Cubic critical, medium range

Percentage Accurate: 31.6% → 95.5%

Time: 10.3s

Alternatives: 7

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: 31.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

2. Taylor expanded in b around inf 96.9%

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

   1. fma-def 96.9%

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

   2. associate-/l* 96.9%

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

   3. unpow2 96.9%

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

   4. fma-def 96.9%

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

4. Simplified 96.9%

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

   1. pow1 96.9%

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

   2. pow-prod-down 96.9%

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

6. Applied egg-rr 96.9%

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

   1. unpow1 96.9%

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

8. Simplified 96.9%

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

   1. unpow2 96.9%

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

   2. associate-*l* 96.9%

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

   3. associate-*l* 96.9%

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

10. Applied egg-rr 96.9%

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

11. Final simplification 96.9%

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

Alternative 2: 93.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

2. Taylor expanded in b around inf 95.1%

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

   1. fma-def 95.1%

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

   2. associate-/l* 95.1%

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

   3. unpow2 95.1%

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

   4. fma-def 95.1%

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

   5. associate-*r/ 95.1%

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

   6. unpow2 95.1%

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

   7. associate-*l* 95.1%

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

4. Simplified 95.1%

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

5. Final simplification 95.1%

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

Alternative 3: 93.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

   1. /-rgt-identity 32.0%

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

   2. metadata-eval 32.0%

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

   3. associate-/r/ 32.0%

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

   4. metadata-eval 32.0%

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

   5. metadata-eval 32.0%

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

   6. times-frac 32.0%

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

   7. *-commutative 32.0%

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

   8. times-frac 32.0%

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

   9. *-commutative 32.0%

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

   10. associate-/r* 32.0%

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

   11. associate-*l/ 32.0%

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

3. Simplified 32.0%

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

4. Taylor expanded in b around inf 94.9%

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

   1. *-commutative 94.9%

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

   2. fma-def 94.9%

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

   3. associate-/l* 94.9%

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

   4. *-commutative 94.9%

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

   5. fma-def 94.9%

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

   6. associate-/l* 94.9%

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

   7. unpow2 94.9%

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

   8. unpow2 94.9%

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

   9. *-commutative 94.9%

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

   10. cube-prod 94.9%

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

6. Simplified 94.9%

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

   1. unpow3 94.9%

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

   2. unswap-sqr 94.9%

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

   3. associate-*l* 94.9%

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

8. Applied egg-rr 94.9%

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

9. Final simplification 94.9%

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

Alternative 4: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))))
   (if (<= t_0 -2e-7) 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 <= -2e-7) {
		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 <= (-2d-7)) 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 <= -2e-7) {
		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 <= -2e-7:
		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 <= -2e-7)
		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 <= -2e-7)
		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, -2e-7], 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)) < -1.9999999999999999e-7

   1. Initial program 67.5%

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

   if -1.9999999999999999e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 14.7%

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

   2. Taylor expanded in b around inf 93.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 90.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

2. Taylor expanded in b around inf 91.4%

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

   1. fma-def 91.4%

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

   2. associate-*r/ 91.4%

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

   3. unpow2 91.4%

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

   4. associate-*l* 91.4%

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

4. Simplified 91.4%

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

5. Final simplification 91.4%

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

Alternative 6: 81.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

2. Taylor expanded in b around inf 80.9%

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

3. Final simplification 80.9%

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

Alternative 7: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 32.0%

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

   1. add-cube-cbrt 32.0%

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

   2. pow3 32.0%

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

   3. neg-mul-1 32.0%

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

   4. fma-def 32.0%

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

   5. associate-*l* 32.0%

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

3. Applied egg-rr 32.0%

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

4. Taylor expanded in c around 0 3.2%

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

   1. pow-base-1 3.2%

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

   2. *-rgt-identity 3.2%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{b + -1 \cdot b}{a}} \]

   3. distribute-rgt1-in 3.2%

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

   4. metadata-eval 3.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

   5. mul0-lft 3.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{a} \]

   6. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 0}{a}} \]

   7. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

6. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

7. Final simplification 3.2%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023224 
(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)))