Cubic critical

Percentage Accurate: 52.7% → 85.1%

Time: 14.8s

Alternatives: 13

Speedup: 16.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+152)
   (fma 0.5 (/ c b) (/ b (/ a -0.6666666666666666)))
   (if (<= b 9.2e-48)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+152) {
		tmp = fma(0.5, (c / b), (b / (a / -0.6666666666666666)));
	} else if (b <= 9.2e-48) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e+152)
		tmp = fma(0.5, Float64(c / b), Float64(b / Float64(a / -0.6666666666666666)));
	elseif (b <= 9.2e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e+152], N[(0.5 * N[(c / b), $MachinePrecision] + N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2e-48], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.29999999999999985e152

   1. Initial program 40.0%

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

      1. /-rgt-identity 40.0%

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

      2. metadata-eval 40.0%

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

      3. associate-/r/ 40.0%

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

      4. metadata-eval 40.0%

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

      5. metadata-eval 40.0%

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

      6. times-frac 40.0%

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

      7. *-commutative 40.0%

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

      8. times-frac 40.0%

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

      9. *-commutative 40.0%

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

      10. associate-/r* 40.0%

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

      11. associate-*l/ 40.0%

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

   3. Simplified 40.0%

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

      1. div-inv 40.0%

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

      2. *-commutative 40.0%

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

      3. fma-udef 40.0%

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

      4. associate-*r* 40.0%

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

      5. add-sqr-sqrt 26.3%

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

      6. hypot-def 57.5%

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

   5. Applied egg-rr 57.5%

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

   6. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + -0.6666666666666666 \cdot \frac{b}{a}} \]

      2. *-commutative 0.0%

         \[\leadsto -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

      3. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} + \frac{b}{a} \cdot -0.6666666666666666 \]

      4. *-commutative 0.0%

         \[\leadsto \frac{-0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      5. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      6. unpow2 0.0%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      7. rem-square-sqrt 98.3%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      8. metadata-eval 98.3%

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

      9. associate-*r/ 98.3%

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

      10. fma-def 98.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)} \]

      11. associate-/r/ 99.7%

          \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}}\right) \]

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)} \]

   if -2.29999999999999985e152 < b < 9.2000000000000003e-48

   1. Initial program 81.1%

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

   if 9.2000000000000003e-48 < b

   1. Initial program 15.8%

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

      1. neg-sub0 15.8%

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

      2. associate-+l- 15.8%

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

      3. sub0-neg 15.8%

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

      4. neg-mul-1 15.8%

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

      5. associate-*r/ 15.8%

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

      6. *-commutative 15.8%

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

      7. metadata-eval 15.8%

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

      8. metadata-eval 15.8%

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

      9. times-frac 15.8%

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

      10. *-commutative 15.8%

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

      11. times-frac 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in b around inf 90.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 2: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+35)
   (fma 0.5 (/ c b) (/ b (/ a -0.6666666666666666)))
   (if (<= b 9e-48)
     (* (- (sqrt (+ (* b b) (* a (* c -3.0)))) b) (/ 0.3333333333333333 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+35) {
		tmp = fma(0.5, (c / b), (b / (a / -0.6666666666666666)));
	} else if (b <= 9e-48) {
		tmp = (sqrt(((b * b) + (a * (c * -3.0)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+35)
		tmp = fma(0.5, Float64(c / b), Float64(b / Float64(a / -0.6666666666666666)));
	elseif (b <= 9e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+35], N[(0.5 * N[(c / b), $MachinePrecision] + N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9e-48], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4999999999999997e35

   1. Initial program 55.5%

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

      1. /-rgt-identity 55.5%

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

      2. metadata-eval 55.5%

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

      3. associate-/r/ 55.5%

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

      4. metadata-eval 55.5%

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

      5. metadata-eval 55.5%

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

      6. times-frac 55.5%

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

      7. *-commutative 55.5%

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

      8. times-frac 55.5%

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

      9. *-commutative 55.5%

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

      10. associate-/r* 55.4%

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

      11. associate-*l/ 55.4%

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

   3. Simplified 55.4%

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

      1. div-inv 55.4%

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

      2. *-commutative 55.4%

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

      3. fma-udef 55.4%

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

      4. associate-*r* 55.4%

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

      5. add-sqr-sqrt 36.2%

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

      6. hypot-def 58.7%

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

   5. Applied egg-rr 58.7%

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

   6. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + -0.6666666666666666 \cdot \frac{b}{a}} \]

      2. *-commutative 0.0%

         \[\leadsto -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

      3. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} + \frac{b}{a} \cdot -0.6666666666666666 \]

      4. *-commutative 0.0%

         \[\leadsto \frac{-0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      5. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      6. unpow2 0.0%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      7. rem-square-sqrt 97.4%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      8. metadata-eval 97.4%

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

      9. associate-*r/ 97.4%

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

      10. fma-def 97.4%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)} \]

      11. associate-/r/ 98.5%

          \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}}\right) \]

   8. Simplified 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)} \]

   if -4.4999999999999997e35 < b < 8.99999999999999977e-48

   1. Initial program 77.5%

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

      1. neg-sub0 77.5%

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

      2. associate-+l- 77.5%

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

      3. sub0-neg 77.5%

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

      4. neg-mul-1 77.5%

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

      5. associate-*r/ 77.5%

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

      6. *-commutative 77.5%

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

      7. metadata-eval 77.5%

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

      8. metadata-eval 77.5%

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

      9. times-frac 77.5%

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

      10. *-commutative 77.5%

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

      11. times-frac 77.4%

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

   3. Simplified 77.4%

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

      1. fma-udef 77.4%

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

   5. Applied egg-rr 77.4%

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

   if 8.99999999999999977e-48 < b

   1. Initial program 15.8%

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

      1. neg-sub0 15.8%

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

      2. associate-+l- 15.8%

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

      3. sub0-neg 15.8%

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

      4. neg-mul-1 15.8%

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

      5. associate-*r/ 15.8%

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

      6. *-commutative 15.8%

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

      7. metadata-eval 15.8%

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

      8. metadata-eval 15.8%

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

      9. times-frac 15.8%

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

      10. *-commutative 15.8%

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

      11. times-frac 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in b around inf 90.4%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 3: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e+152)
   (fma 0.5 (/ c b) (/ b (/ a -0.6666666666666666)))
   (if (<= b 1.08e-47)
     (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e+152) {
		tmp = fma(0.5, (c / b), (b / (a / -0.6666666666666666)));
	} else if (b <= 1.08e-47) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e+152)
		tmp = fma(0.5, Float64(c / b), Float64(b / Float64(a / -0.6666666666666666)));
	elseif (b <= 1.08e-47)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e+152], N[(0.5 * N[(c / b), $MachinePrecision] + N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.08e-47], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0999999999999999e152

   1. Initial program 40.0%

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

      1. /-rgt-identity 40.0%

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

      2. metadata-eval 40.0%

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

      3. associate-/r/ 40.0%

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

      4. metadata-eval 40.0%

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

      5. metadata-eval 40.0%

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

      6. times-frac 40.0%

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

      7. *-commutative 40.0%

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

      8. times-frac 40.0%

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

      9. *-commutative 40.0%

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

      10. associate-/r* 40.0%

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

      11. associate-*l/ 40.0%

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

   3. Simplified 40.0%

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

      1. div-inv 40.0%

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

      2. *-commutative 40.0%

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

      3. fma-udef 40.0%

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

      4. associate-*r* 40.0%

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

      5. add-sqr-sqrt 26.3%

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

      6. hypot-def 57.5%

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

   5. Applied egg-rr 57.5%

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

   6. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + -0.6666666666666666 \cdot \frac{b}{a}} \]

      2. *-commutative 0.0%

         \[\leadsto -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

      3. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} + \frac{b}{a} \cdot -0.6666666666666666 \]

      4. *-commutative 0.0%

         \[\leadsto \frac{-0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      5. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      6. unpow2 0.0%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      7. rem-square-sqrt 98.3%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      8. metadata-eval 98.3%

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

      9. associate-*r/ 98.3%

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

      10. fma-def 98.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)} \]

      11. associate-/r/ 99.7%

          \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}}\right) \]

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)} \]

   if -1.0999999999999999e152 < b < 1.08000000000000005e-47

   1. Initial program 81.1%

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

      1. neg-sub0 81.1%

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

      2. associate-+l- 81.1%

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

      3. sub0-neg 81.1%

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

      4. neg-mul-1 81.1%

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

      5. associate-*r/ 81.1%

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

      6. metadata-eval 81.1%

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

      7. metadata-eval 81.1%

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

      8. times-frac 81.1%

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

      9. *-commutative 81.1%

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

      10. times-frac 81.0%

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

      11. associate-*l/ 81.1%

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

   3. Simplified 81.0%

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

   if 1.08000000000000005e-47 < b

   1. Initial program 15.8%

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

      1. neg-sub0 15.8%

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

      2. associate-+l- 15.8%

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

      3. sub0-neg 15.8%

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

      4. neg-mul-1 15.8%

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

      5. associate-*r/ 15.8%

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

      6. *-commutative 15.8%

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

      7. metadata-eval 15.8%

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

      8. metadata-eval 15.8%

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

      9. times-frac 15.8%

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

      10. *-commutative 15.8%

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

      11. times-frac 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in b around inf 90.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 4: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-137)
   (fma 0.5 (/ c b) (/ b (/ a -0.6666666666666666)))
   (if (<= b 1.25e-47)
     (* (/ 0.3333333333333333 a) (- (sqrt (* (* c a) -3.0)) b))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-137) {
		tmp = fma(0.5, (c / b), (b / (a / -0.6666666666666666)));
	} else if (b <= 1.25e-47) {
		tmp = (0.3333333333333333 / a) * (sqrt(((c * a) * -3.0)) - b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-137)
		tmp = fma(0.5, Float64(c / b), Float64(b / Float64(a / -0.6666666666666666)));
	elseif (b <= 1.25e-47)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-137], N[(0.5 * N[(c / b), $MachinePrecision] + N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-47], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5000000000000003e-137

   1. Initial program 65.1%

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

      1. /-rgt-identity 65.1%

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

      2. metadata-eval 65.1%

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

      3. associate-/r/ 65.1%

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

      4. metadata-eval 65.1%

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

      5. metadata-eval 65.1%

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

      6. times-frac 65.1%

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

      7. *-commutative 65.1%

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

      8. times-frac 65.1%

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

      9. *-commutative 65.1%

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

      10. associate-/r* 65.0%

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

      11. associate-*l/ 65.0%

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

   3. Simplified 65.0%

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

      1. div-inv 64.9%

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

      2. *-commutative 64.9%

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

      3. fma-udef 64.9%

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

      4. associate-*r* 64.9%

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

      5. add-sqr-sqrt 42.2%

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

      6. hypot-def 58.0%

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

   5. Applied egg-rr 58.0%

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

   6. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + -0.6666666666666666 \cdot \frac{b}{a}} \]

      2. *-commutative 0.0%

         \[\leadsto -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

      3. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} + \frac{b}{a} \cdot -0.6666666666666666 \]

      4. *-commutative 0.0%

         \[\leadsto \frac{-0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      5. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      6. unpow2 0.0%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      7. rem-square-sqrt 88.5%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      8. metadata-eval 88.5%

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

      9. associate-*r/ 88.5%

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

      10. fma-def 88.5%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)} \]

      11. associate-/r/ 89.3%

          \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}}\right) \]

   8. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)} \]

   if -5.5000000000000003e-137 < b < 1.25000000000000003e-47

   1. Initial program 71.1%

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

      1. neg-sub0 71.1%

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

      2. associate-+l- 71.1%

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

      3. sub0-neg 71.1%

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

      4. neg-mul-1 71.1%

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

      5. associate-*r/ 71.1%

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

      6. *-commutative 71.1%

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

      7. metadata-eval 71.1%

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

      8. metadata-eval 71.1%

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

      9. times-frac 71.1%

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

      10. *-commutative 71.1%

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

      11. times-frac 71.0%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in b around 0 66.9%

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

   if 1.25000000000000003e-47 < b

   1. Initial program 15.8%

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

      1. neg-sub0 15.8%

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

      2. associate-+l- 15.8%

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

      3. sub0-neg 15.8%

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

      4. neg-mul-1 15.8%

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

      5. associate-*r/ 15.8%

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

      6. *-commutative 15.8%

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

      7. metadata-eval 15.8%

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

      8. metadata-eval 15.8%

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

      9. times-frac 15.8%

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

      10. *-commutative 15.8%

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

      11. times-frac 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in b around inf 90.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 5: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-137)
   (fma 0.5 (/ c b) (/ b (/ a -0.6666666666666666)))
   (if (<= b 1.7e-47)
     (* (/ 0.3333333333333333 a) (- (sqrt (* a (* c -3.0))) b))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-137) {
		tmp = fma(0.5, (c / b), (b / (a / -0.6666666666666666)));
	} else if (b <= 1.7e-47) {
		tmp = (0.3333333333333333 / a) * (sqrt((a * (c * -3.0))) - b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-137)
		tmp = fma(0.5, Float64(c / b), Float64(b / Float64(a / -0.6666666666666666)));
	elseif (b <= 1.7e-47)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(a * Float64(c * -3.0))) - b));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-137], N[(0.5 * N[(c / b), $MachinePrecision] + N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e-47], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.89999999999999999e-137

   1. Initial program 65.1%

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

      1. /-rgt-identity 65.1%

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

      2. metadata-eval 65.1%

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

      3. associate-/r/ 65.1%

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

      4. metadata-eval 65.1%

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

      5. metadata-eval 65.1%

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

      6. times-frac 65.1%

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

      7. *-commutative 65.1%

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

      8. times-frac 65.1%

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

      9. *-commutative 65.1%

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

      10. associate-/r* 65.0%

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

      11. associate-*l/ 65.0%

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

   3. Simplified 65.0%

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

      1. div-inv 64.9%

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

      2. *-commutative 64.9%

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

      3. fma-udef 64.9%

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

      4. associate-*r* 64.9%

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

      5. add-sqr-sqrt 42.2%

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

      6. hypot-def 58.0%

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

   5. Applied egg-rr 58.0%

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

   6. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + -0.6666666666666666 \cdot \frac{b}{a}} \]

      2. *-commutative 0.0%

         \[\leadsto -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

      3. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} + \frac{b}{a} \cdot -0.6666666666666666 \]

      4. *-commutative 0.0%

         \[\leadsto \frac{-0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      5. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      6. unpow2 0.0%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      7. rem-square-sqrt 88.5%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      8. metadata-eval 88.5%

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

      9. associate-*r/ 88.5%

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

      10. fma-def 88.5%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)} \]

      11. associate-/r/ 89.3%

          \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}}\right) \]

   8. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)} \]

   if -1.89999999999999999e-137 < b < 1.7000000000000001e-47

   1. Initial program 71.1%

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

      1. neg-sub0 71.1%

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

      2. associate-+l- 71.1%

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

      3. sub0-neg 71.1%

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

      4. neg-mul-1 71.1%

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

      5. associate-*r/ 71.1%

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

      6. *-commutative 71.1%

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

      7. metadata-eval 71.1%

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

      8. metadata-eval 71.1%

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

      9. times-frac 71.1%

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

      10. *-commutative 71.1%

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

      11. times-frac 71.0%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in b around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-*r* 66.9%

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

   6. Simplified 66.9%

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

   if 1.7000000000000001e-47 < b

   1. Initial program 15.8%

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

      1. neg-sub0 15.8%

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

      2. associate-+l- 15.8%

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

      3. sub0-neg 15.8%

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

      4. neg-mul-1 15.8%

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

      5. associate-*r/ 15.8%

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

      6. *-commutative 15.8%

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

      7. metadata-eval 15.8%

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

      8. metadata-eval 15.8%

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

      9. times-frac 15.8%

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

      10. *-commutative 15.8%

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

      11. times-frac 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in b around inf 90.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 6: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-137)
   (fma 0.5 (/ c b) (/ b (/ a -0.6666666666666666)))
   (if (<= b 1.08e-47)
     (/ (* 0.3333333333333333 (- (sqrt (* a (* c -3.0))) b)) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-137) {
		tmp = fma(0.5, (c / b), (b / (a / -0.6666666666666666)));
	} else if (b <= 1.08e-47) {
		tmp = (0.3333333333333333 * (sqrt((a * (c * -3.0))) - b)) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-137)
		tmp = fma(0.5, Float64(c / b), Float64(b / Float64(a / -0.6666666666666666)));
	elseif (b <= 1.08e-47)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(Float64(a * Float64(c * -3.0))) - b)) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-137], N[(0.5 * N[(c / b), $MachinePrecision] + N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.08e-47], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5000000000000003e-137

   1. Initial program 65.1%

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

      1. /-rgt-identity 65.1%

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

      2. metadata-eval 65.1%

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

      3. associate-/r/ 65.1%

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

      4. metadata-eval 65.1%

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

      5. metadata-eval 65.1%

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

      6. times-frac 65.1%

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

      7. *-commutative 65.1%

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

      8. times-frac 65.1%

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

      9. *-commutative 65.1%

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

      10. associate-/r* 65.0%

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

      11. associate-*l/ 65.0%

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

   3. Simplified 65.0%

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

      1. div-inv 64.9%

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

      2. *-commutative 64.9%

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

      3. fma-udef 64.9%

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

      4. associate-*r* 64.9%

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

      5. add-sqr-sqrt 42.2%

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

      6. hypot-def 58.0%

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

   5. Applied egg-rr 58.0%

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

   6. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + -0.6666666666666666 \cdot \frac{b}{a}} \]

      2. *-commutative 0.0%

         \[\leadsto -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

      3. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} + \frac{b}{a} \cdot -0.6666666666666666 \]

      4. *-commutative 0.0%

         \[\leadsto \frac{-0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      5. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      6. unpow2 0.0%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      7. rem-square-sqrt 88.5%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      8. metadata-eval 88.5%

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

      9. associate-*r/ 88.5%

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

      10. fma-def 88.5%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)} \]

      11. associate-/r/ 89.3%

          \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}}\right) \]

   8. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)} \]

   if -5.5000000000000003e-137 < b < 1.08000000000000005e-47

   1. Initial program 71.1%

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

      1. neg-sub0 71.1%

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

      2. associate-+l- 71.1%

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

      3. sub0-neg 71.1%

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

      4. neg-mul-1 71.1%

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

      5. associate-*r/ 71.1%

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

      6. *-commutative 71.1%

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

      7. metadata-eval 71.1%

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

      8. metadata-eval 71.1%

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

      9. times-frac 71.1%

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

      10. *-commutative 71.1%

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

      11. times-frac 71.0%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in b around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. *-commutative 66.9%

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

      3. *-commutative 66.9%

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

      4. associate-*l* 66.9%

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

   6. Simplified 66.9%

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

      1. associate-*r/ 67.1%

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

      2. associate-*r* 67.0%

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

      3. *-commutative 67.0%

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

      4. associate-*r* 67.0%

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

   8. Applied egg-rr 67.0%

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

   if 1.08000000000000005e-47 < b

   1. Initial program 15.8%

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

      1. neg-sub0 15.8%

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

      2. associate-+l- 15.8%

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

      3. sub0-neg 15.8%

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

      4. neg-mul-1 15.8%

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

      5. associate-*r/ 15.8%

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

      6. *-commutative 15.8%

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

      7. metadata-eval 15.8%

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

      8. metadata-eval 15.8%

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

      9. times-frac 15.8%

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

      10. *-commutative 15.8%

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

      11. times-frac 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in b around inf 90.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 7: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (fma 0.5 (/ c b) (/ b (/ a -0.6666666666666666)))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = fma(0.5, (c / b), (b / (a / -0.6666666666666666)));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = fma(0.5, Float64(c / b), Float64(b / Float64(a / -0.6666666666666666)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(0.5 * N[(c / b), $MachinePrecision] + N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 65.7%

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

      1. /-rgt-identity 65.7%

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

      2. metadata-eval 65.7%

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

      3. associate-/r/ 65.7%

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

      4. metadata-eval 65.7%

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

      5. metadata-eval 65.7%

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

      6. times-frac 65.7%

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

      7. *-commutative 65.7%

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

      8. times-frac 65.7%

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

      9. *-commutative 65.7%

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

      10. associate-/r* 65.6%

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

      11. associate-*l/ 65.6%

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

   3. Simplified 65.7%

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

      1. div-inv 65.6%

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

      2. *-commutative 65.6%

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

      3. fma-udef 65.6%

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

      4. associate-*r* 65.6%

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

      5. add-sqr-sqrt 46.0%

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

      6. hypot-def 61.4%

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

   5. Applied egg-rr 61.4%

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

   6. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + -0.6666666666666666 \cdot \frac{b}{a}} \]

      2. *-commutative 0.0%

         \[\leadsto -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} + \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

      3. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} + \frac{b}{a} \cdot -0.6666666666666666 \]

      4. *-commutative 0.0%

         \[\leadsto \frac{-0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      5. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      6. unpow2 0.0%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      7. rem-square-sqrt 79.4%

         \[\leadsto \frac{\left(-0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{b} + \frac{b}{a} \cdot -0.6666666666666666 \]

      8. metadata-eval 79.4%

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

      9. associate-*r/ 79.4%

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

      10. fma-def 79.4%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)} \]

      11. associate-/r/ 80.1%

          \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}}\right) \]

   8. Simplified 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)} \]

   if -4.999999999999985e-310 < b

   1. Initial program 30.2%

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

      1. neg-sub0 30.2%

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

      2. associate-+l- 30.2%

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

      3. sub0-neg 30.2%

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

      4. neg-mul-1 30.2%

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

      5. associate-*r/ 30.2%

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

      6. *-commutative 30.2%

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

      7. metadata-eval 30.2%

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

      8. metadata-eval 30.2%

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

      9. times-frac 30.2%

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

      10. *-commutative 30.2%

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

      11. times-frac 30.2%

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

   3. Simplified 30.1%

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

   4. Taylor expanded in b around inf 72.7%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{\frac{a}{-0.6666666666666666}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 8: 66.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(-1.5 \cdot \frac{c}{\frac{b}{a}} + b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (/ (* -0.3333333333333333 (+ (* -1.5 (/ c (/ b a))) (* b 2.0))) a)
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.3333333333333333 * ((-1.5 * (c / (b / a))) + (b * 2.0))) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((-0.3333333333333333d0) * (((-1.5d0) * (c / (b / a))) + (b * 2.0d0))) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.3333333333333333 * ((-1.5 * (c / (b / a))) + (b * 2.0))) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (-0.3333333333333333 * ((-1.5 * (c / (b / a))) + (b * 2.0))) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(Float64(-1.5 * Float64(c / Float64(b / a))) + Float64(b * 2.0))) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (-0.3333333333333333 * ((-1.5 * (c / (b / a))) + (b * 2.0))) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(-0.3333333333333333 * N[(N[(-1.5 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 65.7%

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

      1. /-rgt-identity 65.7%

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

      2. metadata-eval 65.7%

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

      3. associate-/r/ 65.7%

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

      4. metadata-eval 65.7%

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

      5. metadata-eval 65.7%

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

      6. times-frac 65.7%

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

      7. *-commutative 65.7%

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

      8. times-frac 65.7%

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

      9. *-commutative 65.7%

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

      10. associate-/r* 65.6%

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

      11. associate-*l/ 65.6%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in b around -inf 74.4%

      \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(-1.5 \cdot \frac{c \cdot a}{b} + 2 \cdot b\right)}}{a} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 71.9%

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(-1.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c \cdot a}{b}\right)\right)} + 2 \cdot b\right)}{a} \]

      2. expm1-udef 71.6%

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(-1.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c \cdot a}{b}\right)} - 1\right)} + 2 \cdot b\right)}{a} \]

      3. associate-/l* 74.7%

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(-1.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{c}{\frac{b}{a}}}\right)} - 1\right) + 2 \cdot b\right)}{a} \]

   6. Applied egg-rr 74.7%

      \[\leadsto \frac{-0.3333333333333333 \cdot \left(-1.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c}{\frac{b}{a}}\right)} - 1\right)} + 2 \cdot b\right)}{a} \]
   7. Step-by-step derivation

      1. expm1-def 75.0%

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(-1.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{\frac{b}{a}}\right)\right)} + 2 \cdot b\right)}{a} \]

      2. expm1-log1p 79.8%

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(-1.5 \cdot \color{blue}{\frac{c}{\frac{b}{a}}} + 2 \cdot b\right)}{a} \]

   8. Simplified 79.8%

      \[\leadsto \frac{-0.3333333333333333 \cdot \left(-1.5 \cdot \color{blue}{\frac{c}{\frac{b}{a}}} + 2 \cdot b\right)}{a} \]

   if -4.999999999999985e-310 < b

   1. Initial program 30.2%

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

      1. neg-sub0 30.2%

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

      2. associate-+l- 30.2%

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

      3. sub0-neg 30.2%

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

      4. neg-mul-1 30.2%

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

      5. associate-*r/ 30.2%

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

      6. *-commutative 30.2%

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

      7. metadata-eval 30.2%

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

      8. metadata-eval 30.2%

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

      9. times-frac 30.2%

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

      10. *-commutative 30.2%

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

      11. times-frac 30.2%

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

   3. Simplified 30.1%

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

   4. Taylor expanded in b around inf 72.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(-1.5 \cdot \frac{c}{\frac{b}{a}} + b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 9: 66.8% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (* (/ c b) -0.5)))

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 / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c / b) * -0.5;
	}
	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 / b) * -0.5
	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 / b) * -0.5);
	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 / b) * -0.5;
	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 / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 65.7%

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

      1. neg-sub0 65.7%

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

      2. associate-+l- 65.7%

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

      3. sub0-neg 65.7%

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

      4. neg-mul-1 65.7%

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

      5. associate-*r/ 65.7%

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

      6. *-commutative 65.7%

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

      7. metadata-eval 65.7%

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

      8. metadata-eval 65.7%

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

      9. times-frac 65.7%

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

      10. *-commutative 65.7%

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

      11. times-frac 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in b around -inf 79.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.2%

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

      1. neg-sub0 30.2%

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

      2. associate-+l- 30.2%

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

      3. sub0-neg 30.2%

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

      4. neg-mul-1 30.2%

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

      5. associate-*r/ 30.2%

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

      6. *-commutative 30.2%

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

      7. metadata-eval 30.2%

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

      8. metadata-eval 30.2%

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

      9. times-frac 30.2%

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

      10. *-commutative 30.2%

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

      11. times-frac 30.2%

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

   3. Simplified 30.1%

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

   4. Taylor expanded in b around inf 72.7%

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

4. Final simplification 76.0%

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

Alternative 10: 66.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-303}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.45e-303) (/ (* b -2.0) (* a 3.0)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.45e-303) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.45d-303) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.45e-303) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.45e-303:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.45e-303)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.45e-303)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.45e-303], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45000000000000007e-303

   1. Initial program 66.0%

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

      1. neg-sub0 66.0%

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

      2. associate-+l- 66.0%

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

      3. sub0-neg 66.0%

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

      4. neg-mul-1 66.0%

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

      5. associate-*r/ 66.0%

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

      6. metadata-eval 66.0%

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

      7. metadata-eval 66.0%

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

      8. times-frac 66.0%

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

      9. *-commutative 66.0%

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

      10. times-frac 66.0%

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

      11. associate-*l/ 66.0%

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

   3. Simplified 66.0%

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

      1. +-commutative 66.0%

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

      2. sub-neg 66.0%

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

      3. cancel-sign-sub-inv 66.0%

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

      4. metadata-eval 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*r* 66.0%

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

      7. add-sqr-sqrt 46.5%

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

      8. hypot-def 61.9%

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

   5. Applied egg-rr 61.9%

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

   6. Taylor expanded in b around -inf 78.7%

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

      1. *-commutative 78.7%

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

   8. Simplified 78.7%

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

   if 1.45000000000000007e-303 < b

   1. Initial program 29.7%

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

      1. neg-sub0 29.7%

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

      2. associate-+l- 29.7%

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

      3. sub0-neg 29.7%

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

      4. neg-mul-1 29.7%

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

      5. associate-*r/ 29.7%

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

      6. *-commutative 29.7%

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

      7. metadata-eval 29.7%

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

      8. metadata-eval 29.7%

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

      9. times-frac 29.7%

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

      10. *-commutative 29.7%

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

      11. times-frac 29.6%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in b around inf 73.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-303}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 11: 66.7% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-303}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.45e-303) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.45e-303) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.45d-303) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.45e-303) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.45e-303:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.45e-303)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.45e-303)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.45e-303], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45000000000000007e-303

   1. Initial program 66.0%

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

      1. neg-sub0 66.0%

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

      2. associate-+l- 66.0%

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

      3. sub0-neg 66.0%

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

      4. neg-mul-1 66.0%

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

      5. associate-*r/ 66.0%

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

      6. *-commutative 66.0%

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

      7. metadata-eval 66.0%

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

      8. metadata-eval 66.0%

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

      9. times-frac 66.0%

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

      10. *-commutative 66.0%

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

      11. times-frac 66.0%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in b around -inf 78.0%

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

      1. *-commutative 78.0%

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

   6. Simplified 78.0%

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

   if 1.45000000000000007e-303 < b

   1. Initial program 29.7%

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

      1. neg-sub0 29.7%

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

      2. associate-+l- 29.7%

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

      3. sub0-neg 29.7%

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

      4. neg-mul-1 29.7%

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

      5. associate-*r/ 29.7%

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

      6. *-commutative 29.7%

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

      7. metadata-eval 29.7%

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

      8. metadata-eval 29.7%

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

      9. times-frac 29.7%

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

      10. *-commutative 29.7%

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

      11. times-frac 29.6%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in b around inf 73.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-303}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 12: 66.7% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-302}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.3e-302) (/ b (/ a -0.6666666666666666)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.3e-302) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.3d-302) then
        tmp = b / (a / (-0.6666666666666666d0))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.3e-302) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.3e-302:
		tmp = b / (a / -0.6666666666666666)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.3e-302)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.3e-302)
		tmp = b / (a / -0.6666666666666666);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.3e-302], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.30000000000000002e-302

   1. Initial program 66.0%

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

      1. /-rgt-identity 66.0%

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

      2. metadata-eval 66.0%

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

      3. associate-/r/ 66.0%

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

      4. metadata-eval 66.0%

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

      5. metadata-eval 66.0%

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

      6. times-frac 66.0%

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

      7. *-commutative 66.0%

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

      8. times-frac 66.0%

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

      9. *-commutative 66.0%

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

      10. associate-/r* 65.9%

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

      11. associate-*l/ 65.9%

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

   3. Simplified 65.9%

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

      1. div-inv 65.8%

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

      2. *-commutative 65.8%

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

      3. fma-udef 65.8%

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

      4. associate-*r* 65.8%

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

      5. add-sqr-sqrt 46.4%

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

      6. hypot-def 61.7%

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

   5. Applied egg-rr 61.7%

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

   6. Taylor expanded in b around -inf 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-/r/ 78.6%

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

   8. Simplified 78.6%

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

   if 2.30000000000000002e-302 < b

   1. Initial program 29.7%

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

      1. neg-sub0 29.7%

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

      2. associate-+l- 29.7%

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

      3. sub0-neg 29.7%

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

      4. neg-mul-1 29.7%

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

      5. associate-*r/ 29.7%

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

      6. *-commutative 29.7%

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

      7. metadata-eval 29.7%

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

      8. metadata-eval 29.7%

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

      9. times-frac 29.7%

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

      10. *-commutative 29.7%

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

      11. times-frac 29.6%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in b around inf 73.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-302}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 13: 34.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.4%

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

   1. neg-sub0 47.4%

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

   2. associate-+l- 47.4%

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

   3. sub0-neg 47.4%

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

   4. neg-mul-1 47.4%

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

   5. associate-*r/ 47.4%

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

   6. *-commutative 47.4%

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

   7. metadata-eval 47.4%

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

   8. metadata-eval 47.4%

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

   9. times-frac 47.4%

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

   10. *-commutative 47.4%

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

   11. times-frac 47.4%

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

3. Simplified 47.3%

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

4. Taylor expanded in b around inf 38.7%

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

5. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2023238 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))