Cubic critical, medium range

Percentage Accurate: 31.5% → 95.5%

Time: 13.5s

Alternatives: 10

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

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))
  (+
   (* -0.5 (/ c b))
   (+
    (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
    (* -1.0546875 (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-1.0546875 * ((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-1.0546875d0) * (((c ** 4.0d0) * (a ** 3.0d0)) / (b ** 7.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-1.0546875 * ((Math.pow(c, 4.0) * Math.pow(a, 3.0)) / Math.pow(b, 7.0)))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(c, 3.0) * math.pow(a, 2.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-1.0546875 * ((math.pow(c, 4.0) * math.pow(a, 3.0)) / math.pow(b, 7.0)))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-1.0546875 * Float64(Float64((c ^ 4.0) * (a ^ 3.0)) / (b ^ 7.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-1.0546875 * (((c ^ 4.0) * (a ^ 3.0)) / (b ^ 7.0)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $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. Taylor expanded in a around 0 96.1%

   \[\leadsto \frac{\color{blue}{-1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + \left(-0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}}{3 \cdot a} \]
3. Step-by-step derivation

   1. fma-def 96.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}, -1.5 \cdot \frac{c \cdot a}{b} + \left(-0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}}{3 \cdot a} \]

   2. associate-/l* 96.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{{a}^{2}}}}, -1.5 \cdot \frac{c \cdot a}{b} + \left(-0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a} \]

   3. associate-/r/ 96.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot {a}^{2}}, -1.5 \cdot \frac{c \cdot a}{b} + \left(-0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a} \]

   4. unpow2 96.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot {a}^{2}, -1.5 \cdot \frac{c \cdot a}{b} + \left(-0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a} \]

   5. associate-/l* 96.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{c}{\frac{{b}^{3}}{c}}} \cdot {a}^{2}, -1.5 \cdot \frac{c \cdot a}{b} + \left(-0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a} \]

   6. unpow2 96.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}, -1.5 \cdot \frac{c \cdot a}{b} + \left(-0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a} \]

   7. fma-def 96.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{fma}\left(-1.5, \frac{c \cdot a}{b}, -0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)}\right)}{3 \cdot a} \]

   8. associate-/l* 96.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.5, \color{blue}{\frac{c}{\frac{b}{a}}}, -0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a} \]

   9. associate-/r/ 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.5, \color{blue}{\frac{c}{b} \cdot a}, -0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a} \]

4. Simplified 96.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.5, \frac{c}{b} \cdot a, \mathsf{fma}\left(-0.5, \frac{{a}^{4}}{b} \cdot \mathsf{fma}\left(5.0625, \frac{{c}^{4}}{{b}^{6}}, \frac{{c}^{4}}{{b}^{6}} \cdot 1.265625\right), -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)\right)\right)}}{3 \cdot a} \]

5. Taylor expanded in c around 0 96.6%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\right)\right)} \]

6. Final simplification 96.6%

   \[\leadsto -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\right)\right) \]

Alternative 2: 95.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

2. Taylor expanded in b around inf 96.6%

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

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

   3. unpow2 96.6%

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

   4. fma-def 96.6%

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

4. Simplified 96.6%

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

5. Taylor expanded in c around 0 96.6%

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

   1. distribute-rgt-out 96.6%

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

   2. associate-*r* 96.6%

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

   3. times-frac 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 3: 94.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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. Taylor expanded in b around inf 95.0%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
3. 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-/l* 95.0%

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

   6. unpow2 95.0%

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

4. Simplified 95.0%

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

5. Final simplification 95.0%

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

Alternative 4: 94.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(a * N[(-0.375 * N[(c * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * -0.5 + N[(N[(-0.5625 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $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. Taylor expanded in b around inf 94.4%

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

   1. fma-def 94.4%

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

   2. associate-/l* 94.4%

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

   3. associate-/r/ 94.4%

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

   4. unpow2 94.4%

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

   5. associate-/l* 94.4%

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

   6. unpow2 94.4%

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

   7. +-commutative 94.4%

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

   8. fma-def 94.4%

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

   9. cube-prod 94.4%

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

   10. associate-/l* 94.4%

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

   11. associate-/r/ 94.6%

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

4. Simplified 94.6%

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

5. Taylor expanded in c around 0 95.0%

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

   1. associate-+r+ 95.0%

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

   2. +-commutative 95.0%

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

   3. *-commutative 95.0%

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

   4. *-commutative 95.0%

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

   5. associate-*r/ 95.0%

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

   6. unpow2 95.0%

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

   7. associate-*l/ 95.0%

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

   8. associate-*l* 95.0%

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

   9. fma-def 95.0%

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

   10. *-commutative 95.0%

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

   11. +-commutative 95.0%

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

   12. *-commutative 95.0%

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

7. Simplified 95.0%

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

8. Final simplification 95.0%

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

Alternative 5: 93.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(-1.125 * N[(N[(a * a), $MachinePrecision] * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.6875 * N[(N[(N[(c * a), $MachinePrecision] * N[(c * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $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. Taylor expanded in b around inf 94.4%

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

   1. fma-def 94.4%

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

   2. associate-/l* 94.4%

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

   3. associate-/r/ 94.4%

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

   4. unpow2 94.4%

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

   5. associate-/l* 94.4%

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

   6. unpow2 94.4%

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

   7. +-commutative 94.4%

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

   8. fma-def 94.4%

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

   9. cube-prod 94.4%

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

   10. associate-/l* 94.4%

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

   11. associate-/r/ 94.6%

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

4. Simplified 94.6%

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

   1. unpow3 94.6%

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

   2. unswap-sqr 94.6%

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

   3. associate-*l* 94.6%

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

6. Applied egg-rr 94.6%

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

7. Final simplification 94.6%

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

Alternative 6: 93.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(N[(-1.125 * N[(a * N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.6875 * N[(N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 / N[(b / N[(c * a), $MachinePrecision]), $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. Taylor expanded in b around inf 94.4%

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

   1. fma-def 94.4%

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

   2. associate-/l* 94.4%

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

   3. associate-/r/ 94.4%

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

   4. unpow2 94.4%

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

   5. associate-/l* 94.4%

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

   6. unpow2 94.4%

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

   7. +-commutative 94.4%

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

   8. fma-def 94.4%

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

   9. cube-prod 94.4%

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

   10. associate-/l* 94.4%

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

   11. associate-/r/ 94.6%

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

4. Simplified 94.6%

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

   1. associate-*l/ 94.4%

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

   2. clear-num 94.4%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \color{blue}{\frac{1}{\frac{b}{c \cdot a}}}\right)\right)}{3 \cdot a} \]

6. Applied egg-rr 94.4%

   \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \color{blue}{\frac{1}{\frac{b}{c \cdot a}}}\right)\right)}{3 \cdot a} \]
7. Step-by-step derivation

   1. fma-udef 94.4%

      \[\leadsto \frac{\color{blue}{-1.125 \cdot \left(\frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right) + \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \frac{1}{\frac{b}{c \cdot a}}\right)}}{3 \cdot a} \]

   2. *-commutative 94.4%

      \[\leadsto \frac{-1.125 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{c}{\frac{{b}^{3}}{c}}\right)} + \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \frac{1}{\frac{b}{c \cdot a}}\right)}{3 \cdot a} \]

   3. associate-*l* 94.4%

      \[\leadsto \frac{-1.125 \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{c}{\frac{{b}^{3}}{c}}\right)\right)} + \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \frac{1}{\frac{b}{c \cdot a}}\right)}{3 \cdot a} \]

   4. un-div-inv 94.4%

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

8. Applied egg-rr 94.4%

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

9. Final simplification 94.4%

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

Alternative 7: 84.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.00047)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (* -0.5 (/ c 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(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / 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(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / 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[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -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. fma-neg 68.1%

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

      2. *-commutative 68.1%

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

      3. distribute-rgt-neg-in 68.1%

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

      4. *-commutative 68.1%

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

      5. distribute-rgt-neg-in 68.1%

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

      6. metadata-eval 68.1%

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

   3. Applied egg-rr 68.1%

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

   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. Taylor expanded in b around inf 90.1%

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

4. Final simplification 85.4%

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

Alternative 8: 84.0% accurate, 0.5× speedup

Math

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

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))
   (* -0.5 (/ c 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 = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: 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 = (-0.5d0) * (c / 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 = -0.5 * (c / 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 = -0.5 * (c / 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(-0.5 * Float64(c / 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 = -0.5 * (c / 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[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -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. Taylor expanded in a around 0 68.0%

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

   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. Taylor expanded in b around inf 90.1%

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

4. Final simplification 85.4%

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

Alternative 9: 90.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(-0.375 * N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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. Taylor expanded in b around inf 91.7%

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

   1. +-commutative 91.7%

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

   2. fma-def 91.7%

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

   3. associate-/l* 91.7%

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

   4. associate-/r/ 91.7%

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

   5. unpow2 91.7%

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

   6. associate-/l* 91.7%

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

4. Simplified 91.7%

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

5. Final simplification 91.7%

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

Alternative 10: 81.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

2. Taylor expanded in b around inf 82.6%

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

3. Final simplification 82.6%

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

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)))