Cubic critical, narrow range

Percentage Accurate: 55.2% → 90.8%

Time: 16.6s

Alternatives: 11

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: 55.2% 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: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot -1.0546875\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (/ (* (pow c 3.0) (* a a)) (pow b 5.0))
  (fma
   -0.5
   (/ c b)
   (+
    (* -0.375 (* (/ a (pow b 3.0)) (* c c)))
    (* (/ (pow (* c a) 4.0) (* a (pow b 7.0))) -1.0546875)))))

C

double code(double a, double b, double c) {
	return fma(-0.5625, ((pow(c, 3.0) * (a * a)) / pow(b, 5.0)), fma(-0.5, (c / b), ((-0.375 * ((a / pow(b, 3.0)) * (c * c))) + ((pow((c * a), 4.0) / (a * pow(b, 7.0))) * -1.0546875))));
}

Julia

function code(a, b, c)
	return fma(-0.5625, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(Float64(a / (b ^ 3.0)) * Float64(c * c))) + Float64(Float64((Float64(c * a) ^ 4.0) / Float64(a * (b ^ 7.0))) * -1.0546875))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

2. Taylor expanded in b around inf 91.6%

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

   1. fma-def 91.6%

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

   2. *-commutative 91.6%

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

   3. unpow2 91.6%

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

   4. fma-def 91.6%

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

   5. fma-def 91.6%

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

4. Simplified 91.6%

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

5. Taylor expanded in c around 0 91.6%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.16666666666666666 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right)\right) \]
6. Step-by-step derivation

   1. distribute-rgt-out 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}\right)\right)\right) \]

   2. associate-*r* 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right)\right) \]

   3. *-commutative 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

   4. times-frac 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{1.265625 + 5.0625}{{b}^{7}}\right)}\right)\right)\right) \]

7. Simplified 91.6%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.16666666666666666 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. fma-udef 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)}\right)\right) \]

   2. associate-/r/ 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \color{blue}{\left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right)} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right) \]

   3. frac-times 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + -0.16666666666666666 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}}\right)\right) \]

9. Applied egg-rr 91.6%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + -0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}}\right)\right) \]
10. Step-by-step derivation

    1. associate-*r/ 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{7}}}\right)\right) \]

11. Applied egg-rr 91.6%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{7}}}\right)\right) \]
12. Step-by-step derivation

    1. associate-*r* 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{\color{blue}{\left(-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}\right) \cdot 6.328125}}{a \cdot {b}^{7}}\right)\right) \]

    2. associate-/l* 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \color{blue}{\frac{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}}\right)\right) \]

    3. associate-*r/ 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \color{blue}{-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}}\right)\right) \]

    4. *-commutative 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}} \cdot -0.16666666666666666}\right)\right) \]

    5. associate-/r/ 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot 6.328125\right)} \cdot -0.16666666666666666\right)\right) \]

    6. associate-*l* 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot \left(6.328125 \cdot -0.16666666666666666\right)}\right)\right) \]

    7. metadata-eval 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot \color{blue}{-1.0546875}\right)\right) \]

13. Simplified 91.6%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot -1.0546875}\right)\right) \]

14. Final simplification 91.6%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot -1.0546875\right)\right) \]

Alternative 2: 88.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, {c}^{3} \cdot \frac{a \cdot a}{{b}^{5}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{-0.5}{\frac{b}{c}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (fma
    -0.5625
    (* (pow c 3.0) (/ (* a a) (pow b 5.0)))
    (fma -0.375 (/ a (/ (pow b 3.0) (* c c))) (/ -0.5 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) * ((a * a) / pow(b, 5.0))), fma(-0.375, (a / (pow(b, 3.0) / (c * c))), (-0.5 / (b / c))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) * Float64(Float64(a * a) / (b ^ 5.0))), fma(-0.375, Float64(a / Float64((b ^ 3.0) / Float64(c * c))), Float64(-0.5 / Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14

   1. Initial program 81.2%

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

      1. neg-sub0 81.2%

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

      2. sqr-neg 81.2%

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

      3. associate-+l- 81.2%

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

      4. sub0-neg 81.2%

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

   3. Simplified 81.3%

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

   if 14 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 93.0%

      \[\leadsto \frac{\color{blue}{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. fma-def 93.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]

      2. cube-prod 93.0%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{\color{blue}{{\left(a \cdot c\right)}^{3}}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]

      3. fma-def 93.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \color{blue}{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}\right)}{3 \cdot a} \]

      4. associate-/l* 93.2%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \color{blue}{\frac{a}{\frac{b}{c}}}, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)\right)}{3 \cdot a} \]

      5. associate-/l* 93.2%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \color{blue}{\frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}\right)\right)}{3 \cdot a} \]

      6. unpow2 93.2%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{\color{blue}{a \cdot a}}{\frac{{b}^{3}}{{c}^{2}}}\right)\right)}{3 \cdot a} \]

      7. unpow2 93.2%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{\color{blue}{c \cdot c}}}\right)\right)}{3 \cdot a} \]

   4. Simplified 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)}}{3 \cdot a} \]

   5. Taylor expanded in a around 0 93.5%

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

      1. fma-def 93.5%

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

      2. associate-/l* 93.5%

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

      3. associate-/r/ 93.5%

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

      4. unpow2 93.5%

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

      5. +-commutative 93.5%

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

      6. fma-def 93.5%

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

      7. associate-/l* 93.5%

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

      8. unpow2 93.5%

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

      9. associate-*r/ 93.5%

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

      10. associate-/l* 93.3%

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

   7. Simplified 93.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, {c}^{3} \cdot \frac{a \cdot a}{{b}^{5}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{-0.5}{\frac{b}{c}}\right)\right)\\ \end{array} \]

Alternative 3: 88.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (fma
    -0.5625
    (/ (* (pow c 3.0) (* a a)) (pow b 5.0))
    (fma -0.5 (/ c b) (* -0.375 (/ a (/ (pow b 3.0) (* c c))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.5625, ((pow(c, 3.0) * (a * a)) / pow(b, 5.0)), fma(-0.5, (c / b), (-0.375 * (a / (pow(b, 3.0) / (c * c))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.5625, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(a / Float64((b ^ 3.0) / Float64(c * c))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14

   1. Initial program 81.2%

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

      1. neg-sub0 81.2%

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

      2. sqr-neg 81.2%

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

      3. associate-+l- 81.2%

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

      4. sub0-neg 81.2%

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

   3. Simplified 81.3%

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

   if 14 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 93.5%

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

      1. fma-def 93.5%

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

      2. *-commutative 93.5%

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

      3. unpow2 93.5%

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

      4. fma-def 93.5%

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

      5. associate-/l* 93.5%

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

      6. unpow2 93.5%

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

   4. Simplified 93.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)\\ \end{array} \]

Alternative 4: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.6875, \frac{\left(c \cdot a\right) \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (/
    (fma
     -1.6875
     (/ (* (* c a) (* (* c a) (* c a))) (pow b 5.0))
     (fma -1.5 (/ a (/ b c)) (* -1.125 (/ (* a a) (/ (pow b 3.0) (* c c))))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-1.6875, (((c * a) * ((c * a) * (c * a))) / pow(b, 5.0)), fma(-1.5, (a / (b / c)), (-1.125 * ((a * a) / (pow(b, 3.0) / (c * c)))))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-1.6875, Float64(Float64(Float64(c * a) * Float64(Float64(c * a) * Float64(c * a))) / (b ^ 5.0)), fma(-1.5, Float64(a / Float64(b / c)), Float64(-1.125 * Float64(Float64(a * a) / Float64((b ^ 3.0) / Float64(c * c)))))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-1.6875 * N[(N[(N[(c * a), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14

   1. Initial program 81.2%

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

      1. neg-sub0 81.2%

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

      2. sqr-neg 81.2%

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

      3. associate-+l- 81.2%

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

      4. sub0-neg 81.2%

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

   3. Simplified 81.3%

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

   if 14 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 93.0%

      \[\leadsto \frac{\color{blue}{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. fma-def 93.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]

      2. cube-prod 93.0%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{\color{blue}{{\left(a \cdot c\right)}^{3}}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]

      3. fma-def 93.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \color{blue}{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}\right)}{3 \cdot a} \]

      4. associate-/l* 93.2%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \color{blue}{\frac{a}{\frac{b}{c}}}, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)\right)}{3 \cdot a} \]

      5. associate-/l* 93.2%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \color{blue}{\frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}\right)\right)}{3 \cdot a} \]

      6. unpow2 93.2%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{\color{blue}{a \cdot a}}{\frac{{b}^{3}}{{c}^{2}}}\right)\right)}{3 \cdot a} \]

      7. unpow2 93.2%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{\color{blue}{c \cdot c}}}\right)\right)}{3 \cdot a} \]

   4. Simplified 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. unpow3 93.2%

         \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)}{3 \cdot a} \]

   6. Applied egg-rr 93.2%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.6875, \frac{\left(c \cdot a\right) \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)}{3 \cdot a}\\ \end{array} \]

Alternative 5: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.00035:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))))
   (if (<= t_0 -0.00035) t_0 (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.00035) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    if (t_0 <= (-0.00035d0)) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.00035) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -0.00035:
		tmp = t_0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -0.00035)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.00035)
		tmp = t_0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -0.00035], t$95$0, N[(-0.5 * N[(c / b), $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)) < -3.49999999999999996e-4

   1. Initial program 74.0%

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

   if -3.49999999999999996e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 41.2%

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

   2. Taylor expanded in b around inf 77.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.00035:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (fma -0.5 (/ c b) (/ (* (* c c) (* a -0.375)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (((c * c) * (a * -0.375)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * Float64(a * -0.375)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14

   1. Initial program 81.2%

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

      1. neg-sub0 81.2%

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

      2. sqr-neg 81.2%

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

      3. associate-+l- 81.2%

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

      4. sub0-neg 81.2%

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

   3. Simplified 81.3%

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

   if 14 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 89.2%

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

      1. fma-def 89.2%

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

      2. associate-*r/ 89.2%

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

      3. associate-*r* 89.2%

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

      4. unpow2 89.2%

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

   4. Simplified 89.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 7: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14.5:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{c \cdot a}{b} + -1.125 \cdot \frac{{\left(a \cdot \frac{c}{b}\right)}^{2}}{b}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.5)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma b b (* a (* c -3.0)))) b))
   (/
    (+ (* -1.5 (/ (* c a) b)) (* -1.125 (/ (pow (* a (/ c b)) 2.0) b)))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.5) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma(b, b, (a * (c * -3.0)))) - b);
	} else {
		tmp = ((-1.5 * ((c * a) / b)) + (-1.125 * (pow((a * (c / b)), 2.0) / b))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.5)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b));
	else
		tmp = Float64(Float64(Float64(-1.5 * Float64(Float64(c * a) / b)) + Float64(-1.125 * Float64((Float64(a * Float64(c / b)) ^ 2.0) / b))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.5], 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[(N[(-1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[Power[N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14.5

   1. Initial program 81.2%

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

      1. neg-sub0 81.2%

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

      2. sqr-neg 81.2%

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

      3. associate-+l- 81.2%

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

      4. sub0-neg 81.2%

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

      5. neg-mul-1 81.2%

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

   3. Simplified 81.2%

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

      1. expm1-log1p-u 81.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{0.3333333333333333}\right)\right)}} \]

   5. Applied egg-rr 81.3%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{0.3333333333333333}\right)\right)}} \]
   6. Step-by-step derivation

      1. div-sub 80.8%

         \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{0.3333333333333333}\right)\right)} - \frac{b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{0.3333333333333333}\right)\right)}} \]

      2. expm1-log1p-u 80.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{\color{blue}{\frac{a}{0.3333333333333333}}} - \frac{b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{0.3333333333333333}\right)\right)} \]

      3. div-inv 79.8%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{\color{blue}{a \cdot \frac{1}{0.3333333333333333}}} - \frac{b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{0.3333333333333333}\right)\right)} \]

      4. metadata-eval 79.8%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a \cdot \color{blue}{3}} - \frac{b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{0.3333333333333333}\right)\right)} \]

      5. expm1-log1p-u 80.1%

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

      6. div-inv 80.7%

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

      7. metadata-eval 80.7%

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

   7. Applied egg-rr 80.7%

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

      1. div-sub 81.3%

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

      2. *-lft-identity 81.3%

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

      3. associate-*l/ 81.3%

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

      4. *-commutative 81.3%

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

      5. associate-/r* 81.3%

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

      6. metadata-eval 81.3%

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

   9. Simplified 81.3%

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

   if 14.5 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 88.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. div-inv 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}\right)}}{3 \cdot a} \]

      2. pow-prod-down 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}\right)}{3 \cdot a} \]

   4. Applied egg-rr 88.8%

      \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot \frac{1}{{b}^{3}}\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. associate-*r/ 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{2} \cdot 1}{{b}^{3}}}}{3 \cdot a} \]

      2. *-rgt-identity 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}}{3 \cdot a} \]

      3. unpow3 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{\left(b \cdot b\right) \cdot b}}}{3 \cdot a} \]

      4. unpow2 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{{b}^{2}} \cdot b}}{3 \cdot a} \]

      5. associate-/r* 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{\frac{{\left(a \cdot c\right)}^{2}}{{b}^{2}}}{b}}}{3 \cdot a} \]

      6. unpow2 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{b \cdot b}}}{b}}{3 \cdot a} \]

      7. unpow2 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{b \cdot b}}{b}}{3 \cdot a} \]

      8. times-frac 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\frac{a \cdot c}{b} \cdot \frac{a \cdot c}{b}}}{b}}{3 \cdot a} \]

      9. associate-*l/ 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot \frac{a \cdot c}{b}}{b}}{3 \cdot a} \]

      10. *-commutative 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\left(c \cdot \frac{a}{b}\right)} \cdot \frac{a \cdot c}{b}}{b}}{3 \cdot a} \]

      11. associate-*l/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\left(c \cdot \frac{a}{b}\right) \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}}{b}}{3 \cdot a} \]

      12. *-commutative 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\left(c \cdot \frac{a}{b}\right) \cdot \color{blue}{\left(c \cdot \frac{a}{b}\right)}}{b}}{3 \cdot a} \]

      13. unpow2 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left(c \cdot \frac{a}{b}\right)}^{2}}}{b}}{3 \cdot a} \]

      14. *-commutative 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left(\frac{a}{b} \cdot c\right)}}^{2}}{b}}{3 \cdot a} \]

      15. associate-/r/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left(\frac{a}{\frac{b}{c}}\right)}}^{2}}{b}}{3 \cdot a} \]

      16. *-rgt-identity 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(\frac{\color{blue}{a \cdot 1}}{\frac{b}{c}}\right)}^{2}}{b}}{3 \cdot a} \]

      17. associate-*r/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left(a \cdot \frac{1}{\frac{b}{c}}\right)}}^{2}}{b}}{3 \cdot a} \]

      18. associate-/r/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot \color{blue}{\left(\frac{1}{b} \cdot c\right)}\right)}^{2}}{b}}{3 \cdot a} \]

      19. associate-*l/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot \color{blue}{\frac{1 \cdot c}{b}}\right)}^{2}}{b}}{3 \cdot a} \]

      20. *-lft-identity 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot \frac{\color{blue}{c}}{b}\right)}^{2}}{b}}{3 \cdot a} \]

   6. Simplified 88.8%

      \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{{\left(a \cdot \frac{c}{b}\right)}^{2}}{b}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14.5:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{c \cdot a}{b} + -1.125 \cdot \frac{{\left(a \cdot \frac{c}{b}\right)}^{2}}{b}}{3 \cdot a}\\ \end{array} \]

Alternative 8: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{c \cdot a}{b} + -1.125 \cdot \frac{{\left(a \cdot \frac{c}{b}\right)}^{2}}{b}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.5)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (/
    (+ (* -1.5 (/ (* c a) b)) (* -1.125 (/ (pow (* a (/ c b)) 2.0) b)))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.5) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = ((-1.5 * ((c * a) / b)) + (-1.125 * (pow((a * (c / b)), 2.0) / b))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(-1.5 * Float64(Float64(c * a) / b)) + Float64(-1.125 * Float64((Float64(a * Float64(c / b)) ^ 2.0) / b))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.5], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[Power[N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14.5

   1. Initial program 81.2%

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

      1. neg-sub0 81.2%

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

      2. sqr-neg 81.2%

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

      3. associate-+l- 81.2%

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

      4. sub0-neg 81.2%

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

   3. Simplified 81.3%

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

   if 14.5 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 88.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. div-inv 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}\right)}}{3 \cdot a} \]

      2. pow-prod-down 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}\right)}{3 \cdot a} \]

   4. Applied egg-rr 88.8%

      \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot \frac{1}{{b}^{3}}\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. associate-*r/ 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{2} \cdot 1}{{b}^{3}}}}{3 \cdot a} \]

      2. *-rgt-identity 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}}{3 \cdot a} \]

      3. unpow3 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{\left(b \cdot b\right) \cdot b}}}{3 \cdot a} \]

      4. unpow2 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{{b}^{2}} \cdot b}}{3 \cdot a} \]

      5. associate-/r* 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{\frac{{\left(a \cdot c\right)}^{2}}{{b}^{2}}}{b}}}{3 \cdot a} \]

      6. unpow2 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{b \cdot b}}}{b}}{3 \cdot a} \]

      7. unpow2 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{b \cdot b}}{b}}{3 \cdot a} \]

      8. times-frac 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\frac{a \cdot c}{b} \cdot \frac{a \cdot c}{b}}}{b}}{3 \cdot a} \]

      9. associate-*l/ 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot \frac{a \cdot c}{b}}{b}}{3 \cdot a} \]

      10. *-commutative 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\left(c \cdot \frac{a}{b}\right)} \cdot \frac{a \cdot c}{b}}{b}}{3 \cdot a} \]

      11. associate-*l/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\left(c \cdot \frac{a}{b}\right) \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}}{b}}{3 \cdot a} \]

      12. *-commutative 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\left(c \cdot \frac{a}{b}\right) \cdot \color{blue}{\left(c \cdot \frac{a}{b}\right)}}{b}}{3 \cdot a} \]

      13. unpow2 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left(c \cdot \frac{a}{b}\right)}^{2}}}{b}}{3 \cdot a} \]

      14. *-commutative 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left(\frac{a}{b} \cdot c\right)}}^{2}}{b}}{3 \cdot a} \]

      15. associate-/r/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left(\frac{a}{\frac{b}{c}}\right)}}^{2}}{b}}{3 \cdot a} \]

      16. *-rgt-identity 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(\frac{\color{blue}{a \cdot 1}}{\frac{b}{c}}\right)}^{2}}{b}}{3 \cdot a} \]

      17. associate-*r/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left(a \cdot \frac{1}{\frac{b}{c}}\right)}}^{2}}{b}}{3 \cdot a} \]

      18. associate-/r/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot \color{blue}{\left(\frac{1}{b} \cdot c\right)}\right)}^{2}}{b}}{3 \cdot a} \]

      19. associate-*l/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot \color{blue}{\frac{1 \cdot c}{b}}\right)}^{2}}{b}}{3 \cdot a} \]

      20. *-lft-identity 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot \frac{\color{blue}{c}}{b}\right)}^{2}}{b}}{3 \cdot a} \]

   6. Simplified 88.8%

      \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{{\left(a \cdot \frac{c}{b}\right)}^{2}}{b}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{c \cdot a}{b} + -1.125 \cdot \frac{{\left(a \cdot \frac{c}{b}\right)}^{2}}{b}}{3 \cdot a}\\ \end{array} \]

Alternative 9: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{c \cdot a}{b} + -1.125 \cdot \frac{{\left(a \cdot \frac{c}{b}\right)}^{2}}{b}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.0)
   (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))
   (/
    (+ (* -1.5 (/ (* c a) b)) (* -1.125 (/ (pow (* a (/ c b)) 2.0) b)))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.0) {
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = ((-1.5 * ((c * a) / b)) + (-1.125 * (pow((a * (c / b)), 2.0) / b))) / (3.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 14.0d0) then
        tmp = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    else
        tmp = (((-1.5d0) * ((c * a) / b)) + ((-1.125d0) * (((a * (c / b)) ** 2.0d0) / b))) / (3.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.0) {
		tmp = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = ((-1.5 * ((c * a) / b)) + (-1.125 * (Math.pow((a * (c / b)), 2.0) / b))) / (3.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 14.0:
		tmp = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	else:
		tmp = ((-1.5 * ((c * a) / b)) + (-1.125 * (math.pow((a * (c / b)), 2.0) / b))) / (3.0 * a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(-1.5 * Float64(Float64(c * a) / b)) + Float64(-1.125 * Float64((Float64(a * Float64(c / b)) ^ 2.0) / b))) / Float64(3.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 14.0)
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	else
		tmp = ((-1.5 * ((c * a) / b)) + (-1.125 * (((a * (c / b)) ^ 2.0) / b))) / (3.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.0], 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], N[(N[(N[(-1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[Power[N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14

   1. Initial program 81.2%

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

   if 14 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 88.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. div-inv 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}\right)}}{3 \cdot a} \]

      2. pow-prod-down 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}\right)}{3 \cdot a} \]

   4. Applied egg-rr 88.8%

      \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot \frac{1}{{b}^{3}}\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. associate-*r/ 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{2} \cdot 1}{{b}^{3}}}}{3 \cdot a} \]

      2. *-rgt-identity 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}}{3 \cdot a} \]

      3. unpow3 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{\left(b \cdot b\right) \cdot b}}}{3 \cdot a} \]

      4. unpow2 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{{b}^{2}} \cdot b}}{3 \cdot a} \]

      5. associate-/r* 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{\frac{{\left(a \cdot c\right)}^{2}}{{b}^{2}}}{b}}}{3 \cdot a} \]

      6. unpow2 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{b \cdot b}}}{b}}{3 \cdot a} \]

      7. unpow2 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{b \cdot b}}{b}}{3 \cdot a} \]

      8. times-frac 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\frac{a \cdot c}{b} \cdot \frac{a \cdot c}{b}}}{b}}{3 \cdot a} \]

      9. associate-*l/ 88.8%

         \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot \frac{a \cdot c}{b}}{b}}{3 \cdot a} \]

      10. *-commutative 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\left(c \cdot \frac{a}{b}\right)} \cdot \frac{a \cdot c}{b}}{b}}{3 \cdot a} \]

      11. associate-*l/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\left(c \cdot \frac{a}{b}\right) \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}}{b}}{3 \cdot a} \]

      12. *-commutative 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\left(c \cdot \frac{a}{b}\right) \cdot \color{blue}{\left(c \cdot \frac{a}{b}\right)}}{b}}{3 \cdot a} \]

      13. unpow2 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left(c \cdot \frac{a}{b}\right)}^{2}}}{b}}{3 \cdot a} \]

      14. *-commutative 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left(\frac{a}{b} \cdot c\right)}}^{2}}{b}}{3 \cdot a} \]

      15. associate-/r/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left(\frac{a}{\frac{b}{c}}\right)}}^{2}}{b}}{3 \cdot a} \]

      16. *-rgt-identity 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(\frac{\color{blue}{a \cdot 1}}{\frac{b}{c}}\right)}^{2}}{b}}{3 \cdot a} \]

      17. associate-*r/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left(a \cdot \frac{1}{\frac{b}{c}}\right)}}^{2}}{b}}{3 \cdot a} \]

      18. associate-/r/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot \color{blue}{\left(\frac{1}{b} \cdot c\right)}\right)}^{2}}{b}}{3 \cdot a} \]

      19. associate-*l/ 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot \color{blue}{\frac{1 \cdot c}{b}}\right)}^{2}}{b}}{3 \cdot a} \]

      20. *-lft-identity 88.8%

          \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot \frac{\color{blue}{c}}{b}\right)}^{2}}{b}}{3 \cdot a} \]

   6. Simplified 88.8%

      \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{{\left(a \cdot \frac{c}{b}\right)}^{2}}{b}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{c \cdot a}{b} + -1.125 \cdot \frac{{\left(a \cdot \frac{c}{b}\right)}^{2}}{b}}{3 \cdot a}\\ \end{array} \]

Alternative 10: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 440:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 440.0)
   (/ (- (sqrt (- (* b b) (* c (/ a 0.3333333333333333)))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 440.0) {
		tmp = (sqrt(((b * b) - (c * (a / 0.3333333333333333)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 440.0d0) then
        tmp = (sqrt(((b * b) - (c * (a / 0.3333333333333333d0)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 440.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a / 0.3333333333333333)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 440.0:
		tmp = (math.sqrt(((b * b) - (c * (a / 0.3333333333333333)))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 440.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a / 0.3333333333333333)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 440.0)
		tmp = (sqrt(((b * b) - (c * (a / 0.3333333333333333)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 440.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 440

   1. Initial program 75.3%

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

      1. add-log-exp 67.7%

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

      2. exp-prod 55.6%

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

      3. *-commutative 55.6%

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

      4. metadata-eval 55.6%

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

      5. div-inv 55.6%

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

   3. Applied egg-rr 55.6%

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

      1. log-pow 60.5%

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

      2. rem-log-exp 75.3%

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

   5. Simplified 75.3%

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

   if 440 < b

   1. Initial program 44.3%

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

   2. Taylor expanded in b around inf 74.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 440:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

2. Taylor expanded in b around inf 64.8%

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

3. Final simplification 64.8%

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

Reproduce

herbie shell --seed 2023279 
(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)))