Cubic critical, medium range

Percentage Accurate: 31.7% → 95.5%

Time: 12.0s

Alternatives: 5

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(c * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.6%

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

   1. /-rgt-identity 28.6%

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

   2. metadata-eval 28.6%

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

   3. associate-/l* 28.6%

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

   4. associate-*r/ 28.6%

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

   5. *-commutative 28.6%

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

   6. associate-*l/ 28.6%

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

   7. associate-*r/ 28.6%

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

   8. metadata-eval 28.6%

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

   9. metadata-eval 28.6%

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

   10. times-frac 28.6%

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

   11. neg-mul-1 28.6%

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

   12. distribute-rgt-neg-in 28.6%

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

   13. times-frac 28.6%

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

   14. metadata-eval 28.6%

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

   15. neg-mul-1 28.6%

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

3. Simplified 28.6%

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

4. Taylor expanded in b around inf 95.2%

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

   1. fma-def 95.2%

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

   2. associate-/l* 95.2%

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

   3. unpow2 95.2%

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

   4. fma-def 95.2%

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

   5. associate-/l* 95.2%

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

   6. unpow2 95.2%

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

   7. fma-def 95.4%

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

6. Simplified 95.4%

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

7. Taylor expanded in c around 0 95.6%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

9. Simplified 95.6%

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

10. Final simplification 95.6%

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

Alternative 2: 93.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

   1. /-rgt-identity 28.6%

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

   2. metadata-eval 28.6%

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

   3. associate-/l* 28.6%

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

   4. associate-*r/ 28.6%

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

   5. *-commutative 28.6%

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

   6. associate-*l/ 28.6%

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

   7. associate-*r/ 28.6%

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

   8. metadata-eval 28.6%

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

   9. metadata-eval 28.6%

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

   10. times-frac 28.6%

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

   11. neg-mul-1 28.6%

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

   12. distribute-rgt-neg-in 28.6%

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

   13. times-frac 28.6%

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

   14. metadata-eval 28.6%

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

   15. neg-mul-1 28.6%

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

3. Simplified 28.6%

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

4. Taylor expanded in b around inf 95.2%

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

   1. fma-def 95.2%

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

   2. associate-/l* 95.2%

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

   3. unpow2 95.2%

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

   4. fma-def 95.2%

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

   5. associate-/l* 95.2%

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

   6. unpow2 95.2%

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

   7. fma-def 95.4%

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

6. Simplified 95.4%

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

7. Taylor expanded in c around 0 94.3%

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

   1. +-commutative 94.3%

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

   2. associate-+l+ 94.3%

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

   3. +-commutative 94.3%

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

   4. fma-def 94.3%

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

   5. +-commutative 94.3%

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

   6. associate-*l/ 94.3%

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

   7. associate-*r* 94.3%

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

   8. associate-*l/ 94.3%

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

   9. unpow2 94.3%

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

   10. associate-*r* 94.3%

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

   11. associate-*r* 94.3%

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

9. Simplified 94.3%

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

10. Final simplification 94.3%

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

Alternative 3: 90.8% accurate, 0.5× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

   1. /-rgt-identity 28.6%

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

   2. metadata-eval 28.6%

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

   3. associate-/l* 28.6%

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

   4. associate-*r/ 28.6%

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

   5. *-commutative 28.6%

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

   6. associate-*l/ 28.6%

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

   7. associate-*r/ 28.6%

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

   8. metadata-eval 28.6%

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

   9. metadata-eval 28.6%

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

   10. times-frac 28.6%

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

   11. neg-mul-1 28.6%

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

   12. distribute-rgt-neg-in 28.6%

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

   13. times-frac 28.6%

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

   14. metadata-eval 28.6%

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

   15. neg-mul-1 28.6%

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

3. Simplified 28.6%

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

4. Taylor expanded in b around inf 91.6%

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

   1. +-commutative 91.6%

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

   2. fma-def 91.6%

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

   3. associate-/l* 91.6%

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

   4. unpow2 91.6%

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

   5. associate-*r/ 91.6%

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

6. Simplified 91.6%

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

7. Final simplification 91.6%

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

Alternative 4: 90.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -0.3333333333333333 * (((1.5 * ((c / b) * a)) + ((1.125 * (a * a)) / (pow(b, 3.0) / (c * c)))) / 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) * (((1.5d0 * ((c / b) * a)) + ((1.125d0 * (a * a)) / ((b ** 3.0d0) / (c * c)))) / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.3333333333333333 * (((1.5 * ((c / b) * a)) + ((1.125 * (a * a)) / (Math.pow(b, 3.0) / (c * c)))) / a);
}

Python

def code(a, b, c):
	return -0.3333333333333333 * (((1.5 * ((c / b) * a)) + ((1.125 * (a * a)) / (math.pow(b, 3.0) / (c * c)))) / a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 * (((1.5 * ((c / b) * a)) + ((1.125 * (a * a)) / ((b ^ 3.0) / (c * c)))) / a);
end

Wolfram

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

Derivation

1. Initial program 28.6%

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

   1. /-rgt-identity 28.6%

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

   2. metadata-eval 28.6%

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

   3. associate-/l* 28.6%

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

   4. associate-*r/ 28.6%

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

   5. *-commutative 28.6%

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

   6. associate-*l/ 28.6%

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

   7. associate-*r/ 28.6%

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

   8. metadata-eval 28.6%

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

   9. metadata-eval 28.6%

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

   10. times-frac 28.6%

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

   11. neg-mul-1 28.6%

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

   12. distribute-rgt-neg-in 28.6%

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

   13. times-frac 28.6%

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

   14. metadata-eval 28.6%

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

   15. neg-mul-1 28.6%

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

3. Simplified 28.6%

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

4. Taylor expanded in b around inf 91.0%

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

   1. fma-def 91.1%

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

   2. associate-*r/ 91.1%

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

   3. *-commutative 91.1%

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

   4. associate-*r* 91.1%

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

   5. unpow2 91.1%

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

   6. unpow2 91.1%

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

6. Simplified 91.1%

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

   1. fma-udef 91.0%

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

   2. associate-/l* 91.0%

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

   3. associate-/r/ 91.1%

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

   4. associate-/l* 91.1%

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

8. Applied egg-rr 91.1%

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

9. Final simplification 91.1%

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

Alternative 5: 81.2% accurate, 23.2× speedup

Math

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

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(Float64(-0.5 * c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

   1. /-rgt-identity 28.6%

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

   2. metadata-eval 28.6%

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

   3. associate-/l* 28.6%

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

   4. associate-*r/ 28.6%

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

   5. *-commutative 28.6%

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

   6. associate-*l/ 28.6%

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

   7. associate-*r/ 28.6%

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

   8. metadata-eval 28.6%

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

   9. metadata-eval 28.6%

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

   10. times-frac 28.6%

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

   11. neg-mul-1 28.6%

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

   12. distribute-rgt-neg-in 28.6%

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

   13. times-frac 28.6%

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

   14. metadata-eval 28.6%

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

   15. neg-mul-1 28.6%

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

3. Simplified 28.6%

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

4. Taylor expanded in b around inf 83.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
5. Step-by-step derivation

   1. associate-*r/ 83.8%

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

6. Simplified 83.8%

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

7. Final simplification 83.8%

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

Reproduce

herbie shell --seed 2023257 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))