5}}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot {c}^{2}\right) + \frac{c}{\frac{b}{-0.5}}} \]

   if 6.5000000000000005e27 < b

   1. Initial program 11.5%

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

   3. Taylor expanded in b around inf 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-*l/ 97.1%

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

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, 2, \frac{a \cdot -1.5}{b} \cdot c\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-71}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot {c}^{2}\right) + \frac{c}{\frac{b}{-0.5}}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, 2, \frac{a \cdot -1.5}{b} \cdot c\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))))
   (if (<= b -2.2e+128)
     (/ (/ (fma b 2.0 (* (/ (* a -1.5) b) c)) a) -3.0)
     (if (<= b 7.5e-89)
       t_0
       (if (<= b 2.2e-66)
         (* c (/ -0.5 b))
         (if (<= b 6.5e+27) t_0 (/ (* c -0.5) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -2.2e+128) {
		tmp = (fma(b, 2.0, (((a * -1.5) / b) * c)) / a) / -3.0;
	} else if (b <= 7.5e-89) {
		tmp = t_0;
	} else if (b <= 2.2e-66) {
		tmp = c * (-0.5 / b);
	} else if (b <= 6.5e+27) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -2.2e+128)
		tmp = Float64(Float64(fma(b, 2.0, Float64(Float64(Float64(a * -1.5) / b) * c)) / a) / -3.0);
	elseif (b <= 7.5e-89)
		tmp = t_0;
	elseif (b <= 2.2e-66)
		tmp = Float64(c * Float64(-0.5 / b));
	elseif (b <= 6.5e+27)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[b, -2.2e+128], N[(N[(N[(b * 2.0 + N[(N[(N[(a * -1.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 7.5e-89], t$95$0, If[LessEqual[b, 2.2e-66], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+27], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.20000000000000017e128

   1. Initial program 44.2%

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

   4. Applied egg-rr 60.7%

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

      1. associate-/r* 60.8%

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

   6. Simplified 60.8%

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

      1. associate-*r/ 60.7%

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

   8. Applied egg-rr 60.7%

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

      1. un-div-inv 60.8%

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

      2. div-sub 60.8%

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

   10. Applied egg-rr 60.8%

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

       1. div-sub 60.8%

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

   12. Simplified 60.8%

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

   13. Taylor expanded in b around -inf 0.0%

       \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b} + 2 \cdot b}}{a}}{-3} \]
   14. Step-by-step derivation

       1. +-commutative 0.0%

          \[\leadsto \frac{\frac{\color{blue}{2 \cdot b + 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}}{a}}{-3} \]

       2. *-commutative 0.0%

          \[\leadsto \frac{\frac{\color{blue}{b \cdot 2} + 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}{a}}{-3} \]

       3. fma-def 0.0%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}\right)}}{a}}{-3} \]

       4. associate-*r/ 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \color{blue}{\frac{0.5 \cdot \left(a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)}{b}}\right)}{a}}{-3} \]

       5. *-commutative 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{0.5 \cdot \color{blue}{\left(\left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot a\right)}}{b}\right)}{a}}{-3} \]

       6. associate-*r* 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\color{blue}{\left(0.5 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right) \cdot a}}{b}\right)}{a}}{-3} \]

       7. *-commutative 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}\right) \cdot a}{b}\right)}{a}}{-3} \]

       8. unpow2 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \left(\color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)} \cdot c\right)\right) \cdot a}{b}\right)}{a}}{-3} \]

       9. rem-square-sqrt 96.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \left(\color{blue}{-3} \cdot c\right)\right) \cdot a}{b}\right)}{a}}{-3} \]

       10. associate-*r* 96.0%

           \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\color{blue}{\left(\left(0.5 \cdot -3\right) \cdot c\right)} \cdot a}{b}\right)}{a}}{-3} \]

       11. metadata-eval 96.0%

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

       12. associate-*r* 96.0%

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

       13. *-commutative 96.0%

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

       14. associate-*r* 96.0%

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

       15. associate-*l/ 96.2%

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

       16. *-commutative 96.2%

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

   15. Simplified 96.2%

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

   if -2.20000000000000017e128 < b < 7.4999999999999999e-89 or 2.2000000000000001e-66 < b < 6.5000000000000005e27

   1. Initial program 86.6%

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

   3. Taylor expanded in a around 0 86.4%

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

   if 7.4999999999999999e-89 < b < 2.2000000000000001e-66

   1. Initial program 7.9%

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

   3. Taylor expanded in b around inf 97.6%

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

      1. associate-/l* 97.6%

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

   5. Simplified 97.6%

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

      1. expm1-log1p-u 56.6%

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

      2. expm1-udef 19.7%

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

      3. times-frac 19.7%

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

      4. metadata-eval 19.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-0.5} \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)} - 1 \]

      5. associate-/r/ 19.7%

         \[\leadsto e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\color{blue}{\frac{a}{b} \cdot c}}{a}\right)} - 1 \]

   7. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 56.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)\right)} \]

      2. expm1-log1p 98.2%

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

      3. associate-/l* 79.5%

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

   9. Simplified 79.5%

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

       1. associate-*r/ 79.5%

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

       2. clear-num 79.8%

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

   11. Applied egg-rr 79.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{c}}{-0.5 \cdot \frac{a}{b}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 79.2%

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

       2. *-commutative 79.2%

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

       3. associate-*r* 79.2%

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

       4. times-frac 79.5%

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

       5. *-commutative 79.5%

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

       6. *-rgt-identity 79.5%

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

       7. *-commutative 79.5%

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

   13. Simplified 79.5%

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

       1. *-commutative 79.5%

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

       2. clear-num 79.5%

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

       3. un-div-inv 79.5%

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

   15. Applied egg-rr 79.5%

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

       1. associate-/r/ 79.5%

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

       2. associate-*l/ 79.5%

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

       3. *-commutative 79.5%

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

       4. *-commutative 79.5%

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

       5. times-frac 79.8%

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

       6. associate-/r/ 98.8%

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

       7. *-inverses 98.8%

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

       8. *-lft-identity 98.8%

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

   17. Simplified 98.8%

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

   if 6.5000000000000005e27 < b

   1. Initial program 11.5%

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

   3. Taylor expanded in b around inf 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-*l/ 97.1%

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

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, 2, \frac{a \cdot -1.5}{b} \cdot c\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, 2, \frac{a \cdot -1.5}{b} \cdot c\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-64}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))))
   (if (<= b -2e+127)
     (/ (/ (fma b 2.0 (* (/ (* a -1.5) b) c)) a) -3.0)
     (if (<= b 2.1e-87)
       t_0
       (if (<= b 2.65e-64)
         (* c (/ -0.5 b))
         (if (<= b 6.5e+27) t_0 (/ (* c -0.5) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -2e+127) {
		tmp = (fma(b, 2.0, (((a * -1.5) / b) * c)) / a) / -3.0;
	} else if (b <= 2.1e-87) {
		tmp = t_0;
	} else if (b <= 2.65e-64) {
		tmp = c * (-0.5 / b);
	} else if (b <= 6.5e+27) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -2e+127)
		tmp = Float64(Float64(fma(b, 2.0, Float64(Float64(Float64(a * -1.5) / b) * c)) / a) / -3.0);
	elseif (b <= 2.1e-87)
		tmp = t_0;
	elseif (b <= 2.65e-64)
		tmp = Float64(c * Float64(-0.5 / b));
	elseif (b <= 6.5e+27)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[b, -2e+127], N[(N[(N[(b * 2.0 + N[(N[(N[(a * -1.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.1e-87], t$95$0, If[LessEqual[b, 2.65e-64], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+27], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.99999999999999991e127

   1. Initial program 44.2%

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

   4. Applied egg-rr 60.7%

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

      1. associate-/r* 60.8%

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

   6. Simplified 60.8%

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

      1. associate-*r/ 60.7%

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

   8. Applied egg-rr 60.7%

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

      1. un-div-inv 60.8%

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

      2. div-sub 60.8%

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

   10. Applied egg-rr 60.8%

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

       1. div-sub 60.8%

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

   12. Simplified 60.8%

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

   13. Taylor expanded in b around -inf 0.0%

       \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b} + 2 \cdot b}}{a}}{-3} \]
   14. Step-by-step derivation

       1. +-commutative 0.0%

          \[\leadsto \frac{\frac{\color{blue}{2 \cdot b + 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}}{a}}{-3} \]

       2. *-commutative 0.0%

          \[\leadsto \frac{\frac{\color{blue}{b \cdot 2} + 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}{a}}{-3} \]

       3. fma-def 0.0%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}\right)}}{a}}{-3} \]

       4. associate-*r/ 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \color{blue}{\frac{0.5 \cdot \left(a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)}{b}}\right)}{a}}{-3} \]

       5. *-commutative 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{0.5 \cdot \color{blue}{\left(\left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot a\right)}}{b}\right)}{a}}{-3} \]

       6. associate-*r* 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\color{blue}{\left(0.5 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right) \cdot a}}{b}\right)}{a}}{-3} \]

       7. *-commutative 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}\right) \cdot a}{b}\right)}{a}}{-3} \]

       8. unpow2 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \left(\color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)} \cdot c\right)\right) \cdot a}{b}\right)}{a}}{-3} \]

       9. rem-square-sqrt 96.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \left(\color{blue}{-3} \cdot c\right)\right) \cdot a}{b}\right)}{a}}{-3} \]

       10. associate-*r* 96.0%

           \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\color{blue}{\left(\left(0.5 \cdot -3\right) \cdot c\right)} \cdot a}{b}\right)}{a}}{-3} \]

       11. metadata-eval 96.0%

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

       12. associate-*r* 96.0%

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

       13. *-commutative 96.0%

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

       14. associate-*r* 96.0%

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

       15. associate-*l/ 96.2%

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

       16. *-commutative 96.2%

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

   15. Simplified 96.2%

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

   if -1.99999999999999991e127 < b < 2.10000000000000007e-87 or 2.6500000000000001e-64 < b < 6.5000000000000005e27

   1. Initial program 86.6%

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

   3. Taylor expanded in a around 0 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*r* 86.5%

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

   5. Simplified 86.5%

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

   if 2.10000000000000007e-87 < b < 2.6500000000000001e-64

   1. Initial program 7.9%

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

   3. Taylor expanded in b around inf 97.6%

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

      1. associate-/l* 97.6%

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

   5. Simplified 97.6%

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

      1. expm1-log1p-u 56.6%

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

      2. expm1-udef 19.7%

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

      3. times-frac 19.7%

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

      4. metadata-eval 19.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-0.5} \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)} - 1 \]

      5. associate-/r/ 19.7%

         \[\leadsto e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\color{blue}{\frac{a}{b} \cdot c}}{a}\right)} - 1 \]

   7. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 56.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)\right)} \]

      2. expm1-log1p 98.2%

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

      3. associate-/l* 79.5%

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

   9. Simplified 79.5%

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

       1. associate-*r/ 79.5%

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

       2. clear-num 79.8%

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

   11. Applied egg-rr 79.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{c}}{-0.5 \cdot \frac{a}{b}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 79.2%

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

       2. *-commutative 79.2%

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

       3. associate-*r* 79.2%

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

       4. times-frac 79.5%

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

       5. *-commutative 79.5%

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

       6. *-rgt-identity 79.5%

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

       7. *-commutative 79.5%

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

   13. Simplified 79.5%

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

       1. *-commutative 79.5%

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

       2. clear-num 79.5%

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

       3. un-div-inv 79.5%

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

   15. Applied egg-rr 79.5%

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

       1. associate-/r/ 79.5%

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

       2. associate-*l/ 79.5%

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

       3. *-commutative 79.5%

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

       4. *-commutative 79.5%

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

       5. times-frac 79.8%

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

       6. associate-/r/ 98.8%

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

       7. *-inverses 98.8%

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

       8. *-lft-identity 98.8%

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

   17. Simplified 98.8%

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

   if 6.5000000000000005e27 < b

   1. Initial program 11.5%

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

   3. Taylor expanded in b around inf 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-*l/ 97.1%

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

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, 2, \frac{a \cdot -1.5}{b} \cdot c\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-64}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, 2, \frac{a \cdot -1.5}{b} \cdot c\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-66}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= b -1.6e+127)
     (/ (/ (fma b 2.0 (* (/ (* a -1.5) b) c)) a) -3.0)
     (if (<= b 5.6e-89)
       t_0
       (if (<= b 8.5e-66)
         (* c (/ -0.5 b))
         (if (<= b 4.4e+32) t_0 (/ (* c -0.5) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -1.6e+127) {
		tmp = (fma(b, 2.0, (((a * -1.5) / b) * c)) / a) / -3.0;
	} else if (b <= 5.6e-89) {
		tmp = t_0;
	} else if (b <= 8.5e-66) {
		tmp = c * (-0.5 / b);
	} else if (b <= 4.4e+32) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -1.6e+127)
		tmp = Float64(Float64(fma(b, 2.0, Float64(Float64(Float64(a * -1.5) / b) * c)) / a) / -3.0);
	elseif (b <= 5.6e-89)
		tmp = t_0;
	elseif (b <= 8.5e-66)
		tmp = Float64(c * Float64(-0.5 / b));
	elseif (b <= 4.4e+32)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.6e+127], N[(N[(N[(b * 2.0 + N[(N[(N[(a * -1.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 5.6e-89], t$95$0, If[LessEqual[b, 8.5e-66], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e+32], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.59999999999999988e127

   1. Initial program 44.2%

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

   4. Applied egg-rr 60.7%

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

      1. associate-/r* 60.8%

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

   6. Simplified 60.8%

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

      1. associate-*r/ 60.7%

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

   8. Applied egg-rr 60.7%

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

      1. un-div-inv 60.8%

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

      2. div-sub 60.8%

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

   10. Applied egg-rr 60.8%

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

       1. div-sub 60.8%

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

   12. Simplified 60.8%

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

   13. Taylor expanded in b around -inf 0.0%

       \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b} + 2 \cdot b}}{a}}{-3} \]
   14. Step-by-step derivation

       1. +-commutative 0.0%

          \[\leadsto \frac{\frac{\color{blue}{2 \cdot b + 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}}{a}}{-3} \]

       2. *-commutative 0.0%

          \[\leadsto \frac{\frac{\color{blue}{b \cdot 2} + 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}{a}}{-3} \]

       3. fma-def 0.0%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}\right)}}{a}}{-3} \]

       4. associate-*r/ 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \color{blue}{\frac{0.5 \cdot \left(a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)}{b}}\right)}{a}}{-3} \]

       5. *-commutative 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{0.5 \cdot \color{blue}{\left(\left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot a\right)}}{b}\right)}{a}}{-3} \]

       6. associate-*r* 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\color{blue}{\left(0.5 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right) \cdot a}}{b}\right)}{a}}{-3} \]

       7. *-commutative 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}\right) \cdot a}{b}\right)}{a}}{-3} \]

       8. unpow2 0.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \left(\color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)} \cdot c\right)\right) \cdot a}{b}\right)}{a}}{-3} \]

       9. rem-square-sqrt 96.0%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\left(0.5 \cdot \left(\color{blue}{-3} \cdot c\right)\right) \cdot a}{b}\right)}{a}}{-3} \]

       10. associate-*r* 96.0%

           \[\leadsto \frac{\frac{\mathsf{fma}\left(b, 2, \frac{\color{blue}{\left(\left(0.5 \cdot -3\right) \cdot c\right)} \cdot a}{b}\right)}{a}}{-3} \]

       11. metadata-eval 96.0%

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

       12. associate-*r* 96.0%

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

       13. *-commutative 96.0%

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

       14. associate-*r* 96.0%

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

       15. associate-*l/ 96.2%

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

       16. *-commutative 96.2%

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

   15. Simplified 96.2%

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

   if -1.59999999999999988e127 < b < 5.5999999999999998e-89 or 8.49999999999999966e-66 < b < 4.40000000000000002e32

   1. Initial program 86.6%

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

   if 5.5999999999999998e-89 < b < 8.49999999999999966e-66

   1. Initial program 7.9%

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

   3. Taylor expanded in b around inf 97.6%

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

      1. associate-/l* 97.6%

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

   5. Simplified 97.6%

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

      1. expm1-log1p-u 56.6%

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

      2. expm1-udef 19.7%

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

      3. times-frac 19.7%

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

      4. metadata-eval 19.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-0.5} \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)} - 1 \]

      5. associate-/r/ 19.7%

         \[\leadsto e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\color{blue}{\frac{a}{b} \cdot c}}{a}\right)} - 1 \]

   7. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 56.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)\right)} \]

      2. expm1-log1p 98.2%

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

      3. associate-/l* 79.5%

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

   9. Simplified 79.5%

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

       1. associate-*r/ 79.5%

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

       2. clear-num 79.8%

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

   11. Applied egg-rr 79.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{c}}{-0.5 \cdot \frac{a}{b}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 79.2%

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

       2. *-commutative 79.2%

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

       3. associate-*r* 79.2%

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

       4. times-frac 79.5%

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

       5. *-commutative 79.5%

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

       6. *-rgt-identity 79.5%

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

       7. *-commutative 79.5%

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

   13. Simplified 79.5%

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

       1. *-commutative 79.5%

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

       2. clear-num 79.5%

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

       3. un-div-inv 79.5%

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

   15. Applied egg-rr 79.5%

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

       1. associate-/r/ 79.5%

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

       2. associate-*l/ 79.5%

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

       3. *-commutative 79.5%

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

       4. *-commutative 79.5%

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

       5. times-frac 79.8%

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

       6. associate-/r/ 98.8%

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

       7. *-inverses 98.8%

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

       8. *-lft-identity 98.8%

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

   17. Simplified 98.8%

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

   if 4.40000000000000002e32 < b

   1. Initial program 11.5%

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

   3. Taylor expanded in b around inf 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-*l/ 97.1%

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

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, 2, \frac{a \cdot -1.5}{b} \cdot c\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-66}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (* b -2.0) (* a 3.0)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(b * -2.0) / 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 <= -4e-310)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(b * -2.0), $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 < -3.999999999999988e-310

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf 70.8%

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

      1. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   if -3.999999999999988e-310 < b

   1. Initial program 40.4%

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

   3. Taylor expanded in b around inf 62.6%

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

      1. *-commutative 62.6%

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

      2. associate-*l/ 62.6%

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

   5. Simplified 62.6%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* (/ b a) -0.6666666666666666) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf 70.7%

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

      1. *-commutative 70.7%

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

   5. Simplified 70.7%

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

   if -3.999999999999988e-310 < b

   1. Initial program 40.4%

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

   3. Taylor expanded in b around inf 48.4%

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

      1. associate-/l* 52.2%

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

   5. Simplified 52.2%

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

      1. expm1-log1p-u 47.3%

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

      2. expm1-udef 22.5%

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

      3. times-frac 22.6%

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

      4. metadata-eval 22.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-0.5} \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)} - 1 \]

      5. associate-/r/ 21.0%

         \[\leadsto e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\color{blue}{\frac{a}{b} \cdot c}}{a}\right)} - 1 \]

   7. Applied egg-rr 21.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 44.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)\right)} \]

      2. expm1-log1p 48.4%

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

      3. associate-/l* 50.8%

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

   9. Simplified 50.8%

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

       1. associate-*r/ 50.8%

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

       2. clear-num 50.5%

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

   11. Applied egg-rr 50.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{c}}{-0.5 \cdot \frac{a}{b}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 50.7%

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

       2. *-commutative 50.7%

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

       3. associate-*r* 50.7%

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

       4. times-frac 52.6%

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

       5. *-commutative 52.6%

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

       6. *-rgt-identity 52.6%

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

       7. *-commutative 52.6%

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

   13. Simplified 52.6%

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

       1. *-commutative 52.6%

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

       2. clear-num 52.3%

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

       3. un-div-inv 52.3%

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

   15. Applied egg-rr 52.3%

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

       1. associate-/r/ 52.6%

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

       2. associate-*l/ 52.6%

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

       3. *-commutative 52.6%

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

       4. *-commutative 52.6%

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

       5. times-frac 57.1%

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

       6. associate-/r/ 62.4%

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

       7. *-inverses 62.4%

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

       8. *-lft-identity 62.4%

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

   17. Simplified 62.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-309}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.5e-309) (* (/ b a) -0.6666666666666666) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.5e-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 <= 4.5d-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 <= 4.5e-309) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.5e-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 <= 4.5e-309)
		tmp = Float64(Float64(b / 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 <= 4.5e-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, 4.5e-309], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5000000000000011e-309

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf 70.7%

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

      1. *-commutative 70.7%

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

   5. Simplified 70.7%

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

   if 4.5000000000000011e-309 < b

   1. Initial program 40.4%

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

   3. Taylor expanded in b around inf 62.6%

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

      1. *-commutative 62.6%

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

      2. associate-*l/ 62.6%

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

   5. Simplified 62.6%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-309}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.8% 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 57.9%

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

3. Taylor expanded in b around inf 24.6%

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

   1. associate-/l* 26.5%

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

5. Simplified 26.5%

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

   1. expm1-log1p-u 24.0%

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

   2. expm1-udef 11.9%

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

   3. times-frac 11.9%

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

   4. metadata-eval 11.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-0.5} \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)} - 1 \]

   5. associate-/r/ 11.1%

      \[\leadsto e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\color{blue}{\frac{a}{b} \cdot c}}{a}\right)} - 1 \]

7. Applied egg-rr 11.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 22.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)\right)} \]

   2. expm1-log1p 24.6%

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

   3. associate-/l* 26.1%

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

9. Simplified 26.1%

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

    1. associate-*r/ 26.1%

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

    2. clear-num 26.0%

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

11. Applied egg-rr 26.0%

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{c}}{-0.5 \cdot \frac{a}{b}}}} \]
12. Step-by-step derivation

    1. associate-/r/ 26.1%

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

    2. *-commutative 26.1%

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

    3. associate-*r* 26.1%

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

    4. times-frac 28.6%

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

    5. *-commutative 28.6%

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

    6. *-rgt-identity 28.6%

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

    7. *-commutative 28.6%

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

13. Simplified 28.6%

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

    1. *-commutative 28.6%

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

    2. clear-num 28.5%

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

    3. un-div-inv 28.5%

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

15. Applied egg-rr 28.5%

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

    1. associate-/r/ 28.6%

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

    2. associate-*l/ 28.6%

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

    3. *-commutative 28.6%

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

    4. *-commutative 28.6%

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

    5. times-frac 30.8%

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

    6. associate-/r/ 31.5%

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

    7. *-inverses 31.5%

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

    8. *-lft-identity 31.5%

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

17. Simplified 31.5%

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

18. Final simplification 31.5%

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

Reproduce

herbie shell --seed 2024026 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))