Cubic critical, wide range

Percentage Accurate: 17.8% → 97.7%

Time: 10.9s

Alternatives: 7

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: 17.8% 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

\[\begin{array}{l} \\ \mathsf{fma}\left(-1.125, \left(c \cdot \frac{c}{{b}^{3}}\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{7}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -1.125
  (* (* c (/ c (pow b 3.0))) (* a 0.3333333333333333))
  (fma
   -0.5
   (/ c b)
   (fma
    -1.6875
    (* (/ (pow c 3.0) (pow b 5.0)) (* a (* a 0.3333333333333333)))
    (/
     (* -0.16666666666666666 (* (pow (* c a) 4.0) 6.328125))
     (* a (pow b 7.0)))))))

C

double code(double a, double b, double c) {
	return fma(-1.125, ((c * (c / pow(b, 3.0))) * (a * 0.3333333333333333)), fma(-0.5, (c / b), fma(-1.6875, ((pow(c, 3.0) / pow(b, 5.0)) * (a * (a * 0.3333333333333333))), ((-0.16666666666666666 * (pow((c * a), 4.0) * 6.328125)) / (a * pow(b, 7.0))))));
}

Julia

function code(a, b, c)
	return fma(-1.125, Float64(Float64(c * Float64(c / (b ^ 3.0))) * Float64(a * 0.3333333333333333)), fma(-0.5, Float64(c / b), fma(-1.6875, Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * Float64(a * 0.3333333333333333))), Float64(Float64(-0.16666666666666666 * Float64((Float64(c * a) ^ 4.0) * 6.328125)) / Float64(a * (b ^ 7.0))))))
end

Wolfram

code[a_, b_, c_] := N[(-1.125 * N[(N[(c * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-1.6875 * N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * N[(a * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.4%

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

   1. neg-sub0 17.4%

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

   2. associate-+l- 17.4%

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

   3. sub0-neg 17.4%

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

   4. neg-mul-1 17.4%

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

   5. associate-*r/ 17.4%

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

   6. *-commutative 17.4%

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

   7. metadata-eval 17.4%

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

   8. metadata-eval 17.4%

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

   9. times-frac 17.4%

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

   10. *-commutative 17.4%

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

   11. times-frac 17.4%

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

3. Simplified 17.4%

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

   1. add-sqr-sqrt 17.4%

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

   2. pow2 17.4%

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

5. Applied egg-rr 17.4%

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

6. Taylor expanded in b around inf 97.6%

   \[\leadsto \color{blue}{-1.125 \cdot \frac{{c}^{2} \cdot \left(a \cdot {\left(\sqrt{0.3333333333333333}\right)}^{2}\right)}{{b}^{3}} + \left(-0.5 \cdot \frac{{\left(\sqrt{0.3333333333333333}\right)}^{2} \cdot \left({\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}} + \left(-1.5 \cdot \frac{c \cdot {\left(\sqrt{0.3333333333333333}\right)}^{2}}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot \left({a}^{2} \cdot {\left(\sqrt{0.3333333333333333}\right)}^{2}\right)}{{b}^{5}}\right)\right)} \]

8. Simplified 98.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \mathsf{fma}\left(5.0625, {\left(c \cdot a\right)}^{4}, {\left(c \cdot a\right)}^{4} \cdot 1.265625\right)}{a \cdot {b}^{7}}\right)\right)\right)} \]

9. Taylor expanded in c around 0 98.2%

   \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{\color{blue}{-0.16666666666666666 \cdot \left({c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)\right)}}{a \cdot {b}^{7}}\right)\right)\right) \]
10. Step-by-step derivation

    1. +-commutative 98.2%

       \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left({c}^{4} \cdot \color{blue}{\left(5.0625 \cdot {a}^{4} + 1.265625 \cdot {a}^{4}\right)}\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    2. distribute-rgt-out 98.2%

       \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left({c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(5.0625 + 1.265625\right)\right)}\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    3. associate-*r* 98.2%

       \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \color{blue}{\left(\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(5.0625 + 1.265625\right)\right)}}{a \cdot {b}^{7}}\right)\right)\right) \]

    4. metadata-eval 98.2%

       \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left({c}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {a}^{4}\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    5. pow-sqr 98.2%

       \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\color{blue}{\left({c}^{2} \cdot {c}^{2}\right)} \cdot {a}^{4}\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    6. metadata-eval 98.2%

       \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\left({c}^{2} \cdot {c}^{2}\right) \cdot {a}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    7. pow-sqr 98.2%

       \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\left({c}^{2} \cdot {c}^{2}\right) \cdot \color{blue}{\left({a}^{2} \cdot {a}^{2}\right)}\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    8. swap-sqr 98.2%

       \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\color{blue}{\left(\left({c}^{2} \cdot {a}^{2}\right) \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)} \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    9. unpow2 98.2%

       \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}\right) \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    10. unpow2 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\left(\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    11. swap-sqr 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)} \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    12. unpow2 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}\right)\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    13. unpow2 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    14. swap-sqr 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    15. unpow2 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left(\color{blue}{{\left(c \cdot a\right)}^{2}} \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    16. unpow2 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\left({\left(c \cdot a\right)}^{2} \cdot \color{blue}{{\left(c \cdot a\right)}^{2}}\right) \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    17. pow-sqr 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left(\color{blue}{{\left(c \cdot a\right)}^{\left(2 \cdot 2\right)}} \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    18. metadata-eval 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{\color{blue}{4}} \cdot \left(5.0625 + 1.265625\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    19. metadata-eval 98.2%

        \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{4} \cdot \color{blue}{6.328125}\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

11. Simplified 98.2%

    \[\leadsto \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{\color{blue}{-0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 6.328125\right)}}{a \cdot {b}^{7}}\right)\right)\right) \]

12. Final simplification 98.2%

    \[\leadsto \mathsf{fma}\left(-1.125, \left(c \cdot \frac{c}{{b}^{3}}\right) \cdot \left(a \cdot 0.3333333333333333\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot \left(a \cdot 0.3333333333333333\right)\right), \frac{-0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

Alternative 2: 97.7% accurate, 0.1× speedup

Math

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

FPCore

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

Julia

function code(a, b, c)
	return fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.16666666666666666, Float64(Float64((Float64(c * a) ^ 4.0) / a) * Float64(6.328125 / (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_] := N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / 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 17.4%

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

   1. /-rgt-identity 17.4%

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

   2. metadata-eval 17.4%

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

   3. associate-/l* 17.4%

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

   4. associate-*r/ 17.4%

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

   5. *-commutative 17.4%

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

   6. associate-*l/ 17.4%

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

   7. associate-*r/ 17.4%

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

   8. metadata-eval 17.4%

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

   9. metadata-eval 17.4%

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

   10. times-frac 17.4%

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

   11. neg-mul-1 17.4%

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

   12. distribute-rgt-neg-in 17.4%

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

   13. times-frac 17.4%

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

   14. metadata-eval 17.4%

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

   15. neg-mul-1 17.4%

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

3. Simplified 17.4%

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

4. Taylor expanded in b around inf 98.2%

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

   1. fma-def 98.2%

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

   2. associate-/l* 98.2%

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

   3. unpow2 98.2%

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

   4. fma-def 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in c around 0 98.2%

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

   1. distribute-rgt-out 98.2%

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

   2. associate-*r* 98.2%

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

   3. times-frac 98.2%

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

9. Simplified 98.2%

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

10. Final simplification 98.2%

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

Alternative 3: 96.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

   1. /-rgt-identity 17.4%

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

   2. metadata-eval 17.4%

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

   3. associate-/l* 17.4%

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

   4. associate-*r/ 17.4%

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

   5. *-commutative 17.4%

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

   6. associate-*l/ 17.4%

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

   7. associate-*r/ 17.4%

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

   8. metadata-eval 17.4%

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

   9. metadata-eval 17.4%

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

   10. times-frac 17.4%

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

   11. neg-mul-1 17.4%

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

   12. distribute-rgt-neg-in 17.4%

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

   13. times-frac 17.4%

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

   14. metadata-eval 17.4%

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

   15. neg-mul-1 17.4%

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

3. Simplified 17.4%

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

4. Taylor expanded in b around inf 96.8%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(1.5 \cdot \frac{c \cdot a}{b} + 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}}{a} \]
5. Step-by-step derivation

   1. fma-def 96.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, 1.5 \cdot \frac{c \cdot a}{b} + 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}}{a} \]

   2. associate-/l* 96.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}}, 1.5 \cdot \frac{c \cdot a}{b} + 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}{a} \]

   3. fma-def 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \color{blue}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}\right)}{a} \]

   4. associate-*r/ 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)}{{b}^{3}}}\right)\right)}{a} \]

   5. associate-*r* 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\color{blue}{\left(1.125 \cdot {c}^{2}\right) \cdot {a}^{2}}}{{b}^{3}}\right)\right)}{a} \]

   6. unpow2 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot {a}^{2}}{{b}^{3}}\right)\right)}{a} \]

   7. unpow2 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{b}^{3}}\right)\right)}{a} \]

6. Simplified 96.9%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)\right)}}{a} \]
7. Step-by-step derivation

   1. add-log-exp 21.9%

      \[\leadsto \color{blue}{\log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)\right)}{a}}\right)} \]

   2. associate-/r/ 21.9%

      \[\leadsto \log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \color{blue}{\frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)\right)}{a}}\right) \]

   3. associate-/l* 21.8%

      \[\leadsto \log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \color{blue}{\frac{c}{\frac{b}{a}}}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)\right)}{a}}\right) \]

   4. associate-*l* 21.8%

      \[\leadsto \log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, \frac{\color{blue}{1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)}}{{b}^{3}}\right)\right)}{a}}\right) \]

8. Applied egg-rr 21.8%

   \[\leadsto \color{blue}{\log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, \frac{1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)}{{b}^{3}}\right)\right)}{a}}\right)} \]

9. Taylor expanded in c around 0 97.6%

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

    1. fma-def 97.6%

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

    2. associate-/l* 97.6%

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

    3. associate-/r/ 97.6%

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

    4. unpow2 97.6%

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

    5. associate-*r/ 97.6%

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

    6. *-commutative 97.6%

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

    7. associate-*r/ 97.3%

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

    8. fma-def 97.3%

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

    9. associate-*l/ 97.3%

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

    10. unpow2 97.3%

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

    11. associate-*r/ 97.3%

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

    12. *-commutative 97.3%

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

    13. associate-*r* 97.3%

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

    14. *-commutative 97.3%

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

    15. associate-*l* 97.3%

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 4: 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 17.4%

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

   1. /-rgt-identity 17.4%

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

   2. metadata-eval 17.4%

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

   3. associate-/l* 17.4%

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

   4. associate-*r/ 17.4%

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

   5. *-commutative 17.4%

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

   6. associate-*l/ 17.4%

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

   7. associate-*r/ 17.4%

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

   8. metadata-eval 17.4%

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

   9. metadata-eval 17.4%

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

   10. times-frac 17.4%

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

   11. neg-mul-1 17.4%

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

   12. distribute-rgt-neg-in 17.4%

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

   13. times-frac 17.4%

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

   14. metadata-eval 17.4%

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

   15. neg-mul-1 17.4%

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

3. Simplified 17.4%

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

4. Taylor expanded in b around inf 97.6%

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

   1. fma-def 97.6%

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

   2. associate-/l* 97.6%

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

   3. unpow2 97.6%

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

   4. +-commutative 97.6%

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

   5. fma-def 97.6%

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

   6. associate-/l* 97.6%

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

   7. unpow2 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Alternative 5: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{a} \cdot \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, \frac{c}{\frac{b}{a}} \cdot \left(\frac{a \cdot \left(c \cdot 1.125\right)}{b \cdot b} + 1.5\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ -0.3333333333333333 a)
  (fma
   1.6875
   (/ (pow (* c a) 3.0) (pow b 5.0))
   (* (/ c (/ b a)) (+ (/ (* a (* c 1.125)) (* b b)) 1.5)))))

C

double code(double a, double b, double c) {
	return (-0.3333333333333333 / a) * fma(1.6875, (pow((c * a), 3.0) / pow(b, 5.0)), ((c / (b / a)) * (((a * (c * 1.125)) / (b * b)) + 1.5)));
}

Julia

function code(a, b, c)
	return Float64(Float64(-0.3333333333333333 / a) * fma(1.6875, Float64((Float64(c * a) ^ 3.0) / (b ^ 5.0)), Float64(Float64(c / Float64(b / a)) * Float64(Float64(Float64(a * Float64(c * 1.125)) / Float64(b * b)) + 1.5))))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(1.6875 * N[(N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * N[(c * 1.125), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.4%

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

   1. /-rgt-identity 17.4%

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

   2. metadata-eval 17.4%

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

   3. associate-/l* 17.4%

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

   4. associate-*r/ 17.4%

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

   5. *-commutative 17.4%

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

   6. associate-*l/ 17.4%

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

   7. associate-*r/ 17.4%

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

   8. metadata-eval 17.4%

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

   9. metadata-eval 17.4%

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

   10. times-frac 17.4%

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

   11. neg-mul-1 17.4%

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

   12. distribute-rgt-neg-in 17.4%

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

   13. times-frac 17.4%

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

   14. metadata-eval 17.4%

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

   15. neg-mul-1 17.4%

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

3. Simplified 17.4%

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

4. Taylor expanded in b around inf 96.8%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(1.5 \cdot \frac{c \cdot a}{b} + 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}}{a} \]
5. Step-by-step derivation

   1. fma-def 96.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, 1.5 \cdot \frac{c \cdot a}{b} + 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}}{a} \]

   2. associate-/l* 96.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}}, 1.5 \cdot \frac{c \cdot a}{b} + 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}{a} \]

   3. fma-def 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \color{blue}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}\right)}{a} \]

   4. associate-*r/ 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)}{{b}^{3}}}\right)\right)}{a} \]

   5. associate-*r* 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\color{blue}{\left(1.125 \cdot {c}^{2}\right) \cdot {a}^{2}}}{{b}^{3}}\right)\right)}{a} \]

   6. unpow2 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot {a}^{2}}{{b}^{3}}\right)\right)}{a} \]

   7. unpow2 96.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{b}^{3}}\right)\right)}{a} \]

6. Simplified 96.9%

   \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)\right)}}{a} \]
7. Step-by-step derivation

   1. add-log-exp 21.9%

      \[\leadsto \color{blue}{\log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{3}}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)\right)}{a}}\right)} \]

   2. associate-/r/ 21.9%

      \[\leadsto \log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \color{blue}{\frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}}, \mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)\right)}{a}}\right) \]

   3. associate-/l* 21.8%

      \[\leadsto \log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \color{blue}{\frac{c}{\frac{b}{a}}}, \frac{\left(1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)\right)}{a}}\right) \]

   4. associate-*l* 21.8%

      \[\leadsto \log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, \frac{\color{blue}{1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)}}{{b}^{3}}\right)\right)}{a}}\right) \]

8. Applied egg-rr 21.8%

   \[\leadsto \color{blue}{\log \left(e^{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, \frac{1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)}{{b}^{3}}\right)\right)}{a}}\right)} \]
9. Step-by-step derivation

   1. add-log-exp 96.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, \frac{1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)}{{b}^{3}}\right)\right)}{a}} \]

   2. associate-/r/ 97.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \color{blue}{\frac{c}{b} \cdot a}, \frac{1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)}{{b}^{3}}\right)\right)}{a} \]

   3. associate-/l* 97.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, \color{blue}{\frac{1.125}{\frac{{b}^{3}}{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}}}\right)\right)}{a} \]

   4. unswap-sqr 97.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, \frac{1.125}{\frac{{b}^{3}}{\color{blue}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}}\right)\right)}{a} \]

10. Applied egg-rr 97.0%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, \frac{1.125}{\frac{{b}^{3}}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}\right)\right)}{a}} \]
11. Step-by-step derivation

    1. associate-*r/ 97.2%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, \frac{1.125}{\frac{{b}^{3}}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}\right)\right)}{a}} \]

    2. associate-/l* 97.0%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{a}{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, \frac{1.125}{\frac{{b}^{3}}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}\right)\right)}}} \]

    3. associate-/r/ 97.0%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, \mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, \frac{1.125}{\frac{{b}^{3}}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}\right)\right)} \]

    4. associate-*l/ 97.0%

       \[\leadsto \frac{-0.3333333333333333}{a} \cdot \mathsf{fma}\left(1.6875, \color{blue}{\frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}}, \mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, \frac{1.125}{\frac{{b}^{3}}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}\right)\right) \]

    5. cube-prod 97.0%

       \[\leadsto \frac{-0.3333333333333333}{a} \cdot \mathsf{fma}\left(1.6875, \frac{\color{blue}{{\left(c \cdot a\right)}^{3}}}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, \frac{1.125}{\frac{{b}^{3}}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}\right)\right) \]

    6. fma-udef 96.9%

       \[\leadsto \frac{-0.3333333333333333}{a} \cdot \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, \color{blue}{1.5 \cdot \left(\frac{c}{b} \cdot a\right) + \frac{1.125}{\frac{{b}^{3}}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}}\right) \]

    7. +-commutative 96.9%

       \[\leadsto \frac{-0.3333333333333333}{a} \cdot \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, \color{blue}{\frac{1.125}{\frac{{b}^{3}}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}} + 1.5 \cdot \left(\frac{c}{b} \cdot a\right)}\right) \]

12. Simplified 96.8%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, \frac{c}{\frac{b}{a}} \cdot \left(\frac{\left(1.125 \cdot c\right) \cdot a}{b \cdot b} + 1.5\right)\right)} \]

13. Final simplification 96.8%

    \[\leadsto \frac{-0.3333333333333333}{a} \cdot \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, \frac{c}{\frac{b}{a}} \cdot \left(\frac{a \cdot \left(c \cdot 1.125\right)}{b \cdot b} + 1.5\right)\right) \]

Alternative 6: 95.4% 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 17.4%

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

   1. /-rgt-identity 17.4%

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

   2. metadata-eval 17.4%

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

   3. associate-/l* 17.4%

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

   4. associate-*r/ 17.4%

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

   5. *-commutative 17.4%

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

   6. associate-*l/ 17.4%

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

   7. associate-*r/ 17.4%

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

   8. metadata-eval 17.4%

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

   9. metadata-eval 17.4%

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

   10. times-frac 17.4%

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

   11. neg-mul-1 17.4%

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

   12. distribute-rgt-neg-in 17.4%

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

   13. times-frac 17.4%

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

   14. metadata-eval 17.4%

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

   15. neg-mul-1 17.4%

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

3. Simplified 17.4%

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

4. Taylor expanded in b around inf 96.0%

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

   1. +-commutative 96.0%

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

   2. fma-def 96.0%

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

   3. associate-/l* 96.0%

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

   4. unpow2 96.0%

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

6. Simplified 96.0%

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

   1. fma-udef 96.0%

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

   2. associate-/r/ 96.0%

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

8. Applied egg-rr 96.0%

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

9. Final simplification 96.0%

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

Alternative 7: 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 17.4%

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

   1. /-rgt-identity 17.4%

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

   2. metadata-eval 17.4%

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

   3. associate-/l* 17.4%

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

   4. associate-*r/ 17.4%

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

   5. *-commutative 17.4%

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

   6. associate-*l/ 17.4%

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

   7. associate-*r/ 17.4%

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

   8. metadata-eval 17.4%

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

   9. metadata-eval 17.4%

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

   10. times-frac 17.4%

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

   11. neg-mul-1 17.4%

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

   12. distribute-rgt-neg-in 17.4%

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

   13. times-frac 17.4%

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

   14. metadata-eval 17.4%

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

   15. neg-mul-1 17.4%

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

3. Simplified 17.4%

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

4. Taylor expanded in b around inf 91.3%

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

5. Final simplification 91.3%

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

Reproduce

herbie shell --seed 2023194 
(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)))