Cubic critical, medium range

Percentage Accurate: 31.2% → 95.6%

Time: 14.2s

Alternatives: 9

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.2% 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: 95.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))
  (fma
   -0.5
   (/ c b)
   (fma
    -0.375
    (/ a (/ (pow b 3.0) (* c c)))
    (*
     -0.16666666666666666
     (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	return fma(-0.5625, ((a * a) / (pow(b, 5.0) / pow(c, 3.0))), fma(-0.5, (c / b), fma(-0.375, (a / (pow(b, 3.0) / (c * c))), (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0)))))));
}

Julia

function code(a, b, c)
	return fma(-0.5625, Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))), fma(-0.5, Float64(c / b), fma(-0.375, Float64(a / Float64((b ^ 3.0) / Float64(c * c))), Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $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] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.1%

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

   1. sqr-neg 29.1%

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

   2. sqr-neg 29.1%

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

   3. associate-*l* 29.1%

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

3. Simplified 29.1%

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

4. Taylor expanded in b around inf 95.1%

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

   1. fma-def 95.1%

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

   2. associate-/l* 95.1%

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

   3. unpow2 95.1%

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

   4. fma-def 95.1%

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

   5. fma-def 95.1%

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

6. Simplified 95.1%

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

7. Taylor expanded in c around 0 95.1%

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

   1. distribute-rgt-out 95.1%

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

   2. associate-*r* 95.1%

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

   3. *-commutative 95.1%

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

   4. times-frac 95.1%

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

9. Simplified 95.1%

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

10. Final simplification 95.1%

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

Alternative 2: 95.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5625, a \cdot \left({c}^{3} \cdot \frac{a}{{b}^{5}}\right), \mathsf{fma}\left(-0.375, \frac{a \cdot c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\right) + \frac{-0.5}{3 \cdot {b}^{7}} \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{a}{6.328125}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (fma
   -0.5625
   (* a (* (pow c 3.0) (/ a (pow b 5.0))))
   (fma -0.375 (/ (* a c) (/ (pow b 3.0) c)) (* -0.5 (/ c b))))
  (* (/ -0.5 (* 3.0 (pow b 7.0))) (/ (pow (* a c) 4.0) (/ a 6.328125)))))

C

double code(double a, double b, double c) {
	return fma(-0.5625, (a * (pow(c, 3.0) * (a / pow(b, 5.0)))), fma(-0.375, ((a * c) / (pow(b, 3.0) / c)), (-0.5 * (c / b)))) + ((-0.5 / (3.0 * pow(b, 7.0))) * (pow((a * c), 4.0) / (a / 6.328125)));
}

Julia

function code(a, b, c)
	return Float64(fma(-0.5625, Float64(a * Float64((c ^ 3.0) * Float64(a / (b ^ 5.0)))), fma(-0.375, Float64(Float64(a * c) / Float64((b ^ 3.0) / c)), Float64(-0.5 * Float64(c / b)))) + Float64(Float64(-0.5 / Float64(3.0 * (b ^ 7.0))) * Float64((Float64(a * c) ^ 4.0) / Float64(a / 6.328125))))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[(3.0 * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(a / 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.1%

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

   1. neg-sub0 29.1%

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

   2. sqr-neg 29.1%

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

   3. associate-+l- 29.1%

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

   4. sub0-neg 29.1%

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

   5. neg-mul-1 29.1%

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

3. Simplified 29.2%

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

   1. div-inv 29.2%

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

   2. metadata-eval 29.2%

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

   3. *-commutative 29.2%

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

   4. add-sqr-sqrt 29.2%

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

   5. pow2 29.2%

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

5. Applied egg-rr 29.2%

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

6. Taylor expanded in b around inf 94.0%

   \[\leadsto \color{blue}{-1.6875 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5} \cdot {\left(\sqrt{3}\right)}^{2}} + \left(-1.5 \cdot \frac{c}{b \cdot {\left(\sqrt{3}\right)}^{2}} + \left(-1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3} \cdot {\left(\sqrt{3}\right)}^{2}} + -0.5 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot \left({b}^{7} \cdot {\left(\sqrt{3}\right)}^{2}\right)}\right)\right)} \]

8. Simplified 94.8%

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

9. Taylor expanded in b around 0 95.1%

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

10. Final simplification 95.1%

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

Alternative 3: 93.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. neg-sub0 29.1%

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

   2. sqr-neg 29.1%

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

   3. associate-+l- 29.1%

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

   4. sub0-neg 29.1%

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

   5. neg-mul-1 29.1%

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

3. Simplified 29.2%

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

   1. clear-num 29.2%

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

   2. inv-pow 29.2%

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

5. Applied egg-rr 29.2%

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

6. Taylor expanded in b around inf 93.8%

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

   1. fma-def 93.8%

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

   2. associate-/l* 93.8%

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

   3. unpow2 93.8%

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

   4. associate-*l/ 93.8%

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

   5. *-commutative 93.8%

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

   6. associate-/r/ 93.8%

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

   7. +-commutative 93.8%

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

   8. fma-def 93.8%

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

   9. unpow2 93.8%

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

   10. associate-*r* 93.8%

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

   11. associate-/l* 93.8%

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

   12. associate-*r/ 93.8%

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

   13. associate-/l* 93.5%

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

8. Simplified 93.5%

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

9. Final simplification 93.5%

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

Alternative 4: 94.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $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. Initial program 29.1%

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

   1. sqr-neg 29.1%

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

   2. sqr-neg 29.1%

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

   3. associate-*l* 29.1%

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

3. Simplified 29.1%

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

4. Taylor expanded in b around inf 93.8%

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

   1. fma-def 93.8%

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

   2. associate-/l* 93.8%

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

   3. unpow2 93.8%

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

   4. fma-def 93.8%

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

   5. associate-/l* 93.8%

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

   6. unpow2 93.8%

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

6. Simplified 93.8%

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

7. Final simplification 93.8%

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

Alternative 5: 93.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. sqr-neg 29.1%

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

   2. sqr-neg 29.1%

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

   3. associate-*l* 29.1%

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

3. Simplified 29.1%

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

4. Taylor expanded in b around inf 93.3%

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

   1. fma-def 93.3%

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

   2. cube-prod 93.3%

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

   3. fma-def 93.3%

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

   4. associate-/l* 93.3%

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

   5. associate-/l* 93.3%

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

   6. unpow2 93.3%

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

   7. unpow2 93.3%

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

6. Simplified 93.3%

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

7. Taylor expanded in a around 0 93.8%

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

   1. +-commutative 93.8%

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

   2. associate-*r/ 93.8%

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

   3. associate-*l/ 93.4%

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

   4. associate-+l+ 93.4%

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

   5. *-commutative 93.4%

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

   6. +-commutative 93.4%

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

   7. fma-def 93.5%

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

   8. +-commutative 93.5%

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

   9. fma-def 93.5%

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

9. Simplified 93.5%

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

    1. fma-udef 93.4%

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

    2. associate-*r/ 93.4%

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

    3. associate-/r/ 93.4%

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

    4. *-commutative 93.4%

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

11. Applied egg-rr 93.4%

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

12. Final simplification 93.4%

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

Alternative 6: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -8e-8)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -8e-8) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} 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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-8d-8)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    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 (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -8e-8) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -8e-8:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -8e-8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -8e-8)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -8e-8], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], 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)) < -8.0000000000000002e-8

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. sqr-neg 67.0%

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

      3. associate-*l* 67.0%

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

   3. Simplified 67.0%

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

   if -8.0000000000000002e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 14.6%

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

      1. sqr-neg 14.6%

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

      2. sqr-neg 14.6%

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

      3. associate-*l* 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in b around inf 93.5%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 91.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. sqr-neg 29.1%

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

   2. sqr-neg 29.1%

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

   3. associate-*l* 29.1%

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

3. Simplified 29.1%

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

4. Taylor expanded in b around inf 91.3%

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

   1. +-commutative 91.3%

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

   2. fma-def 91.3%

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

   3. associate-/l* 91.3%

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

   4. associate-/r/ 91.3%

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

   5. unpow2 91.3%

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

6. Simplified 91.3%

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

7. Final simplification 91.3%

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

Alternative 8: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+ (* -1.5 (* c (/ a b))) (* -1.125 (/ (* a a) (/ (pow b 3.0) (* c c)))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return ((-1.5 * (c * (a / b))) + (-1.125 * ((a * a) / (pow(b, 3.0) / (c * c))))) / (a * 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 = (((-1.5d0) * (c * (a / b))) + ((-1.125d0) * ((a * a) / ((b ** 3.0d0) / (c * c))))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((-1.5 * (c * (a / b))) + (-1.125 * ((a * a) / (Math.pow(b, 3.0) / (c * c))))) / (a * 3.0);
}

Python

def code(a, b, c):
	return ((-1.5 * (c * (a / b))) + (-1.125 * ((a * a) / (math.pow(b, 3.0) / (c * c))))) / (a * 3.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-1.5 * Float64(c * Float64(a / b))) + Float64(-1.125 * Float64(Float64(a * a) / Float64((b ^ 3.0) / Float64(c * c))))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-1.5 * (c * (a / b))) + (-1.125 * ((a * a) / ((b ^ 3.0) / (c * c))))) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-1.5 * N[(c * N[(a / b), $MachinePrecision]), $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] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.1%

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

   1. sqr-neg 29.1%

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

   2. sqr-neg 29.1%

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

   3. associate-*l* 29.1%

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

3. Simplified 29.1%

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

4. Taylor expanded in b around inf 90.8%

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

   1. fma-def 90.9%

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

   2. associate-/l* 90.8%

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

   3. associate-/l* 90.8%

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

   4. unpow2 90.8%

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

   5. unpow2 90.8%

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

6. Simplified 90.8%

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

   1. fma-udef 90.8%

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

   2. associate-/r/ 90.8%

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

8. Applied egg-rr 90.8%

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

9. Final simplification 90.8%

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

Alternative 9: 81.5% 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 29.1%

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

   1. sqr-neg 29.1%

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

   2. sqr-neg 29.1%

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

   3. associate-*l* 29.1%

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

3. Simplified 29.1%

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

4. Taylor expanded in b around inf 82.9%

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

5. Final simplification 82.9%

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

Reproduce

herbie shell --seed 2023272 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))