Cubic critical

Percentage Accurate: 52.3% → 84.9%

Time: 15.7s

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: 52.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+83}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(b, 2, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+83)
   (/ (* -0.3333333333333333 (fma b 2.0 (* -1.5 (/ c (/ b a))))) a)
   (if (<= b 3.7e-81)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+83) {
		tmp = (-0.3333333333333333 * fma(b, 2.0, (-1.5 * (c / (b / a))))) / a;
	} else if (b <= 3.7e-81) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e+83)
		tmp = Float64(Float64(-0.3333333333333333 * fma(b, 2.0, Float64(-1.5 * Float64(c / Float64(b / a))))) / a);
	elseif (b <= 3.7e-81)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e+83], N[(N[(-0.3333333333333333 * N[(b * 2.0 + N[(-1.5 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.7e-81], 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 * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.29999999999999995e83

   1. Initial program 52.8%

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

      1. /-rgt-identity 52.8%

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

      2. metadata-eval 52.8%

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

      3. associate-/r/ 52.8%

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

      4. metadata-eval 52.8%

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

      5. metadata-eval 52.8%

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

      6. times-frac 52.8%

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

      7. *-commutative 52.8%

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

      8. times-frac 52.8%

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

      9. *-commutative 52.8%

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

      10. associate-/r* 52.8%

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

      11. associate-*l/ 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in b around -inf 92.4%

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

      1. +-commutative 92.4%

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

      2. *-commutative 92.4%

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

      3. fma-def 92.4%

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

      4. associate-/l* 95.5%

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

   6. Simplified 95.5%

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

   if -2.29999999999999995e83 < b < 3.69999999999999986e-81

   1. Initial program 87.5%

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

   if 3.69999999999999986e-81 < b

   1. Initial program 15.0%

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

      1. /-rgt-identity 15.0%

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

      2. metadata-eval 15.0%

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

      3. associate-/r/ 15.0%

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

      4. metadata-eval 15.0%

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

      5. metadata-eval 15.0%

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

      6. times-frac 15.0%

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

      7. *-commutative 15.0%

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

      8. times-frac 15.0%

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

      9. associate-/r* 15.0%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in b around inf 24.6%

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

      1. clear-num 24.6%

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

      2. inv-pow 24.6%

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

   6. Applied egg-rr 24.6%

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

      1. unpow-1 24.6%

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

   8. Simplified 24.6%

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

   9. Taylor expanded in b around 0 85.5%

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

       1. *-commutative 85.5%

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

       2. associate-*l/ 85.5%

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

   11. Simplified 85.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+83}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(b, 2, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(b, 2, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+114)
   (/ (* -0.3333333333333333 (fma b 2.0 (* -1.5 (/ c (/ b a))))) a)
   (if (<= b 3.1e-81)
     (/ (* -0.3333333333333333 (- b (sqrt (- (* b b) (* c (* a 3.0)))))) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+114) {
		tmp = (-0.3333333333333333 * fma(b, 2.0, (-1.5 * (c / (b / a))))) / a;
	} else if (b <= 3.1e-81) {
		tmp = (-0.3333333333333333 * (b - sqrt(((b * b) - (c * (a * 3.0)))))) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e+114)
		tmp = Float64(Float64(-0.3333333333333333 * fma(b, 2.0, Float64(-1.5 * Float64(c / Float64(b / a))))) / a);
	elseif (b <= 3.1e-81)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))))) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+114], N[(N[(-0.3333333333333333 * N[(b * 2.0 + N[(-1.5 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.1e-81], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2e114

   1. Initial program 49.0%

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

      1. /-rgt-identity 49.0%

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

      2. metadata-eval 49.0%

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

      3. associate-/r/ 49.0%

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

      4. metadata-eval 49.0%

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

      5. metadata-eval 49.0%

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

      6. times-frac 49.0%

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

      7. *-commutative 49.0%

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

      8. times-frac 49.0%

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

      9. *-commutative 49.0%

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

      10. associate-/r* 49.0%

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

      11. associate-*l/ 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in b around -inf 91.8%

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

      1. +-commutative 91.8%

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

      2. *-commutative 91.8%

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

      3. fma-def 91.8%

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

      4. associate-/l* 95.1%

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

   6. Simplified 95.1%

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

   if -1.2e114 < b < 3.09999999999999988e-81

   1. Initial program 88.0%

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

      1. /-rgt-identity 88.0%

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

      2. metadata-eval 88.0%

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

      3. associate-/r/ 88.0%

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

      4. metadata-eval 88.0%

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

      5. metadata-eval 88.0%

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

      6. times-frac 88.0%

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

      7. *-commutative 88.0%

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

      8. times-frac 87.9%

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

      9. *-commutative 87.9%

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

      10. associate-/r* 87.8%

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

      11. associate-*l/ 87.9%

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

   3. Simplified 87.8%

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

      1. fma-udef 87.8%

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

      2. *-commutative 87.8%

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

      3. metadata-eval 87.8%

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

      4. cancel-sign-sub-inv 87.8%

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

      5. associate-*r* 87.9%

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

      6. *-commutative 87.9%

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

      7. *-commutative 87.9%

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

   5. Applied egg-rr 87.9%

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

   if 3.09999999999999988e-81 < b

   1. Initial program 15.0%

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

      1. /-rgt-identity 15.0%

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

      2. metadata-eval 15.0%

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

      3. associate-/r/ 15.0%

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

      4. metadata-eval 15.0%

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

      5. metadata-eval 15.0%

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

      6. times-frac 15.0%

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

      7. *-commutative 15.0%

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

      8. times-frac 15.0%

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

      9. associate-/r* 15.0%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in b around inf 24.6%

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

      1. clear-num 24.6%

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

      2. inv-pow 24.6%

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

   6. Applied egg-rr 24.6%

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

      1. unpow-1 24.6%

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

   8. Simplified 24.6%

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

   9. Taylor expanded in b around 0 85.5%

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

       1. *-commutative 85.5%

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

       2. associate-*l/ 85.5%

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

   11. Simplified 85.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(b, 2, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(b, 2, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e+82)
   (/ (* -0.3333333333333333 (fma b 2.0 (* -1.5 (/ c (/ b a))))) a)
   (if (<= b 1.7e-81)
     (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e+82) {
		tmp = (-0.3333333333333333 * fma(b, 2.0, (-1.5 * (c / (b / a))))) / a;
	} else if (b <= 1.7e-81) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e+82)
		tmp = Float64(Float64(-0.3333333333333333 * fma(b, 2.0, Float64(-1.5 * Float64(c / Float64(b / a))))) / a);
	elseif (b <= 1.7e-81)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e+82], N[(N[(-0.3333333333333333 * N[(b * 2.0 + N[(-1.5 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.7e-81], 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 * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.19999999999999975e82

   1. Initial program 52.8%

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

      1. /-rgt-identity 52.8%

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

      2. metadata-eval 52.8%

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

      3. associate-/r/ 52.8%

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

      4. metadata-eval 52.8%

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

      5. metadata-eval 52.8%

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

      6. times-frac 52.8%

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

      7. *-commutative 52.8%

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

      8. times-frac 52.8%

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

      9. *-commutative 52.8%

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

      10. associate-/r* 52.8%

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

      11. associate-*l/ 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in b around -inf 92.4%

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

      1. +-commutative 92.4%

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

      2. *-commutative 92.4%

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

      3. fma-def 92.4%

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

      4. associate-/l* 95.5%

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

   6. Simplified 95.5%

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

   if -3.19999999999999975e82 < b < 1.6999999999999999e-81

   1. Initial program 87.5%

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

      1. neg-sub0 87.5%

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

      2. associate-+l- 87.5%

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

      3. sub0-neg 87.5%

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

      4. neg-mul-1 87.5%

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

      5. associate-*r/ 87.5%

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

      6. metadata-eval 87.5%

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

      7. metadata-eval 87.5%

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

      8. times-frac 87.5%

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

      9. *-commutative 87.5%

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

      10. times-frac 87.3%

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

      11. associate-*l/ 87.5%

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

   3. Simplified 87.3%

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

   if 1.6999999999999999e-81 < b

   1. Initial program 15.0%

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

      1. /-rgt-identity 15.0%

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

      2. metadata-eval 15.0%

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

      3. associate-/r/ 15.0%

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

      4. metadata-eval 15.0%

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

      5. metadata-eval 15.0%

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

      6. times-frac 15.0%

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

      7. *-commutative 15.0%

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

      8. times-frac 15.0%

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

      9. associate-/r* 15.0%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in b around inf 24.6%

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

      1. clear-num 24.6%

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

      2. inv-pow 24.6%

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

   6. Applied egg-rr 24.6%

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

      1. unpow-1 24.6%

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

   8. Simplified 24.6%

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

   9. Taylor expanded in b around 0 85.5%

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

       1. *-commutative 85.5%

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

       2. associate-*l/ 85.5%

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

   11. Simplified 85.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(b, 2, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-81}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-38)
   (+ (* (/ b a) -0.6666666666666666) (* 0.5 (/ c b)))
   (if (<= b 1.5e-81)
     (* (- b (sqrt (* c (* a -3.0)))) (/ -0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-38) {
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b));
	} else if (b <= 1.5e-81) {
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.2d-38)) then
        tmp = ((b / a) * (-0.6666666666666666d0)) + (0.5d0 * (c / b))
    else if (b <= 1.5d-81) then
        tmp = (b - sqrt((c * (a * (-3.0d0))))) * ((-0.3333333333333333d0) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-38) {
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b));
	} else if (b <= 1.5e-81) {
		tmp = (b - Math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-38:
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b))
	elif b <= 1.5e-81:
		tmp = (b - math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-38)
		tmp = Float64(Float64(Float64(b / a) * -0.6666666666666666) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.5e-81)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.2e-38)
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b));
	elseif (b <= 1.5e-81)
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-38], N[(N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.5e-81], N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.20000000000000026e-38

   1. Initial program 65.9%

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

      1. /-rgt-identity 65.9%

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

      2. metadata-eval 65.9%

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

      3. associate-/r/ 65.9%

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

      4. metadata-eval 65.9%

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

      5. metadata-eval 65.9%

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

      6. times-frac 65.9%

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

      7. *-commutative 65.9%

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

      8. times-frac 65.9%

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

      9. *-commutative 65.9%

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

      10. associate-/r* 65.8%

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

      11. associate-*l/ 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in b around -inf 91.5%

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

   if -4.20000000000000026e-38 < b < 1.4999999999999999e-81

   1. Initial program 83.4%

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

      1. /-rgt-identity 83.4%

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

      2. metadata-eval 83.4%

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

      3. associate-/r/ 83.4%

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

      4. metadata-eval 83.4%

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

      5. metadata-eval 83.4%

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

      6. times-frac 83.4%

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

      7. *-commutative 83.4%

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

      8. times-frac 83.1%

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

      9. associate-/r* 83.0%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in b around 0 79.5%

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

      1. *-commutative 79.5%

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

      2. metadata-eval 79.5%

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

      3. distribute-rgt-neg-in 79.5%

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

      4. associate-*r* 79.6%

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

      5. distribute-rgt-neg-in 79.6%

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

      6. metadata-eval 79.6%

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

      7. distribute-rgt-neg-in 79.6%

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

      8. distribute-lft-neg-in 79.6%

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

      9. distribute-lft-neg-out 79.6%

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

      10. remove-double-neg 79.6%

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

   6. Simplified 79.6%

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

   if 1.4999999999999999e-81 < b

   1. Initial program 15.5%

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

      1. /-rgt-identity 15.5%

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

      2. metadata-eval 15.5%

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

      3. associate-/r/ 15.5%

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

      4. metadata-eval 15.5%

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

      5. metadata-eval 15.5%

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

      6. times-frac 15.5%

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

      7. *-commutative 15.5%

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

      8. times-frac 15.5%

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

      9. associate-/r* 15.6%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in b around inf 25.0%

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

      1. clear-num 25.0%

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

      2. inv-pow 25.0%

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

   6. Applied egg-rr 25.0%

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

      1. unpow-1 25.0%

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

   8. Simplified 25.0%

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

   9. Taylor expanded in b around 0 85.2%

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

       1. *-commutative 85.2%

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

       2. associate-*l/ 85.2%

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

   11. Simplified 85.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-81}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-39}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.65e-39)
   (+ (* (/ b a) -0.6666666666666666) (* 0.5 (/ c b)))
   (if (<= b 1.45e-81)
     (/ (* -0.3333333333333333 (- b (sqrt (* c (* a -3.0))))) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e-39) {
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b));
	} else if (b <= 1.45e-81) {
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.65d-39)) then
        tmp = ((b / a) * (-0.6666666666666666d0)) + (0.5d0 * (c / b))
    else if (b <= 1.45d-81) then
        tmp = ((-0.3333333333333333d0) * (b - sqrt((c * (a * (-3.0d0)))))) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e-39) {
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b));
	} else if (b <= 1.45e-81) {
		tmp = (-0.3333333333333333 * (b - Math.sqrt((c * (a * -3.0))))) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.65e-39:
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b))
	elif b <= 1.45e-81:
		tmp = (-0.3333333333333333 * (b - math.sqrt((c * (a * -3.0))))) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.65e-39)
		tmp = Float64(Float64(Float64(b / a) * -0.6666666666666666) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.45e-81)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(Float64(c * Float64(a * -3.0))))) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.65e-39)
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b));
	elseif (b <= 1.45e-81)
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.65e-39], N[(N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e-81], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.65000000000000002e-39

   1. Initial program 65.9%

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

      1. /-rgt-identity 65.9%

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

      2. metadata-eval 65.9%

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

      3. associate-/r/ 65.9%

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

      4. metadata-eval 65.9%

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

      5. metadata-eval 65.9%

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

      6. times-frac 65.9%

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

      7. *-commutative 65.9%

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

      8. times-frac 65.9%

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

      9. *-commutative 65.9%

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

      10. associate-/r* 65.8%

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

      11. associate-*l/ 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in b around -inf 91.5%

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

   if -2.65000000000000002e-39 < b < 1.44999999999999994e-81

   1. Initial program 83.4%

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

      1. /-rgt-identity 83.4%

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

      2. metadata-eval 83.4%

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

      3. associate-/r/ 83.4%

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

      4. metadata-eval 83.4%

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

      5. metadata-eval 83.4%

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

      6. times-frac 83.4%

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

      7. *-commutative 83.4%

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

      8. times-frac 83.1%

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

      9. *-commutative 83.1%

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

      10. associate-/r* 83.0%

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

      11. associate-*l/ 83.2%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in b around 0 79.7%

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

      1. *-commutative 79.5%

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

      2. metadata-eval 79.5%

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

      3. distribute-rgt-neg-in 79.5%

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

      4. associate-*r* 79.6%

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

      5. distribute-rgt-neg-in 79.6%

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

      6. metadata-eval 79.6%

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

      7. distribute-rgt-neg-in 79.6%

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

      8. distribute-lft-neg-in 79.6%

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

      9. distribute-lft-neg-out 79.6%

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

      10. remove-double-neg 79.6%

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

   6. Simplified 79.8%

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

   if 1.44999999999999994e-81 < b

   1. Initial program 15.5%

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

      1. /-rgt-identity 15.5%

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

      2. metadata-eval 15.5%

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

      3. associate-/r/ 15.5%

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

      4. metadata-eval 15.5%

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

      5. metadata-eval 15.5%

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

      6. times-frac 15.5%

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

      7. *-commutative 15.5%

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

      8. times-frac 15.5%

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

      9. associate-/r* 15.6%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in b around inf 25.0%

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

      1. clear-num 25.0%

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

      2. inv-pow 25.0%

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

   6. Applied egg-rr 25.0%

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

      1. unpow-1 25.0%

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

   8. Simplified 25.0%

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

   9. Taylor expanded in b around 0 85.2%

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

       1. *-commutative 85.2%

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

       2. associate-*l/ 85.2%

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

   11. Simplified 85.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-39}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(b, 2, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.7e-38)
   (/ (* -0.3333333333333333 (fma b 2.0 (* -1.5 (/ c (/ b a))))) a)
   (if (<= b 1.5e-81)
     (/ (* -0.3333333333333333 (- b (sqrt (* c (* a -3.0))))) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.7e-38) {
		tmp = (-0.3333333333333333 * fma(b, 2.0, (-1.5 * (c / (b / a))))) / a;
	} else if (b <= 1.5e-81) {
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.7e-38)
		tmp = Float64(Float64(-0.3333333333333333 * fma(b, 2.0, Float64(-1.5 * Float64(c / Float64(b / a))))) / a);
	elseif (b <= 1.5e-81)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(Float64(c * Float64(a * -3.0))))) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.7e-38], N[(N[(-0.3333333333333333 * N[(b * 2.0 + N[(-1.5 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.5e-81], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.70000000000000005e-38

   1. Initial program 65.9%

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

      1. /-rgt-identity 65.9%

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

      2. metadata-eval 65.9%

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

      3. associate-/r/ 65.9%

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

      4. metadata-eval 65.9%

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

      5. metadata-eval 65.9%

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

      6. times-frac 65.9%

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

      7. *-commutative 65.9%

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

      8. times-frac 65.9%

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

      9. *-commutative 65.9%

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

      10. associate-/r* 65.8%

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

      11. associate-*l/ 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in b around -inf 89.4%

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

      1. +-commutative 89.4%

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

      2. *-commutative 89.4%

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

      3. fma-def 89.4%

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

      4. associate-/l* 91.6%

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

   6. Simplified 91.6%

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

   if -2.70000000000000005e-38 < b < 1.4999999999999999e-81

   1. Initial program 83.4%

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

      1. /-rgt-identity 83.4%

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

      2. metadata-eval 83.4%

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

      3. associate-/r/ 83.4%

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

      4. metadata-eval 83.4%

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

      5. metadata-eval 83.4%

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

      6. times-frac 83.4%

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

      7. *-commutative 83.4%

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

      8. times-frac 83.1%

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

      9. *-commutative 83.1%

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

      10. associate-/r* 83.0%

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

      11. associate-*l/ 83.2%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in b around 0 79.7%

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

      1. *-commutative 79.5%

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

      2. metadata-eval 79.5%

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

      3. distribute-rgt-neg-in 79.5%

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

      4. associate-*r* 79.6%

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

      5. distribute-rgt-neg-in 79.6%

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

      6. metadata-eval 79.6%

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

      7. distribute-rgt-neg-in 79.6%

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

      8. distribute-lft-neg-in 79.6%

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

      9. distribute-lft-neg-out 79.6%

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

      10. remove-double-neg 79.6%

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

   6. Simplified 79.8%

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

   if 1.4999999999999999e-81 < b

   1. Initial program 15.5%

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

      1. /-rgt-identity 15.5%

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

      2. metadata-eval 15.5%

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

      3. associate-/r/ 15.5%

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

      4. metadata-eval 15.5%

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

      5. metadata-eval 15.5%

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

      6. times-frac 15.5%

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

      7. *-commutative 15.5%

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

      8. times-frac 15.5%

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

      9. associate-/r* 15.6%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in b around inf 25.0%

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

      1. clear-num 25.0%

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

      2. inv-pow 25.0%

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

   6. Applied egg-rr 25.0%

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

      1. unpow-1 25.0%

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

   8. Simplified 25.0%

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

   9. Taylor expanded in b around 0 85.2%

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

       1. *-commutative 85.2%

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

       2. associate-*l/ 85.2%

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

   11. Simplified 85.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(b, 2, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 66.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-297}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.05e-297)
   (* (/ -0.3333333333333333 a) (* b 2.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-297) {
		tmp = (-0.3333333333333333 / a) * (b * 2.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.05d-297) then
        tmp = ((-0.3333333333333333d0) / a) * (b * 2.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-297) {
		tmp = (-0.3333333333333333 / a) * (b * 2.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.05e-297:
		tmp = (-0.3333333333333333 / a) * (b * 2.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.05e-297)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b * 2.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.05e-297)
		tmp = (-0.3333333333333333 / a) * (b * 2.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.05e-297], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.05000000000000007e-297

   1. Initial program 72.7%

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

      1. /-rgt-identity 72.7%

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

      2. metadata-eval 72.7%

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

      3. associate-/r/ 72.7%

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

      4. metadata-eval 72.7%

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

      5. metadata-eval 72.7%

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

      6. times-frac 72.7%

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

      7. *-commutative 72.7%

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

      8. times-frac 72.6%

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

      9. associate-/r* 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in b around -inf 68.6%

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

      1. *-commutative 68.6%

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

   6. Simplified 68.6%

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

   if 1.05000000000000007e-297 < b

   1. Initial program 32.2%

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

      1. /-rgt-identity 32.2%

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

      2. metadata-eval 32.2%

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

      3. associate-/r/ 32.2%

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

      4. metadata-eval 32.2%

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

      5. metadata-eval 32.2%

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

      6. times-frac 32.2%

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

      7. *-commutative 32.2%

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

      8. times-frac 32.1%

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

      9. associate-/r* 32.1%

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

   3. Simplified 32.9%

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

   4. Taylor expanded in b around inf 19.4%

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

      1. clear-num 19.4%

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

      2. inv-pow 19.4%

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

   6. Applied egg-rr 19.4%

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

      1. unpow-1 19.4%

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

   8. Simplified 19.4%

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

   9. Taylor expanded in b around 0 67.2%

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

       1. *-commutative 67.2%

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

       2. associate-*l/ 67.2%

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

   11. Simplified 67.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-297}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8: 66.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-303) (* (/ b 3.0) (/ -2.0 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-303) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.2d-303) then
        tmp = (b / 3.0d0) * ((-2.0d0) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-303) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-303:
		tmp = (b / 3.0) * (-2.0 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-303)
		tmp = Float64(Float64(b / 3.0) * Float64(-2.0 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.2e-303)
		tmp = (b / 3.0) * (-2.0 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-303], N[(N[(b / 3.0), $MachinePrecision] * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.2e-303

   1. Initial program 72.7%

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

      1. neg-sub0 72.7%

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

      2. associate-+l- 72.7%

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

      3. sub0-neg 72.7%

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

      4. neg-mul-1 72.7%

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

      5. associate-*r/ 72.7%

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

      6. metadata-eval 72.7%

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

      7. metadata-eval 72.7%

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

      8. times-frac 72.7%

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

      9. *-commutative 72.7%

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

      10. times-frac 72.6%

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

      11. associate-*l/ 72.7%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in b around -inf 68.7%

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

      1. *-commutative 68.7%

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

   6. Simplified 68.7%

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

      1. times-frac 68.6%

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

   8. Applied egg-rr 68.6%

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

   if 4.2e-303 < b

   1. Initial program 32.2%

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

      1. /-rgt-identity 32.2%

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

      2. metadata-eval 32.2%

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

      3. associate-/r/ 32.2%

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

      4. metadata-eval 32.2%

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

      5. metadata-eval 32.2%

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

      6. times-frac 32.2%

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

      7. *-commutative 32.2%

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

      8. times-frac 32.1%

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

      9. associate-/r* 32.1%

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

   3. Simplified 32.9%

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

   4. Taylor expanded in b around inf 19.4%

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

      1. clear-num 19.4%

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

      2. inv-pow 19.4%

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

   6. Applied egg-rr 19.4%

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

      1. unpow-1 19.4%

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

   8. Simplified 19.4%

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

   9. Taylor expanded in b around 0 67.2%

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

       1. *-commutative 67.2%

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

       2. associate-*l/ 67.2%

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

   11. Simplified 67.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9: 66.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-301}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.25e-301) (/ (* b -2.0) (* a 3.0)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.25e-301) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.25d-301) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.25e-301) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.25e-301:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.25e-301)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.25e-301)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.25e-301], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25000000000000003e-301

   1. Initial program 72.7%

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

      1. neg-sub0 72.7%

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

      2. associate-+l- 72.7%

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

      3. sub0-neg 72.7%

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

      4. neg-mul-1 72.7%

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

      5. associate-*r/ 72.7%

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

      6. metadata-eval 72.7%

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

      7. metadata-eval 72.7%

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

      8. times-frac 72.7%

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

      9. *-commutative 72.7%

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

      10. times-frac 72.6%

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

      11. associate-*l/ 72.7%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in b around -inf 68.7%

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

      1. *-commutative 68.7%

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

   6. Simplified 68.7%

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

   if 1.25000000000000003e-301 < b

   1. Initial program 32.2%

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

      1. /-rgt-identity 32.2%

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

      2. metadata-eval 32.2%

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

      3. associate-/r/ 32.2%

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

      4. metadata-eval 32.2%

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

      5. metadata-eval 32.2%

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

      6. times-frac 32.2%

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

      7. *-commutative 32.2%

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

      8. times-frac 32.1%

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

      9. associate-/r* 32.1%

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

   3. Simplified 32.9%

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

   4. Taylor expanded in b around inf 19.4%

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

      1. clear-num 19.4%

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

      2. inv-pow 19.4%

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

   6. Applied egg-rr 19.4%

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

      1. unpow-1 19.4%

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

   8. Simplified 19.4%

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

   9. Taylor expanded in b around 0 67.2%

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

       1. *-commutative 67.2%

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

       2. associate-*l/ 67.2%

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

   11. Simplified 67.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-301}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 66.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.32e-303) (* (/ b a) -0.6666666666666666) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.32e-303) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.32d-303) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.32e-303) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.32e-303:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.32e-303)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.32e-303)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.32e-303], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.32000000000000005e-303

   1. Initial program 72.7%

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

      1. /-rgt-identity 72.7%

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

      2. metadata-eval 72.7%

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

      3. associate-/r/ 72.7%

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

      4. metadata-eval 72.7%

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

      5. metadata-eval 72.7%

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

      6. times-frac 72.7%

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

      7. *-commutative 72.7%

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

      8. times-frac 72.6%

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

      9. *-commutative 72.6%

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

      10. associate-/r* 72.5%

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

      11. associate-*l/ 72.5%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in b around -inf 68.5%

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

      1. *-commutative 68.5%

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

   6. Simplified 68.5%

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

   if 1.32000000000000005e-303 < b

   1. Initial program 32.2%

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

      1. /-rgt-identity 32.2%

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

      2. metadata-eval 32.2%

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

      3. associate-/r/ 32.2%

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

      4. metadata-eval 32.2%

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

      5. metadata-eval 32.2%

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

      6. times-frac 32.2%

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

      7. *-commutative 32.2%

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

      8. times-frac 32.1%

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

      9. *-commutative 32.1%

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

      10. associate-/r* 32.1%

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

      11. associate-*l/ 32.2%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in b around inf 67.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 66.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4e-303) (/ b (/ a -0.6666666666666666)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-303) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.4d-303) then
        tmp = b / (a / (-0.6666666666666666d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-303) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4e-303:
		tmp = b / (a / -0.6666666666666666)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4e-303)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.4e-303)
		tmp = b / (a / -0.6666666666666666);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4e-303], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4000000000000001e-303

   1. Initial program 72.7%

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

      1. /-rgt-identity 72.7%

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

      2. metadata-eval 72.7%

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

      3. associate-/r/ 72.7%

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

      4. metadata-eval 72.7%

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

      5. metadata-eval 72.7%

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

      6. times-frac 72.7%

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

      7. *-commutative 72.7%

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

      8. times-frac 72.6%

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

      9. *-commutative 72.6%

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

      10. associate-/r* 72.5%

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

      11. associate-*l/ 72.5%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in b around -inf 68.5%

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

      1. *-commutative 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in b around 0 68.5%

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

      1. *-commutative 68.5%

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

      2. associate-/r/ 68.6%

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

   9. Simplified 68.6%

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

   if 2.4000000000000001e-303 < b

   1. Initial program 32.2%

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

      1. /-rgt-identity 32.2%

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

      2. metadata-eval 32.2%

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

      3. associate-/r/ 32.2%

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

      4. metadata-eval 32.2%

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

      5. metadata-eval 32.2%

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

      6. times-frac 32.2%

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

      7. *-commutative 32.2%

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

      8. times-frac 32.1%

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

      9. *-commutative 32.1%

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

      10. associate-/r* 32.1%

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

      11. associate-*l/ 32.2%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in b around inf 67.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12: 66.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.32e-303) (/ b (/ a -0.6666666666666666)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.32e-303) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.32d-303) then
        tmp = b / (a / (-0.6666666666666666d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.32e-303) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.32e-303:
		tmp = b / (a / -0.6666666666666666)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.32e-303)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.32e-303)
		tmp = b / (a / -0.6666666666666666);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.32e-303], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.32000000000000005e-303

   1. Initial program 72.7%

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

      1. /-rgt-identity 72.7%

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

      2. metadata-eval 72.7%

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

      3. associate-/r/ 72.7%

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

      4. metadata-eval 72.7%

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

      5. metadata-eval 72.7%

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

      6. times-frac 72.7%

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

      7. *-commutative 72.7%

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

      8. times-frac 72.6%

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

      9. *-commutative 72.6%

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

      10. associate-/r* 72.5%

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

      11. associate-*l/ 72.5%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in b around -inf 68.5%

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

      1. *-commutative 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in b around 0 68.5%

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

      1. *-commutative 68.5%

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

      2. associate-/r/ 68.6%

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

   9. Simplified 68.6%

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

   if 1.32000000000000005e-303 < b

   1. Initial program 32.2%

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

      1. /-rgt-identity 32.2%

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

      2. metadata-eval 32.2%

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

      3. associate-/r/ 32.2%

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

      4. metadata-eval 32.2%

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

      5. metadata-eval 32.2%

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

      6. times-frac 32.2%

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

      7. *-commutative 32.2%

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

      8. times-frac 32.1%

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

      9. associate-/r* 32.1%

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

   3. Simplified 32.9%

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

   4. Taylor expanded in b around inf 19.4%

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

      1. clear-num 19.4%

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

      2. inv-pow 19.4%

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

   6. Applied egg-rr 19.4%

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

      1. unpow-1 19.4%

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

   8. Simplified 19.4%

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

   9. Taylor expanded in b around 0 67.2%

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

       1. *-commutative 67.2%

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

       2. associate-*l/ 67.2%

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

   11. Simplified 67.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 13: 34.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. /-rgt-identity 52.9%

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

   2. metadata-eval 52.9%

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

   3. associate-/r/ 52.9%

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

   4. metadata-eval 52.9%

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

   5. metadata-eval 52.9%

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

   6. times-frac 52.9%

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

   7. *-commutative 52.9%

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

   8. times-frac 52.8%

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

   9. *-commutative 52.8%

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

   10. associate-/r* 52.8%

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

   11. associate-*l/ 52.8%

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

3. Simplified 53.2%

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

4. Taylor expanded in b around inf 34.1%

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

5. Final simplification 34.1%

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

Reproduce

herbie shell --seed 2023182 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))