Cubic critical, wide range

Percentage Accurate: 17.6% → 97.7%

Time: 10.6s

Alternatives: 6

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.6% 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: 97.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) (* (* a a) -1.125))))
   (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 = (c * c) * ((a * a) * -1.125);
	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(Float64(c * c) * Float64(Float64(a * a) * -1.125))
	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[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -1.125), $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 16.8%

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

   1. /-rgt-identity 16.8%

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

   2. metadata-eval 16.8%

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

   3. associate-/l* 16.8%

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

   4. associate-*r/ 16.8%

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

   5. *-commutative 16.8%

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

   6. associate-*l/ 16.8%

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

   7. associate-*r/ 16.8%

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

   8. metadata-eval 16.8%

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

   9. metadata-eval 16.8%

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

   10. times-frac 16.8%

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

   11. neg-mul-1 16.8%

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

   12. distribute-rgt-neg-in 16.8%

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

   13. times-frac 16.8%

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

   14. metadata-eval 16.8%

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

   15. neg-mul-1 16.8%

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

3. Simplified 16.8%

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

4. Taylor expanded in b around inf 98.6%

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

   1. fma-def 98.6%

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

   2. associate-/l* 98.6%

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

   3. unpow2 98.6%

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

   4. fma-def 98.6%

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

6. Simplified 98.6%

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

   1. pow1 98.6%

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

   2. pow-prod-down 98.6%

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

8. Applied egg-rr 98.6%

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

   1. unpow1 98.6%

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

10. Simplified 98.6%

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

    1. unpow2 98.6%

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

    2. associate-*l* 98.6%

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

    3. associate-*l* 98.6%

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

12. Applied egg-rr 98.6%

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

13. Final simplification 98.6%

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

Alternative 2: 97.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

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[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.8%

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

   1. /-rgt-identity 16.8%

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

   2. metadata-eval 16.8%

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

   3. associate-/l* 16.8%

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

   4. associate-*r/ 16.8%

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

   5. *-commutative 16.8%

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

   6. associate-*l/ 16.8%

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

   7. associate-*r/ 16.8%

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

   8. metadata-eval 16.8%

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

   9. metadata-eval 16.8%

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

   10. times-frac 16.8%

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

   11. neg-mul-1 16.8%

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

   12. distribute-rgt-neg-in 16.8%

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

   13. times-frac 16.8%

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

   14. metadata-eval 16.8%

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

   15. neg-mul-1 16.8%

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

3. Simplified 16.8%

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

4. Taylor expanded in b around inf 97.9%

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

   1. fma-def 97.9%

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

   2. associate-/l* 97.9%

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

   3. unpow2 97.9%

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

   4. +-commutative 97.9%

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

   5. fma-def 97.9%

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

   6. associate-/l* 97.9%

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

   7. unpow2 97.9%

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

6. Simplified 97.9%

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

7. Final simplification 97.9%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * (a * ((c * c) / (b ** 3.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. /-rgt-identity 16.8%

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

   2. metadata-eval 16.8%

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

   3. associate-/l* 16.8%

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

   4. associate-*r/ 16.8%

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

   5. *-commutative 16.8%

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

   6. associate-*l/ 16.8%

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

   7. associate-*r/ 16.8%

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

   8. metadata-eval 16.8%

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

   9. metadata-eval 16.8%

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

   10. times-frac 16.8%

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

   11. neg-mul-1 16.8%

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

   12. distribute-rgt-neg-in 16.8%

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

   13. times-frac 16.8%

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

   14. metadata-eval 16.8%

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

   15. neg-mul-1 16.8%

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

3. Simplified 16.8%

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

4. Taylor expanded in b around inf 96.3%

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

   1. +-commutative 96.3%

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

   2. fma-def 96.3%

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

   3. associate-/l* 96.3%

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

   4. unpow2 96.3%

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

6. Simplified 96.3%

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

   1. fma-udef 96.3%

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

   2. associate-/r/ 96.3%

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

8. Applied egg-rr 96.3%

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

9. Final simplification 96.3%

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

Alternative 4: 94.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{\left(a \cdot \frac{c}{b}\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 1.125}{b \cdot b} + 1.5\right)}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  -0.3333333333333333
  (/ (* (* a (/ c b)) (+ (/ (* (* c a) 1.125) (* b b)) 1.5)) a)))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 * (((a * (c / b)) * ((((c * a) * 1.125) / (b * b)) + 1.5)) / 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 = (-0.3333333333333333d0) * (((a * (c / b)) * ((((c * a) * 1.125d0) / (b * b)) + 1.5d0)) / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.3333333333333333 * (((a * (c / b)) * ((((c * a) * 1.125) / (b * b)) + 1.5)) / a);
}

Python

def code(a, b, c):
	return -0.3333333333333333 * (((a * (c / b)) * ((((c * a) * 1.125) / (b * b)) + 1.5)) / a)

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 * Float64(Float64(Float64(a * Float64(c / b)) * Float64(Float64(Float64(Float64(c * a) * 1.125) / Float64(b * b)) + 1.5)) / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 * (((a * (c / b)) * ((((c * a) * 1.125) / (b * b)) + 1.5)) / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 * N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(c * a), $MachinePrecision] * 1.125), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.8%

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

   1. /-rgt-identity 16.8%

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

   2. metadata-eval 16.8%

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

   3. associate-/l* 16.8%

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

   4. associate-*r/ 16.8%

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

   5. *-commutative 16.8%

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

   6. associate-*l/ 16.8%

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

   7. associate-*r/ 16.8%

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

   8. metadata-eval 16.8%

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

   9. metadata-eval 16.8%

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

   10. times-frac 16.8%

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

   11. neg-mul-1 16.8%

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

   12. distribute-rgt-neg-in 16.8%

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

   13. times-frac 16.8%

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

   14. metadata-eval 16.8%

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

   15. neg-mul-1 16.8%

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

3. Simplified 16.8%

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

4. Taylor expanded in b around inf 95.6%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1.5 \cdot \frac{c \cdot a}{b} + 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}}}{a} \]
5. Step-by-step derivation

   1. fma-def 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}}{a} \]

   2. associate-*r/ 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)}{{b}^{3}}}\right)}{a} \]

   3. associate-*r* 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\color{blue}{\left(1.125 \cdot {c}^{2}\right) \cdot {a}^{2}}}{{b}^{3}}\right)}{a} \]

   4. unpow2 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot {a}^{2}}{{b}^{3}}\right)}{a} \]

   5. unpow2 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{b}^{3}}\right)}{a} \]

6. Simplified 95.7%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)}}{a} \]

7. Taylor expanded in c around 0 95.6%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1.5 \cdot \frac{c \cdot a}{b} + 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}}}{a} \]
8. Step-by-step derivation

   1. +-commutative 95.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + 1.5 \cdot \frac{c \cdot a}{b}}}{a} \]

   2. unpow2 95.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{1.125 \cdot \frac{\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}}{{b}^{3}} + 1.5 \cdot \frac{c \cdot a}{b}}{a} \]

   3. unpow2 95.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{1.125 \cdot \frac{\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{b}^{3}} + 1.5 \cdot \frac{c \cdot a}{b}}{a} \]

   4. swap-sqr 95.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{1.125 \cdot \frac{\color{blue}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}{{b}^{3}} + 1.5 \cdot \frac{c \cdot a}{b}}{a} \]

   5. associate-*r/ 95.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{1.125 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{3}}} + 1.5 \cdot \frac{c \cdot a}{b}}{a} \]

   6. unpow3 95.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{\frac{1.125 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(b \cdot b\right) \cdot b}} + 1.5 \cdot \frac{c \cdot a}{b}}{a} \]

   7. associate-*r* 95.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\left(1.125 \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}}{\left(b \cdot b\right) \cdot b} + 1.5 \cdot \frac{c \cdot a}{b}}{a} \]

   8. times-frac 95.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} \cdot \frac{c \cdot a}{b}} + 1.5 \cdot \frac{c \cdot a}{b}}{a} \]

   9. distribute-rgt-out 95.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{c \cdot a}{b} \cdot \left(\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} + 1.5\right)}}{a} \]

   10. associate-/l* 95.6%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{c}{\frac{b}{a}}} \cdot \left(\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} + 1.5\right)}{a} \]

   11. associate-/r/ 95.7%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\left(\frac{c}{b} \cdot a\right)} \cdot \left(\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} + 1.5\right)}{a} \]

9. Simplified 95.7%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\left(\frac{c}{b} \cdot a\right) \cdot \left(\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} + 1.5\right)}}{a} \]

10. Final simplification 95.7%

    \[\leadsto -0.3333333333333333 \cdot \frac{\left(a \cdot \frac{c}{b}\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 1.125}{b \cdot b} + 1.5\right)}{a} \]

Alternative 5: 94.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{a} \cdot \left(\left(a \cdot \frac{c}{b}\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 1.125}{b \cdot b} + 1.5\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ -0.3333333333333333 a)
  (* (* a (/ c b)) (+ (/ (* (* c a) 1.125) (* b b)) 1.5))))

C

double code(double a, double b, double c) {
	return (-0.3333333333333333 / a) * ((a * (c / b)) * ((((c * a) * 1.125) / (b * b)) + 1.5));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.3333333333333333d0) / a) * ((a * (c / b)) * ((((c * a) * 1.125d0) / (b * b)) + 1.5d0))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.3333333333333333 / a) * ((a * (c / b)) * ((((c * a) * 1.125) / (b * b)) + 1.5));
}

Python

def code(a, b, c):
	return (-0.3333333333333333 / a) * ((a * (c / b)) * ((((c * a) * 1.125) / (b * b)) + 1.5))

Julia

function code(a, b, c)
	return Float64(Float64(-0.3333333333333333 / a) * Float64(Float64(a * Float64(c / b)) * Float64(Float64(Float64(Float64(c * a) * 1.125) / Float64(b * b)) + 1.5)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.3333333333333333 / a) * ((a * (c / b)) * ((((c * a) * 1.125) / (b * b)) + 1.5));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(c * a), $MachinePrecision] * 1.125), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.8%

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

   1. /-rgt-identity 16.8%

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

   2. metadata-eval 16.8%

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

   3. associate-/l* 16.8%

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

   4. associate-*r/ 16.8%

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

   5. *-commutative 16.8%

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

   6. associate-*l/ 16.8%

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

   7. associate-*r/ 16.8%

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

   8. metadata-eval 16.8%

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

   9. metadata-eval 16.8%

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

   10. times-frac 16.8%

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

   11. neg-mul-1 16.8%

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

   12. distribute-rgt-neg-in 16.8%

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

   13. times-frac 16.8%

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

   14. metadata-eval 16.8%

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

   15. neg-mul-1 16.8%

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

3. Simplified 16.8%

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

4. Taylor expanded in b around inf 95.6%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1.5 \cdot \frac{c \cdot a}{b} + 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}}}{a} \]
5. Step-by-step derivation

   1. fma-def 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}}{a} \]

   2. associate-*r/ 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)}{{b}^{3}}}\right)}{a} \]

   3. associate-*r* 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\color{blue}{\left(1.125 \cdot {c}^{2}\right) \cdot {a}^{2}}}{{b}^{3}}\right)}{a} \]

   4. unpow2 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot {a}^{2}}{{b}^{3}}\right)}{a} \]

   5. unpow2 95.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{b}^{3}}\right)}{a} \]

6. Simplified 95.7%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)}}{a} \]
7. Step-by-step derivation

   1. associate-*r/ 95.8%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)}{a}} \]

   2. associate-*l* 95.8%

      \[\leadsto \frac{-0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\color{blue}{1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)}}{{b}^{3}}\right)}{a} \]

   3. unswap-sqr 95.8%

      \[\leadsto \frac{-0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{1.125 \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}}{{b}^{3}}\right)}{a} \]

8. Applied egg-rr 95.8%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{1.125 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{3}}\right)}{a}} \]
9. Step-by-step derivation

   1. associate-/l* 95.8%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{a}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{1.125 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{3}}\right)}}} \]

   2. associate-/r/ 95.6%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{1.125 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{3}}\right)} \]

   3. fma-def 95.6%

      \[\leadsto \frac{-0.3333333333333333}{a} \cdot \color{blue}{\left(1.5 \cdot \frac{c \cdot a}{b} + \frac{1.125 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{3}}\right)} \]

   4. +-commutative 95.6%

      \[\leadsto \frac{-0.3333333333333333}{a} \cdot \color{blue}{\left(\frac{1.125 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{3}} + 1.5 \cdot \frac{c \cdot a}{b}\right)} \]

   5. associate-*r* 95.6%

      \[\leadsto \frac{-0.3333333333333333}{a} \cdot \left(\frac{\color{blue}{\left(1.125 \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}}{{b}^{3}} + 1.5 \cdot \frac{c \cdot a}{b}\right) \]

   6. unpow3 95.6%

      \[\leadsto \frac{-0.3333333333333333}{a} \cdot \left(\frac{\left(1.125 \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{\color{blue}{\left(b \cdot b\right) \cdot b}} + 1.5 \cdot \frac{c \cdot a}{b}\right) \]

   7. times-frac 95.6%

      \[\leadsto \frac{-0.3333333333333333}{a} \cdot \left(\color{blue}{\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} \cdot \frac{c \cdot a}{b}} + 1.5 \cdot \frac{c \cdot a}{b}\right) \]

   8. distribute-rgt-out 95.6%

      \[\leadsto \frac{-0.3333333333333333}{a} \cdot \color{blue}{\left(\frac{c \cdot a}{b} \cdot \left(\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} + 1.5\right)\right)} \]

   9. associate-/l* 95.6%

      \[\leadsto \frac{-0.3333333333333333}{a} \cdot \left(\color{blue}{\frac{c}{\frac{b}{a}}} \cdot \left(\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} + 1.5\right)\right) \]

   10. associate-/r/ 95.7%

       \[\leadsto \frac{-0.3333333333333333}{a} \cdot \left(\color{blue}{\left(\frac{c}{b} \cdot a\right)} \cdot \left(\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} + 1.5\right)\right) \]

10. Simplified 95.7%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(\left(\frac{c}{b} \cdot a\right) \cdot \left(\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot b} + 1.5\right)\right)} \]

11. Final simplification 95.7%

    \[\leadsto \frac{-0.3333333333333333}{a} \cdot \left(\left(a \cdot \frac{c}{b}\right) \cdot \left(\frac{\left(c \cdot a\right) \cdot 1.125}{b \cdot b} + 1.5\right)\right) \]

Alternative 6: 90.5% 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 16.8%

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

   1. /-rgt-identity 16.8%

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

   2. metadata-eval 16.8%

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

   3. associate-/l* 16.8%

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

   4. associate-*r/ 16.8%

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

   5. *-commutative 16.8%

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

   6. associate-*l/ 16.8%

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

   7. associate-*r/ 16.8%

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

   8. metadata-eval 16.8%

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

   9. metadata-eval 16.8%

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

   10. times-frac 16.8%

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

   11. neg-mul-1 16.8%

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

   12. distribute-rgt-neg-in 16.8%

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

   13. times-frac 16.8%

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

   14. metadata-eval 16.8%

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

   15. neg-mul-1 16.8%

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

3. Simplified 16.8%

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

4. Taylor expanded in b around inf 91.2%

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

5. Final simplification 91.2%

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

Reproduce

herbie shell --seed 2023238 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))