Cubic critical, medium range

Percentage Accurate: 31.5% → 95.5%

Time: 15.5s

Alternatives: 12

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: 31.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. /-rgt-identity 29.1%

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

   2. metadata-eval 29.1%

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

   3. associate-/l* 29.1%

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

   4. associate-*r/ 29.1%

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

   5. *-commutative 29.1%

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

   6. associate-*l/ 29.1%

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

   7. associate-*r/ 29.1%

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

   8. metadata-eval 29.1%

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

   9. metadata-eval 29.1%

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

   10. times-frac 29.1%

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

   11. neg-mul-1 29.1%

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

   12. distribute-rgt-neg-in 29.1%

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

   13. times-frac 29.1%

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

   14. metadata-eval 29.1%

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

   15. neg-mul-1 29.1%

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

3. Simplified 29.2%

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

4. Taylor expanded in b around inf 96.6%

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

   1. fma-def 96.6%

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

   2. unpow2 96.6%

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

   3. fma-def 96.6%

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

6. Simplified 96.6%

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

7. Taylor expanded in c around 0 96.6%

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

   1. distribute-rgt-out 96.6%

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

   2. associate-*r* 96.6%

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

   3. times-frac 96.6%

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

9. Simplified 96.6%

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

10. Final simplification 96.6%

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

Alternative 2: 95.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. neg-sub0 29.1%

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

   2. associate-+l- 29.1%

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

   3. sub0-neg 29.1%

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

   4. neg-mul-1 29.1%

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

   5. associate-*r/ 29.1%

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

   6. metadata-eval 29.1%

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

   7. metadata-eval 29.1%

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

   8. times-frac 29.1%

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

   9. *-commutative 29.1%

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

   10. times-frac 29.1%

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

   11. associate-*l/ 29.1%

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

3. Simplified 29.2%

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

   1. fma-udef 29.1%

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

   2. associate-*r* 29.1%

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

   3. *-commutative 29.1%

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

   4. *-commutative 29.1%

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

   5. metadata-eval 29.1%

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

   6. cancel-sign-sub-inv 29.1%

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

   7. add-cbrt-cube 28.9%

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

   8. add-sqr-sqrt 28.9%

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

   9. cancel-sign-sub-inv 28.9%

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

   10. metadata-eval 28.9%

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

   11. *-commutative 28.9%

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

   12. associate-*r* 28.9%

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

   13. fma-udef 29.0%

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

   14. add-sqr-sqrt 29.0%

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

   15. cancel-sign-sub-inv 29.0%

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

   16. metadata-eval 29.0%

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

   17. *-commutative 29.0%

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

   18. associate-*r* 29.0%

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

5. Applied egg-rr 30.4%

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

   1. unpow1/3 29.2%

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

   2. fma-def 29.1%

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

   3. +-commutative 29.1%

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

   4. associate-*r* 29.1%

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

   5. metadata-eval 29.1%

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

   6. distribute-rgt-neg-in 29.1%

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

   7. *-commutative 29.1%

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

   8. associate-*r* 29.1%

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

   9. distribute-rgt-neg-in 29.1%

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

   10. fma-def 29.1%

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

   11. distribute-rgt-neg-in 29.1%

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

   12. metadata-eval 29.1%

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

7. Simplified 29.1%

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

8. Taylor expanded in b around inf 96.6%

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

   1. fma-def 96.6%

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

   2. associate-*l/ 96.6%

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

   3. unpow2 96.6%

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

   4. fma-def 96.6%

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

10. Simplified 96.4%

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

11. Taylor expanded in c around 0 96.4%

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

    1. +-commutative 96.4%

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

    2. distribute-rgt-out 96.4%

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

    3. associate-*r* 96.4%

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

    4. times-frac 96.4%

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

13. Simplified 96.4%

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

14. Final simplification 96.4%

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

Alternative 3: 93.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. neg-sub0 29.1%

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

   2. associate-+l- 29.1%

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

   3. sub0-neg 29.1%

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

   4. neg-mul-1 29.1%

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

   5. associate-*r/ 29.1%

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

   6. metadata-eval 29.1%

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

   7. metadata-eval 29.1%

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

   8. times-frac 29.1%

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

   9. *-commutative 29.1%

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

   10. times-frac 29.1%

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

   11. associate-*l/ 29.1%

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

3. Simplified 29.2%

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

   1. fma-udef 29.1%

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

   2. associate-*r* 29.1%

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

   3. *-commutative 29.1%

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

   4. *-commutative 29.1%

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

   5. metadata-eval 29.1%

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

   6. cancel-sign-sub-inv 29.1%

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

   7. add-cbrt-cube 28.9%

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

   8. add-sqr-sqrt 28.9%

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

   9. cancel-sign-sub-inv 28.9%

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

   10. metadata-eval 28.9%

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

   11. *-commutative 28.9%

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

   12. associate-*r* 28.9%

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

   13. fma-udef 29.0%

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

   14. add-sqr-sqrt 29.0%

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

   15. cancel-sign-sub-inv 29.0%

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

   16. metadata-eval 29.0%

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

   17. *-commutative 29.0%

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

   18. associate-*r* 29.0%

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

5. Applied egg-rr 30.4%

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

   1. unpow1/3 29.2%

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

   2. fma-def 29.1%

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

   3. +-commutative 29.1%

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

   4. associate-*r* 29.1%

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

   5. metadata-eval 29.1%

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

   6. distribute-rgt-neg-in 29.1%

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

   7. *-commutative 29.1%

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

   8. associate-*r* 29.1%

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

   9. distribute-rgt-neg-in 29.1%

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

   10. fma-def 29.1%

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

   11. distribute-rgt-neg-in 29.1%

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

   12. metadata-eval 29.1%

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

7. Simplified 29.1%

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

8. Taylor expanded in c around 0 95.0%

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

   1. fma-def 95.0%

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

   2. associate-*l/ 95.0%

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

   3. unpow2 95.0%

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

   4. +-commutative 95.0%

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

   5. associate-*l/ 95.0%

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

   6. unpow2 95.0%

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

   7. fma-def 95.0%

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

   8. unpow2 95.0%

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

   9. associate-*l/ 95.0%

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

   10. *-commutative 95.0%

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

   11. unpow2 95.0%

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

   12. associate-*r/ 95.0%

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

   13. associate-*l/ 94.8%

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

   14. *-commutative 94.8%

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

10. Simplified 94.8%

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

11. Final simplification 94.8%

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

Alternative 4: 94.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. /-rgt-identity 29.1%

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

   2. metadata-eval 29.1%

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

   3. associate-/l* 29.1%

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

   4. associate-*r/ 29.1%

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

   5. *-commutative 29.1%

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

   6. associate-*l/ 29.1%

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

   7. associate-*r/ 29.1%

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

   8. metadata-eval 29.1%

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

   9. metadata-eval 29.1%

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

   10. times-frac 29.1%

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

   11. neg-mul-1 29.1%

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

   12. distribute-rgt-neg-in 29.1%

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

   13. times-frac 29.1%

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

   14. metadata-eval 29.1%

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

   15. neg-mul-1 29.1%

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

3. Simplified 29.2%

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

4. Taylor expanded in b around inf 95.0%

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

   1. fma-def 95.0%

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

   2. associate-/l* 95.0%

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

   3. unpow2 95.0%

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

   4. fma-def 95.0%

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

   5. associate-*r/ 95.0%

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

   6. *-commutative 95.0%

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

   7. unpow2 95.0%

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

6. Simplified 95.0%

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

7. Final simplification 95.0%

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

Alternative 5: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -6\\ \frac{-0.125 \cdot \frac{{t_0}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{t_0}^{3}}{{b}^{5}} + 0.5 \cdot \frac{t_0}{b}\right)}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* 3.0 (* c a)) (* (* c a) -6.0))))
   (/
    (+
     (* -0.125 (/ (pow t_0 2.0) (pow b 3.0)))
     (+ (* 0.0625 (/ (pow t_0 3.0) (pow b 5.0))) (* 0.5 (/ t_0 b))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (3.0 * (c * a)) + ((c * a) * -6.0);
	return ((-0.125 * (pow(t_0, 2.0) / pow(b, 3.0))) + ((0.0625 * (pow(t_0, 3.0) / pow(b, 5.0))) + (0.5 * (t_0 / b)))) / (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
    real(8) :: t_0
    t_0 = (3.0d0 * (c * a)) + ((c * a) * (-6.0d0))
    code = (((-0.125d0) * ((t_0 ** 2.0d0) / (b ** 3.0d0))) + ((0.0625d0 * ((t_0 ** 3.0d0) / (b ** 5.0d0))) + (0.5d0 * (t_0 / b)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (3.0 * (c * a)) + ((c * a) * -6.0);
	return ((-0.125 * (Math.pow(t_0, 2.0) / Math.pow(b, 3.0))) + ((0.0625 * (Math.pow(t_0, 3.0) / Math.pow(b, 5.0))) + (0.5 * (t_0 / b)))) / (3.0 * a);
}

Python

def code(a, b, c):
	t_0 = (3.0 * (c * a)) + ((c * a) * -6.0)
	return ((-0.125 * (math.pow(t_0, 2.0) / math.pow(b, 3.0))) + ((0.0625 * (math.pow(t_0, 3.0) / math.pow(b, 5.0))) + (0.5 * (t_0 / b)))) / (3.0 * a)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(3.0 * Float64(c * a)) + Float64(Float64(c * a) * -6.0))
	return Float64(Float64(Float64(-0.125 * Float64((t_0 ^ 2.0) / (b ^ 3.0))) + Float64(Float64(0.0625 * Float64((t_0 ^ 3.0) / (b ^ 5.0))) + Float64(0.5 * Float64(t_0 / b)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (3.0 * (c * a)) + ((c * a) * -6.0);
	tmp = ((-0.125 * ((t_0 ^ 2.0) / (b ^ 3.0))) + ((0.0625 * ((t_0 ^ 3.0) / (b ^ 5.0))) + (0.5 * (t_0 / b)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(-0.125 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0625 * N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.1%

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

   1. neg-sub0 29.1%

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

   2. associate-+l- 29.1%

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

   3. sub0-neg 29.1%

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

   4. neg-mul-1 29.1%

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

   5. associate-*r/ 29.1%

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

   6. metadata-eval 29.1%

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

   7. metadata-eval 29.1%

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

   8. times-frac 29.1%

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

   9. *-commutative 29.1%

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

   10. times-frac 29.1%

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

   11. associate-*l/ 29.1%

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

3. Simplified 29.1%

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

   1. *-commutative 29.1%

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

   2. prod-diff 29.2%

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

   3. fma-neg 29.1%

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

   4. cancel-sign-sub-inv 29.1%

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

   5. metadata-eval 29.1%

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

   6. *-commutative 29.1%

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

   7. fma-udef 29.2%

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

   8. associate-*r* 29.2%

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

   9. metadata-eval 29.2%

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

5. Applied egg-rr 29.2%

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

6. Taylor expanded in b around inf 94.4%

   \[\leadsto \frac{\color{blue}{-0.125 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{b}\right)}}{3 \cdot a} \]

7. Final simplification 94.4%

   \[\leadsto \frac{-0.125 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -6\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -6\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -6}{b}\right)}{3 \cdot a} \]

Alternative 6: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -6\\ t_1 := \left(c \cdot a\right) \cdot -3\\ 0.16666666666666666 \cdot \frac{t_0}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{t_0}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{t_1 \cdot t_1}{a \cdot {b}^{3}}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* 3.0 (* c a)) (* (* c a) -6.0))) (t_1 (* (* c a) -3.0)))
   (+
    (* 0.16666666666666666 (/ t_0 (* a b)))
    (+
     (* 0.020833333333333332 (/ (pow t_0 3.0) (* a (pow b 5.0))))
     (* -0.041666666666666664 (/ (* t_1 t_1) (* a (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = (3.0 * (c * a)) + ((c * a) * -6.0);
	double t_1 = (c * a) * -3.0;
	return (0.16666666666666666 * (t_0 / (a * b))) + ((0.020833333333333332 * (pow(t_0, 3.0) / (a * pow(b, 5.0)))) + (-0.041666666666666664 * ((t_1 * t_1) / (a * 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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (3.0d0 * (c * a)) + ((c * a) * (-6.0d0))
    t_1 = (c * a) * (-3.0d0)
    code = (0.16666666666666666d0 * (t_0 / (a * b))) + ((0.020833333333333332d0 * ((t_0 ** 3.0d0) / (a * (b ** 5.0d0)))) + ((-0.041666666666666664d0) * ((t_1 * t_1) / (a * (b ** 3.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (3.0 * (c * a)) + ((c * a) * -6.0);
	double t_1 = (c * a) * -3.0;
	return (0.16666666666666666 * (t_0 / (a * b))) + ((0.020833333333333332 * (Math.pow(t_0, 3.0) / (a * Math.pow(b, 5.0)))) + (-0.041666666666666664 * ((t_1 * t_1) / (a * Math.pow(b, 3.0)))));
}

Python

def code(a, b, c):
	t_0 = (3.0 * (c * a)) + ((c * a) * -6.0)
	t_1 = (c * a) * -3.0
	return (0.16666666666666666 * (t_0 / (a * b))) + ((0.020833333333333332 * (math.pow(t_0, 3.0) / (a * math.pow(b, 5.0)))) + (-0.041666666666666664 * ((t_1 * t_1) / (a * math.pow(b, 3.0)))))

Julia

function code(a, b, c)
	t_0 = Float64(Float64(3.0 * Float64(c * a)) + Float64(Float64(c * a) * -6.0))
	t_1 = Float64(Float64(c * a) * -3.0)
	return Float64(Float64(0.16666666666666666 * Float64(t_0 / Float64(a * b))) + Float64(Float64(0.020833333333333332 * Float64((t_0 ^ 3.0) / Float64(a * (b ^ 5.0)))) + Float64(-0.041666666666666664 * Float64(Float64(t_1 * t_1) / Float64(a * (b ^ 3.0))))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (3.0 * (c * a)) + ((c * a) * -6.0);
	t_1 = (c * a) * -3.0;
	tmp = (0.16666666666666666 * (t_0 / (a * b))) + ((0.020833333333333332 * ((t_0 ^ 3.0) / (a * (b ^ 5.0)))) + (-0.041666666666666664 * ((t_1 * t_1) / (a * (b ^ 3.0)))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]}, N[(N[(0.16666666666666666 * N[(t$95$0 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.020833333333333332 * N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.041666666666666664 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 29.1%

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

   1. neg-sub0 29.1%

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

   2. associate-+l- 29.1%

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

   3. sub0-neg 29.1%

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

   4. neg-mul-1 29.1%

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

   5. associate-*r/ 29.1%

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

   6. metadata-eval 29.1%

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

   7. metadata-eval 29.1%

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

   8. times-frac 29.1%

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

   9. *-commutative 29.1%

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

   10. times-frac 29.1%

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

   11. associate-*l/ 29.1%

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

3. Simplified 29.1%

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

   1. *-commutative 29.1%

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

   2. prod-diff 29.2%

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

   3. fma-neg 29.1%

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

   4. cancel-sign-sub-inv 29.1%

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

   5. metadata-eval 29.1%

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

   6. *-commutative 29.1%

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

   7. fma-udef 29.2%

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

   8. associate-*r* 29.2%

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

   9. metadata-eval 29.2%

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

5. Applied egg-rr 29.2%

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

6. Taylor expanded in b around inf 94.4%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{2}}{a \cdot {b}^{3}}\right)} \]
7. Step-by-step derivation

   1. unpow2 94.4%

      \[\leadsto 0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{\color{blue}{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right) \cdot \left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}}{a \cdot {b}^{3}}\right) \]

   2. distribute-rgt-out 94.4%

      \[\leadsto 0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot \left(3 + -6\right)\right)} \cdot \left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}{a \cdot {b}^{3}}\right) \]

   3. *-commutative 94.4%

      \[\leadsto 0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{\left(\color{blue}{\left(a \cdot c\right)} \cdot \left(3 + -6\right)\right) \cdot \left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}{a \cdot {b}^{3}}\right) \]

   4. metadata-eval 94.4%

      \[\leadsto 0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{\left(\left(a \cdot c\right) \cdot \color{blue}{-3}\right) \cdot \left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}{a \cdot {b}^{3}}\right) \]

   5. distribute-rgt-out 94.4%

      \[\leadsto 0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{\left(\left(a \cdot c\right) \cdot -3\right) \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(3 + -6\right)\right)}}{a \cdot {b}^{3}}\right) \]

   6. *-commutative 94.4%

      \[\leadsto 0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{\left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\color{blue}{\left(a \cdot c\right)} \cdot \left(3 + -6\right)\right)}{a \cdot {b}^{3}}\right) \]

   7. metadata-eval 94.4%

      \[\leadsto 0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{\left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot \color{blue}{-3}\right)}{a \cdot {b}^{3}}\right) \]

8. Applied egg-rr 94.4%

   \[\leadsto 0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}}{a \cdot {b}^{3}}\right) \]

9. Final simplification 94.4%

   \[\leadsto 0.16666666666666666 \cdot \frac{3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -6}{a \cdot b} + \left(0.020833333333333332 \cdot \frac{{\left(3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -6\right)}^{3}}{a \cdot {b}^{5}} + -0.041666666666666664 \cdot \frac{\left(\left(c \cdot a\right) \cdot -3\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right)}{a \cdot {b}^{3}}\right) \]

Alternative 7: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.00047:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.00047)
   (* -0.3333333333333333 (/ (- b (sqrt (fma b b (* a (* c -3.0))))) a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.00047) {
		tmp = -0.3333333333333333 * ((b - sqrt(fma(b, b, (a * (c * -3.0))))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.00047)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00047], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4.69999999999999986e-4

   1. Initial program 68.0%

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

      1. /-rgt-identity 68.0%

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

      2. metadata-eval 68.0%

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

      3. associate-/l* 68.0%

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

      4. associate-*r/ 68.0%

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

      5. *-commutative 68.0%

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

      6. associate-*l/ 68.0%

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

      7. associate-*r/ 68.0%

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

      8. metadata-eval 68.0%

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

      9. metadata-eval 68.0%

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

      10. times-frac 68.0%

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

      11. neg-mul-1 68.0%

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

      12. distribute-rgt-neg-in 68.0%

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

      13. times-frac 68.0%

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

      14. metadata-eval 68.0%

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

      15. neg-mul-1 68.0%

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

   3. Simplified 68.1%

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

   if -4.69999999999999986e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.7%

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

      1. /-rgt-identity 18.7%

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

      2. metadata-eval 18.7%

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

      3. associate-/l* 18.7%

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

      4. associate-*r/ 18.7%

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

      5. *-commutative 18.7%

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

      6. associate-*l/ 18.7%

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

      7. associate-*r/ 18.7%

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

      8. metadata-eval 18.7%

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

      9. metadata-eval 18.7%

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

      10. times-frac 18.7%

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

      11. neg-mul-1 18.7%

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

      12. distribute-rgt-neg-in 18.7%

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

      13. times-frac 18.7%

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

      14. metadata-eval 18.7%

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

      15. neg-mul-1 18.7%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in b around inf 90.1%

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

      1. associate-*r/ 90.1%

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

   6. Simplified 90.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.00047:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.00047:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.00047)
   (/ (* -0.3333333333333333 (- b (sqrt (fma b b (* (* c a) -3.0))))) a)
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.00047) {
		tmp = (-0.3333333333333333 * (b - sqrt(fma(b, b, ((c * a) * -3.0))))) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.00047)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))))) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00047], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4.69999999999999986e-4

   1. Initial program 68.0%

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

      1. /-rgt-identity 68.0%

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

      2. metadata-eval 68.0%

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

      3. associate-/r/ 68.0%

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

      4. metadata-eval 68.0%

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

      5. metadata-eval 68.0%

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

      6. times-frac 68.0%

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

      7. *-commutative 68.0%

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

      8. times-frac 68.0%

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

      9. *-commutative 68.0%

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

      10. associate-/r* 68.1%

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

      11. associate-*l/ 68.1%

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

   3. Simplified 68.1%

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

   if -4.69999999999999986e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.7%

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

      1. /-rgt-identity 18.7%

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

      2. metadata-eval 18.7%

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

      3. associate-/l* 18.7%

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

      4. associate-*r/ 18.7%

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

      5. *-commutative 18.7%

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

      6. associate-*l/ 18.7%

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

      7. associate-*r/ 18.7%

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

      8. metadata-eval 18.7%

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

      9. metadata-eval 18.7%

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

      10. times-frac 18.7%

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

      11. neg-mul-1 18.7%

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

      12. distribute-rgt-neg-in 18.7%

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

      13. times-frac 18.7%

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

      14. metadata-eval 18.7%

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

      15. neg-mul-1 18.7%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in b around inf 90.1%

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

      1. associate-*r/ 90.1%

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

   6. Simplified 90.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.00047:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.00047:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.00047)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.00047) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)) <= (-0.00047d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.00047) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.00047:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.00047)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.00047)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00047], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4.69999999999999986e-4

   1. Initial program 68.0%

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

      1. neg-sub0 68.0%

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

      2. associate-+l- 68.0%

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

      3. sub0-neg 68.0%

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

      4. neg-mul-1 68.0%

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

      5. associate-*r/ 68.0%

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

      6. metadata-eval 68.0%

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

      7. metadata-eval 68.0%

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

      8. times-frac 68.0%

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

      9. *-commutative 68.0%

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

      10. times-frac 68.0%

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

      11. associate-*l/ 68.0%

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

   3. Simplified 68.0%

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

   if -4.69999999999999986e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.7%

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

      1. /-rgt-identity 18.7%

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

      2. metadata-eval 18.7%

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

      3. associate-/l* 18.7%

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

      4. associate-*r/ 18.7%

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

      5. *-commutative 18.7%

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

      6. associate-*l/ 18.7%

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

      7. associate-*r/ 18.7%

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

      8. metadata-eval 18.7%

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

      9. metadata-eval 18.7%

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

      10. times-frac 18.7%

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

      11. neg-mul-1 18.7%

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

      12. distribute-rgt-neg-in 18.7%

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

      13. times-frac 18.7%

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

      14. metadata-eval 18.7%

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

      15. neg-mul-1 18.7%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in b around inf 90.1%

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

      1. associate-*r/ 90.1%

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

   6. Simplified 90.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.00047:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 90.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. /-rgt-identity 29.1%

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

   2. metadata-eval 29.1%

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

   3. associate-/l* 29.1%

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

   4. associate-*r/ 29.1%

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

   5. *-commutative 29.1%

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

   6. associate-*l/ 29.1%

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

   7. associate-*r/ 29.1%

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

   8. metadata-eval 29.1%

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

   9. metadata-eval 29.1%

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

   10. times-frac 29.1%

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

   11. neg-mul-1 29.1%

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

   12. distribute-rgt-neg-in 29.1%

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

   13. times-frac 29.1%

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

   14. metadata-eval 29.1%

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

   15. neg-mul-1 29.1%

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

3. Simplified 29.2%

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

4. Taylor expanded in b around inf 91.7%

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

   1. fma-def 91.7%

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

   2. associate-*r/ 91.7%

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

   3. *-commutative 91.7%

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

   4. unpow2 91.7%

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

6. Simplified 91.7%

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

7. Final simplification 91.7%

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

Alternative 11: 81.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 = c * ((-0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. neg-sub0 29.1%

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

   2. associate-+l- 29.1%

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

   3. sub0-neg 29.1%

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

   4. neg-mul-1 29.1%

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

   5. associate-*r/ 29.1%

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

   6. metadata-eval 29.1%

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

   7. metadata-eval 29.1%

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

   8. times-frac 29.1%

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

   9. *-commutative 29.1%

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

   10. times-frac 29.1%

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

   11. associate-*l/ 29.1%

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

3. Simplified 29.2%

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

4. Taylor expanded in b around inf 20.3%

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

5. Taylor expanded in c around 0 82.6%

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

   1. associate-*r/ 82.6%

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

   2. associate-/l* 82.4%

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

7. Simplified 82.4%

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

   1. associate-/r/ 82.4%

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

9. Applied egg-rr 82.4%

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

10. Final simplification 82.4%

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

Alternative 12: 81.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 = (c * (-0.5d0)) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. /-rgt-identity 29.1%

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

   2. metadata-eval 29.1%

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

   3. associate-/l* 29.1%

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

   4. associate-*r/ 29.1%

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

   5. *-commutative 29.1%

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

   6. associate-*l/ 29.1%

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

   7. associate-*r/ 29.1%

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

   8. metadata-eval 29.1%

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

   9. metadata-eval 29.1%

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

   10. times-frac 29.1%

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

   11. neg-mul-1 29.1%

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

   12. distribute-rgt-neg-in 29.1%

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

   13. times-frac 29.1%

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

   14. metadata-eval 29.1%

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

   15. neg-mul-1 29.1%

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

3. Simplified 29.2%

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

4. Taylor expanded in b around inf 82.6%

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

   1. associate-*r/ 82.6%

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

6. Simplified 82.6%

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

7. Final simplification 82.6%

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

Reproduce

herbie shell --seed 2023187 
(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)))