Cubic critical, narrow range

Percentage Accurate: 55.6% → 90.4%

Time: 15.7s

Alternatives: 10

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.6% 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: 90.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{-1}{b}\right)}^{-3}\\ \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{a \cdot \frac{t_0 \cdot \frac{-1}{a}}{{b}^{3}} + \mathsf{fma}\left(t_0, \frac{-1}{a}, -1\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (/ -1.0 b) -3.0)))
   (fma
    -0.5625
    (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
    (fma
     -0.16666666666666666
     (/
      (+
       (pow (* -1.125 (* (* a a) (* c c))) 2.0)
       (* 5.0625 (* (pow c 4.0) (pow a 4.0))))
      (* a (pow b 7.0)))
     (fma
      -0.5
      (/ c b)
      (*
       -0.375
       (/
        (* c c)
        (+
         (* a (/ (* t_0 (/ -1.0 a)) (pow b 3.0)))
         (fma t_0 (/ -1.0 a) -1.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((-1.0 / b), -3.0);
	return fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.16666666666666666, ((pow((-1.125 * ((a * a) * (c * c))), 2.0) + (5.0625 * (pow(c, 4.0) * pow(a, 4.0)))) / (a * pow(b, 7.0))), fma(-0.5, (c / b), (-0.375 * ((c * c) / ((a * ((t_0 * (-1.0 / a)) / pow(b, 3.0))) + fma(t_0, (-1.0 / a), -1.0)))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(-1.0 / b) ^ -3.0
	return fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.16666666666666666, Float64(Float64((Float64(-1.125 * Float64(Float64(a * a) * Float64(c * c))) ^ 2.0) + Float64(5.0625 * Float64((c ^ 4.0) * (a ^ 4.0)))) / Float64(a * (b ^ 7.0))), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(c * c) / Float64(Float64(a * Float64(Float64(t_0 * Float64(-1.0 / a)) / (b ^ 3.0))) + fma(t_0, Float64(-1.0 / a), -1.0)))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(-1.0 / b), $MachinePrecision], -3.0], $MachinePrecision]}, N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(-1.125 * N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(5.0625 * N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[(a * N[(N[(t$95$0 * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(-1.0 / a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.5%

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

   1. neg-sub0 54.5%

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

   2. associate-+l- 54.5%

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

   3. sub0-neg 54.5%

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

   4. neg-mul-1 54.5%

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

   5. associate-*r/ 54.5%

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

   6. *-commutative 54.5%

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

   7. metadata-eval 54.5%

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

   8. metadata-eval 54.5%

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

   9. times-frac 54.5%

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

   10. *-commutative 54.5%

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

   11. times-frac 54.5%

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

3. Simplified 54.7%

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

4. Taylor expanded in b around inf 90.9%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]
5. Step-by-step derivation

   1. fma-def 90.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   2. associate-/l* 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

   3. unpow2 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

   4. fma-def 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)}\right) \]

6. Simplified 90.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{b}^{3}}{a}\right)\right)}}\right)\right)\right) \]

   2. expm1-udef 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{b}^{3}}{a}\right)} - 1}}\right)\right)\right) \]

8. Applied egg-rr 90.9%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{b}^{3}}{a}\right)} - 1}}\right)\right)\right) \]

9. Taylor expanded in b around -inf 0.0%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\color{blue}{\left(e^{-3 \cdot \log \left(\frac{-1}{b}\right) + \log \left(-\frac{1}{a}\right)} + \frac{e^{-3 \cdot \log \left(\frac{-1}{b}\right) + \log \left(-\frac{1}{a}\right)} \cdot a}{{b}^{3}}\right) - 1}}\right)\right)\right) \]
10. Step-by-step derivation

    1. +-commutative 0.0%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\color{blue}{\left(\frac{e^{-3 \cdot \log \left(\frac{-1}{b}\right) + \log \left(-\frac{1}{a}\right)} \cdot a}{{b}^{3}} + e^{-3 \cdot \log \left(\frac{-1}{b}\right) + \log \left(-\frac{1}{a}\right)}\right)} - 1}\right)\right)\right) \]

    2. associate--l+ 0.0%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\color{blue}{\frac{e^{-3 \cdot \log \left(\frac{-1}{b}\right) + \log \left(-\frac{1}{a}\right)} \cdot a}{{b}^{3}} + \left(e^{-3 \cdot \log \left(\frac{-1}{b}\right) + \log \left(-\frac{1}{a}\right)} - 1\right)}}\right)\right)\right) \]

11. Simplified 90.9%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\color{blue}{\frac{{\left(\frac{-1}{b}\right)}^{-3} \cdot \frac{-1}{a}}{{b}^{3}} \cdot a + \mathsf{fma}\left({\left(\frac{-1}{b}\right)}^{-3}, \frac{-1}{a}, -1\right)}}\right)\right)\right) \]

12. Final simplification 90.9%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{a \cdot \frac{{\left(\frac{-1}{b}\right)}^{-3} \cdot \frac{-1}{a}}{{b}^{3}} + \mathsf{fma}\left({\left(\frac{-1}{b}\right)}^{-3}, \frac{-1}{a}, -1\right)}\right)\right)\right) \]

Alternative 2: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\\ \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right) + t_0 \cdot t_0}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -1.125 (* c (* c (* a a))))))
   (fma
    -0.5625
    (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
    (fma
     -0.16666666666666666
     (/
      (+ (* 5.0625 (* (pow c 4.0) (pow a 4.0))) (* t_0 t_0))
      (* a (pow b 7.0)))
     (fma -0.5 (/ c b) (* -0.375 (/ (* c c) (/ (pow b 3.0) a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -1.125 * (c * (c * (a * a)));
	return fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.16666666666666666, (((5.0625 * (pow(c, 4.0) * pow(a, 4.0))) + (t_0 * t_0)) / (a * pow(b, 7.0))), fma(-0.5, (c / b), (-0.375 * ((c * c) / (pow(b, 3.0) / a))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(-1.125 * Float64(c * Float64(c * Float64(a * a))))
	return fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.16666666666666666, Float64(Float64(Float64(5.0625 * Float64((c ^ 4.0) * (a ^ 4.0))) + Float64(t_0 * t_0)) / Float64(a * (b ^ 7.0))), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(c * c) / Float64((b ^ 3.0) / a))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.125 * N[(c * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(5.0625 * N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.5%

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

   1. neg-sub0 54.5%

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

   2. associate-+l- 54.5%

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

   3. sub0-neg 54.5%

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

   4. neg-mul-1 54.5%

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

   5. associate-*r/ 54.5%

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

   6. *-commutative 54.5%

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

   7. metadata-eval 54.5%

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

   8. metadata-eval 54.5%

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

   9. times-frac 54.5%

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

   10. *-commutative 54.5%

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

   11. times-frac 54.5%

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

3. Simplified 54.7%

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

4. Taylor expanded in b around inf 90.9%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]
5. Step-by-step derivation

   1. fma-def 90.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   2. associate-/l* 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

   3. unpow2 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

   4. fma-def 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)}\right) \]

6. Simplified 90.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right)} \]
7. Step-by-step derivation

   1. unpow2 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

   2. associate-*l* 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(-1.125 \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}\right) \cdot \left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

   3. associate-*l* 90.9%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(-1.125 \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}\right) + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

8. Applied egg-rr 90.9%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{\left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right)} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

9. Final simplification 90.9%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right) + \left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

Alternative 3: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.06:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, -0.5 \cdot \frac{c}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.06)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.3333333333333333 a))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma -0.375 (/ (* c c) (/ (pow b 3.0) a)) (* -0.5 (/ c b)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -3.0)));
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.06) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.3333333333333333 / a);
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.375, ((c * c) / (pow(b, 3.0) / a)), (-0.5 * (c / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.06)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.3333333333333333 / a));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.375, Float64(Float64(c * c) / Float64((b ^ 3.0) / a)), Float64(-0.5 * Float64(c / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.06], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.059999999999999998

   1. Initial program 81.3%

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

      1. neg-sub0 81.3%

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

      2. associate-+l- 81.3%

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

      3. sub0-neg 81.3%

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

      4. neg-mul-1 81.3%

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

      5. associate-*r/ 81.3%

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

      6. *-commutative 81.3%

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

      7. metadata-eval 81.3%

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

      8. metadata-eval 81.3%

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

      9. times-frac 81.3%

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

      10. *-commutative 81.3%

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

      11. times-frac 81.3%

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

   3. Simplified 81.5%

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

      1. add-cbrt-cube 81.4%

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

      2. pow3 81.4%

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

   5. Applied egg-rr 81.4%

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

      1. flip-- 81.1%

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

      2. add-sqr-sqrt 81.8%

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

      3. rem-cbrt-cube 81.9%

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

      4. associate-*r* 81.9%

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

      5. rem-cbrt-cube 81.9%

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

      6. associate-*r* 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. associate-*r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-*r* 81.9%

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

   9. Simplified 81.9%

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

   if -0.059999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 47.7%

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

      1. neg-sub0 47.7%

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

      2. associate-+l- 47.7%

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

      3. sub0-neg 47.7%

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

      4. neg-mul-1 47.7%

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

      5. associate-*r/ 47.7%

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

      6. *-commutative 47.7%

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

      7. metadata-eval 47.7%

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

      8. metadata-eval 47.7%

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

      9. times-frac 47.7%

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

      10. *-commutative 47.7%

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

      11. times-frac 47.7%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in b around inf 92.3%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
   5. Step-by-step derivation

      1. fma-def 92.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      2. associate-/l* 92.3%

         \[\leadsto \mathsf{fma}\left(-0.5625, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) \]

      3. unpow2 92.3%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) \]

      4. +-commutative 92.3%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.5 \cdot \frac{c}{b}}\right) \]

      5. fma-def 92.3%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{\mathsf{fma}\left(-0.375, \frac{{c}^{2} \cdot a}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)}\right) \]

      6. associate-/l* 92.3%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}}, -0.5 \cdot \frac{c}{b}\right)\right) \]

      7. unpow2 92.3%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}}, -0.5 \cdot \frac{c}{b}\right)\right) \]

   6. Simplified 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, -0.5 \cdot \frac{c}{b}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, -0.5 \cdot \frac{c}{b}\right)\right)\\ \end{array} \]

Alternative 4: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.006)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.3333333333333333 a))
     (+ (* -0.5 (/ c b)) (* -0.375 (* a (/ (* c c) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -3.0)));
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.006) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.3333333333333333 / a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.006)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.006], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.0060000000000000001

   1. Initial program 79.3%

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

      1. neg-sub0 79.3%

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

      2. associate-+l- 79.3%

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

      3. sub0-neg 79.3%

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

      4. neg-mul-1 79.3%

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

      5. associate-*r/ 79.3%

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

      6. *-commutative 79.3%

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

      7. metadata-eval 79.3%

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

      8. metadata-eval 79.3%

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

      9. times-frac 79.3%

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

      10. *-commutative 79.3%

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

      11. times-frac 79.3%

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

   3. Simplified 79.4%

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

      1. add-cbrt-cube 79.3%

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

      2. pow3 79.3%

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

   5. Applied egg-rr 79.3%

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

      1. flip-- 79.1%

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

      2. add-sqr-sqrt 80.1%

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

      3. rem-cbrt-cube 80.2%

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

      4. associate-*r* 80.2%

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

      5. rem-cbrt-cube 80.2%

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

      6. associate-*r* 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. associate-*r* 80.2%

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

      2. +-commutative 80.2%

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

      3. associate-*r* 80.2%

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

   9. Simplified 80.2%

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

   if -0.0060000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 45.0%

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

      1. neg-sub0 45.0%

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

      2. associate-+l- 45.0%

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

      3. sub0-neg 45.0%

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

      4. neg-mul-1 45.0%

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

      5. associate-*r/ 45.0%

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

      6. *-commutative 45.0%

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

      7. metadata-eval 45.0%

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

      8. metadata-eval 45.0%

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

      9. times-frac 45.0%

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

      10. *-commutative 45.0%

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

      11. times-frac 45.0%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. +-commutative 88.9%

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

      2. fma-def 88.9%

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

      3. associate-/l* 88.9%

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

      4. unpow2 88.9%

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

   6. Simplified 88.9%

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

      1. fma-udef 88.9%

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

      2. associate-/r/ 88.9%

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

   8. Applied egg-rr 88.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{1}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.006)
   (* (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ 1.0 (/ a 0.3333333333333333)))
   (+ (* -0.5 (/ c b)) (* -0.375 (* a (/ (* c c) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.006) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * (1.0 / (a / 0.3333333333333333));
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.006)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * Float64(1.0 / Float64(a / 0.3333333333333333)));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.006], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.0060000000000000001

   1. Initial program 79.3%

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

      1. neg-sub0 79.3%

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

      2. associate-+l- 79.3%

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

      3. sub0-neg 79.3%

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

      4. neg-mul-1 79.3%

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

      5. associate-*r/ 79.3%

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

      6. *-commutative 79.3%

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

      7. metadata-eval 79.3%

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

      8. metadata-eval 79.3%

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

      9. times-frac 79.3%

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

      10. *-commutative 79.3%

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

      11. times-frac 79.3%

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

   3. Simplified 79.4%

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

      1. clear-num 79.4%

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

      2. inv-pow 79.4%

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

   5. Applied egg-rr 79.4%

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

      1. unpow-1 79.4%

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

   7. Simplified 79.4%

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

   if -0.0060000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 45.0%

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

      1. neg-sub0 45.0%

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

      2. associate-+l- 45.0%

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

      3. sub0-neg 45.0%

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

      4. neg-mul-1 45.0%

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

      5. associate-*r/ 45.0%

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

      6. *-commutative 45.0%

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

      7. metadata-eval 45.0%

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

      8. metadata-eval 45.0%

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

      9. times-frac 45.0%

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

      10. *-commutative 45.0%

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

      11. times-frac 45.0%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. +-commutative 88.9%

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

      2. fma-def 88.9%

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

      3. associate-/l* 88.9%

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

      4. unpow2 88.9%

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

   6. Simplified 88.9%

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

      1. fma-udef 88.9%

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

      2. associate-/r/ 88.9%

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

   8. Applied egg-rr 88.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{1}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 6: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.006)
   (* -0.3333333333333333 (/ (- b (sqrt (fma b b (* a (* c -3.0))))) a))
   (+ (* -0.5 (/ c b)) (* -0.375 (* a (/ (* c c) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.006) {
		tmp = -0.3333333333333333 * ((b - sqrt(fma(b, b, (a * (c * -3.0))))) / a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.006)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.006], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.0060000000000000001

   1. Initial program 79.3%

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

      1. /-rgt-identity 79.3%

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

      2. metadata-eval 79.3%

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

      3. associate-/l* 79.3%

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

      4. associate-*r/ 79.3%

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

      5. *-commutative 79.3%

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

      6. associate-*l/ 79.3%

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

      7. associate-*r/ 79.3%

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

      8. metadata-eval 79.3%

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

      9. metadata-eval 79.3%

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

      10. times-frac 79.3%

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

      11. neg-mul-1 79.3%

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

      12. distribute-rgt-neg-in 79.3%

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

      13. times-frac 79.2%

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

      14. metadata-eval 79.2%

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

      15. neg-mul-1 79.2%

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

   3. Simplified 79.3%

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

   if -0.0060000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 45.0%

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

      1. neg-sub0 45.0%

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

      2. associate-+l- 45.0%

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

      3. sub0-neg 45.0%

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

      4. neg-mul-1 45.0%

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

      5. associate-*r/ 45.0%

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

      6. *-commutative 45.0%

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

      7. metadata-eval 45.0%

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

      8. metadata-eval 45.0%

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

      9. times-frac 45.0%

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

      10. *-commutative 45.0%

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

      11. times-frac 45.0%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. +-commutative 88.9%

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

      2. fma-def 88.9%

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

      3. associate-/l* 88.9%

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

      4. unpow2 88.9%

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

   6. Simplified 88.9%

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

      1. fma-udef 88.9%

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

      2. associate-/r/ 88.9%

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

   8. Applied egg-rr 88.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 7: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.006)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma b b (* a (* c -3.0)))) b))
   (+ (* -0.5 (/ c b)) (* -0.375 (* a (/ (* c c) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.006) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma(b, b, (a * (c * -3.0)))) - b);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.006)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.006], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.0060000000000000001

   1. Initial program 79.3%

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

      1. neg-sub0 79.3%

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

      2. associate-+l- 79.3%

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

      3. sub0-neg 79.3%

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

      4. neg-mul-1 79.3%

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

      5. associate-*r/ 79.3%

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

      6. *-commutative 79.3%

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

      7. metadata-eval 79.3%

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

      8. metadata-eval 79.3%

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

      9. times-frac 79.3%

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

      10. *-commutative 79.3%

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

      11. times-frac 79.3%

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

   3. Simplified 79.4%

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

   if -0.0060000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 45.0%

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

      1. neg-sub0 45.0%

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

      2. associate-+l- 45.0%

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

      3. sub0-neg 45.0%

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

      4. neg-mul-1 45.0%

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

      5. associate-*r/ 45.0%

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

      6. *-commutative 45.0%

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

      7. metadata-eval 45.0%

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

      8. metadata-eval 45.0%

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

      9. times-frac 45.0%

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

      10. *-commutative 45.0%

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

      11. times-frac 45.0%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. +-commutative 88.9%

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

      2. fma-def 88.9%

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

      3. associate-/l* 88.9%

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

      4. unpow2 88.9%

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

   6. Simplified 88.9%

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

      1. fma-udef 88.9%

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

      2. associate-/r/ 88.9%

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

   8. Applied egg-rr 88.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 8: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.006)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* 3.0 a))
   (+ (* -0.5 (/ c b)) (* -0.375 (* a (/ (* c c) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.006) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / pow(b, 3.0))));
	}
	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 (((sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)) <= (-0.006d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (3.0d0 * a)
    else
        tmp = ((-0.5d0) * (c / b)) + ((-0.375d0) * (a * ((c * c) / (b ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.006) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / Math.pow(b, 3.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.006:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a)
	else:
		tmp = (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / math.pow(b, 3.0))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.006)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.006)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	else
		tmp = (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / (b ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.006], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.0060000000000000001

   1. Initial program 79.3%

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

      1. neg-sub0 79.3%

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

      2. associate-+l- 79.3%

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

      3. sub0-neg 79.3%

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

      4. neg-mul-1 79.3%

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

      5. associate-*r/ 79.3%

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

      6. metadata-eval 79.3%

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

      7. metadata-eval 79.3%

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

      8. times-frac 79.3%

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

      9. *-commutative 79.3%

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

      10. times-frac 79.3%

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

      11. associate-*l/ 79.3%

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

   3. Simplified 79.3%

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

   if -0.0060000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 45.0%

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

      1. neg-sub0 45.0%

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

      2. associate-+l- 45.0%

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

      3. sub0-neg 45.0%

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

      4. neg-mul-1 45.0%

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

      5. associate-*r/ 45.0%

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

      6. *-commutative 45.0%

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

      7. metadata-eval 45.0%

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

      8. metadata-eval 45.0%

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

      9. times-frac 45.0%

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

      10. *-commutative 45.0%

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

      11. times-frac 45.0%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. +-commutative 88.9%

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

      2. fma-def 88.9%

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

      3. associate-/l* 88.9%

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

      4. unpow2 88.9%

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

   6. Simplified 88.9%

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

      1. fma-udef 88.9%

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

      2. associate-/r/ 88.9%

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

   8. Applied egg-rr 88.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.006:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 9: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (* a (/ (* c c) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / pow(b, 3.0))));
}

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)) + ((-0.375d0) * (a * ((c * c) / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * (a * ((c * c) / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. neg-sub0 54.5%

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

   2. associate-+l- 54.5%

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

   3. sub0-neg 54.5%

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

   4. neg-mul-1 54.5%

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

   5. associate-*r/ 54.5%

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

   6. *-commutative 54.5%

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

   7. metadata-eval 54.5%

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

   8. metadata-eval 54.5%

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

   9. times-frac 54.5%

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

   10. *-commutative 54.5%

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

   11. times-frac 54.5%

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

3. Simplified 54.7%

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

4. Taylor expanded in b around inf 81.7%

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

   1. +-commutative 81.7%

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

   2. fma-def 81.7%

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

   3. associate-/l* 81.7%

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

   4. unpow2 81.7%

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

6. Simplified 81.7%

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

   1. fma-udef 81.7%

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

   2. associate-/r/ 81.7%

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

8. Applied egg-rr 81.7%

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

9. Final simplification 81.7%

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

Alternative 10: 64.1% 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 54.5%

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

   1. neg-sub0 54.5%

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

   2. associate-+l- 54.5%

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

   3. sub0-neg 54.5%

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

   4. neg-mul-1 54.5%

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

   5. associate-*r/ 54.5%

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

   6. *-commutative 54.5%

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

   7. metadata-eval 54.5%

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

   8. metadata-eval 54.5%

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

   9. times-frac 54.5%

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

   10. *-commutative 54.5%

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

   11. times-frac 54.5%

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

3. Simplified 54.7%

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

4. Taylor expanded in b around inf 64.8%

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

5. Final simplification 64.8%

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

Reproduce

herbie shell --seed 2023256 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))