Cubic critical

Percentage Accurate: 52.6% → 85.4%

Time: 10.9s

Alternatives: 9

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-0.375, c \cdot \frac{a}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e+143)
   (/ b (* a -1.5))
   (if (<= b 8.6e-43)
     (/ (/ (- b (sqrt (fma b b (* (* a c) -3.0)))) a) -3.0)
     (* c (fma -0.375 (* c (/ a (* b (* b b)))) (/ -0.5 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+143) {
		tmp = b / (a * -1.5);
	} else if (b <= 8.6e-43) {
		tmp = ((b - sqrt(fma(b, b, ((a * c) * -3.0)))) / a) / -3.0;
	} else {
		tmp = c * fma(-0.375, (c * (a / (b * (b * b)))), (-0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e+143)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 8.6e-43)
		tmp = Float64(Float64(Float64(b - sqrt(fma(b, b, Float64(Float64(a * c) * -3.0)))) / a) / -3.0);
	else
		tmp = Float64(c * fma(-0.375, Float64(c * Float64(a / Float64(b * Float64(b * b)))), Float64(-0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e+143], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-43], N[(N[(N[(b - N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(c * N[(-0.375 * N[(c * N[(a / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5999999999999999e143

   1. Initial program 39.0%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.1

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

   5. Simplified 93.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.2

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

   7. Applied egg-rr 93.2%

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

      1. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-2}{3}}}} \]

      2. un-div-inv N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      6. metadata-eval 93.4

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

   9. Applied egg-rr 93.4%

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

   if -3.5999999999999999e143 < b < 8.59999999999999927e-43

   1. Initial program 83.8%

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

   4. Applied egg-rr 83.9%

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

   if 8.59999999999999927e-43 < b

   1. Initial program 12.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto c \cdot \frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      9. associate-*r/ N/A

         \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}}} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto c \cdot \left(\frac{\color{blue}{\frac{-3}{8} \cdot \left(a \cdot c\right)}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{{b}^{3}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

   5. Simplified 85.9%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.375, c \cdot \frac{a}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-43}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-0.375, c \cdot \frac{a}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e+143)
   (/ b (* a -1.5))
   (if (<= b 8.6e-43)
     (* (- b (sqrt (fma b b (* (* a c) -3.0)))) (/ -0.3333333333333333 a))
     (* c (fma -0.375 (* c (/ a (* b (* b b)))) (/ -0.5 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+143) {
		tmp = b / (a * -1.5);
	} else if (b <= 8.6e-43) {
		tmp = (b - sqrt(fma(b, b, ((a * c) * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = c * fma(-0.375, (c * (a / (b * (b * b)))), (-0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e+143)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 8.6e-43)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(Float64(a * c) * -3.0)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(c * fma(-0.375, Float64(c * Float64(a / Float64(b * Float64(b * b)))), Float64(-0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e+143], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-43], N[(N[(b - N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.375 * N[(c * N[(a / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5999999999999999e143

   1. Initial program 39.0%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.1

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

   5. Simplified 93.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.2

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

   7. Applied egg-rr 93.2%

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

      1. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-2}{3}}}} \]

      2. un-div-inv N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      6. metadata-eval 93.4

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

   9. Applied egg-rr 93.4%

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

   if -3.5999999999999999e143 < b < 8.59999999999999927e-43

   1. Initial program 83.8%

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

   4. Applied egg-rr 83.8%

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

   if 8.59999999999999927e-43 < b

   1. Initial program 12.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto c \cdot \frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      9. associate-*r/ N/A

         \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}}} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto c \cdot \left(\frac{\color{blue}{\frac{-3}{8} \cdot \left(a \cdot c\right)}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{{b}^{3}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

   5. Simplified 85.9%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.375, c \cdot \frac{a}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-43}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-0.375, c \cdot \frac{a}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ b (* a -1.5))
   (if (<= b 1.15e-61)
     (/ (- (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 <= -1e+153) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.15e-61) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+153)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.15d-61) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+153) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.15e-61) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+153:
		tmp = b / (a * -1.5)
	elif b <= 1.15e-61:
		tmp = (math.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 <= -1e+153)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.15e-61)
		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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+153)
		tmp = b / (a * -1.5);
	elseif (b <= 1.15e-61)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+153], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.15e-61], 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 < -1e153

   1. Initial program 31.7%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 92.4

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

   5. Simplified 92.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 92.4

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

   7. Applied egg-rr 92.4%

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

      1. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-2}{3}}}} \]

      2. un-div-inv N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      6. metadata-eval 92.6

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

   9. Applied egg-rr 92.6%

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

   if -1e153 < b < 1.14999999999999996e-61

   1. Initial program 84.7%

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

   if 1.14999999999999996e-61 < b

   1. Initial program 13.6%

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

   4. Applied egg-rr 13.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 85.2

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

   7. Simplified 85.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-61}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e+143)
   (/ b (* a -1.5))
   (if (<= b 1.15e-61)
     (* (- b (sqrt (fma b b (* (* a c) -3.0)))) (/ -0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+143) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.15e-61) {
		tmp = (b - sqrt(fma(b, b, ((a * c) * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e+143)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.15e-61)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(Float64(a * c) * -3.0)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e+143], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.15e-61], N[(N[(b - N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -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 < -3.5999999999999999e143

   1. Initial program 39.0%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.1

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

   5. Simplified 93.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.2

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

   7. Applied egg-rr 93.2%

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

      1. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-2}{3}}}} \]

      2. un-div-inv N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      6. metadata-eval 93.4

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

   9. Applied egg-rr 93.4%

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

   if -3.5999999999999999e143 < b < 1.14999999999999996e-61

   1. Initial program 84.3%

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

   4. Applied egg-rr 84.3%

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

   if 1.14999999999999996e-61 < b

   1. Initial program 13.6%

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

   4. Applied egg-rr 13.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 85.2

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

   7. Simplified 85.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-61}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-272) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 78.3%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 55.4

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

   5. Simplified 55.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 55.5

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

   7. Applied egg-rr 55.5%

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

      1. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-2}{3}}}} \]

      2. un-div-inv N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      6. metadata-eval 55.6

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

   9. Applied egg-rr 55.6%

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

   if 5.0999999999999998e-272 < b

   1. Initial program 24.6%

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

   4. Applied egg-rr 24.6%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 72.0

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

   7. Simplified 72.0%

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

Alternative 6: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (/ b (* a -1.5)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = b / (a * -1.5);
	} 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 <= 5.1d-272) then
        tmp = b / (a * (-1.5d0))
    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 <= 5.1e-272) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(b / Float64(a * -1.5));
	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 <= 5.1e-272)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 78.3%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 55.4

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

   5. Simplified 55.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 55.5

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

   7. Applied egg-rr 55.5%

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

      1. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-2}{3}}}} \]

      2. un-div-inv N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      6. metadata-eval 55.6

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

   9. Applied egg-rr 55.6%

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

   if 5.0999999999999998e-272 < b

   1. Initial program 24.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 72.0

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

   5. Simplified 72.0%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-272) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 78.3%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 55.4

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

   5. Simplified 55.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 55.5

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

   7. Applied egg-rr 55.5%

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

   if 5.0999999999999998e-272 < b

   1. Initial program 24.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 72.0

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

   5. Simplified 72.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-272) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 78.3%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 55.4

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

   5. Simplified 55.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 55.5

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

   7. Applied egg-rr 55.5%

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

   if 5.0999999999999998e-272 < b

   1. Initial program 24.6%

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

   4. Applied egg-rr 20.1%

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

   5. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{2}{3} + \frac{-1}{2} \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{a} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{2}{3} + \frac{-1}{2} \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{a} \]

      2. +-commutative N/A

         \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b}^{2}} + \frac{2}{3}\right)}}{a} \]

      3. *-commutative N/A

         \[\leadsto \frac{b \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{2}} \cdot \frac{-1}{2}} + \frac{2}{3}\right)}{a} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{b \cdot \color{blue}{\mathsf{fma}\left(\frac{a \cdot c}{{b}^{2}}, \frac{-1}{2}, \frac{2}{3}\right)}}{a} \]

      5. associate-/l* N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2}, \frac{2}{3}\right)}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2}, \frac{2}{3}\right)}{a} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(a \cdot \color{blue}{\frac{c}{{b}^{2}}}, \frac{-1}{2}, \frac{2}{3}\right)}{a} \]

      8. unpow2 N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(a \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2}, \frac{2}{3}\right)}{a} \]

      9. lower-*.f64 3.2

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

   7. Simplified 3.2%

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

   8. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(\frac{2}{3} \cdot \frac{b}{a \cdot c} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\frac{2}{3} \cdot \frac{b}{a \cdot c} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{b}{a \cdot c} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{2}{3}, \frac{b}{a \cdot c}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{2}{3}, \color{blue}{\frac{b}{a \cdot c}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{b}{\color{blue}{a \cdot c}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{b}{a \cdot c}, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{b}{a \cdot c}, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{b}\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{b}{a \cdot c}, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b}}\right) \]

      9. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{b}{a \cdot c}, \frac{\color{blue}{\frac{-1}{2}}}{b}\right) \]

      10. lower-/.f64 3.3

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

   10. Simplified 3.3%

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

   11. Taylor expanded in b around 0

       \[\leadsto c \cdot \color{blue}{\frac{\frac{-1}{2}}{b}} \]
   12. Step-by-step derivation

       1. lower-/.f64 71.8

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

   13. Simplified 71.8%

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

Alternative 9: 36.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{-0.6666666666666666}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* b (/ -0.6666666666666666 a)))

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

public static double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / a);
}

Python

def code(a, b, c):
	return b * (-0.6666666666666666 / a)

Julia

function code(a, b, c)
	return Float64(b * Float64(-0.6666666666666666 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b * (-0.6666666666666666 / a);
end

Wolfram

code[a_, b_, c_] := N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in b around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 29.1

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

5. Simplified 29.1%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 29.1

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

7. Applied egg-rr 29.1%

   \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))