Cubic critical, medium range

Percentage Accurate: 30.7% → 95.8%

Time: 12.4s

Alternatives: 13

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: 30.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return fma(-0.5625, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), fma(-0.5, Float64(c / b), fma(-0.375, Float64(a / Float64((b ^ 3.0) / Float64(c * c))), Float64(-0.16666666666666666 * Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 6.328125)))))))
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[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

2. Taylor expanded in b around inf 94.3%

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

   1. fma-def 94.3%

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

   2. cube-prod 94.3%

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

   3. fma-def 94.4%

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

   4. associate-/l* 94.3%

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

   5. fma-def 94.3%

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

4. Simplified 94.3%

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

5. Taylor expanded in a around 0 94.9%

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

   1. fma-def 94.9%

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

   2. *-commutative 94.9%

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

   3. unpow2 94.9%

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

   4. fma-def 94.9%

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

   5. fma-def 94.9%

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

   6. associate-/l* 94.9%

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

   7. unpow2 94.9%

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

7. Simplified 94.9%

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

8. Final simplification 94.9%

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

Alternative 2: 95.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

2. Taylor expanded in b around inf 94.3%

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

   1. fma-def 94.3%

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

   2. cube-prod 94.3%

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

   3. fma-def 94.4%

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

   4. associate-/l* 94.3%

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

   5. fma-def 94.3%

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

4. Simplified 94.3%

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

5. Taylor expanded in c around 0 94.9%

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

7. Simplified 94.7%

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

8. Final simplification 94.7%

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

Alternative 3: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{a}{0.3333333333333333}}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -75000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{t_0 \cdot {t_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (cbrt (/ a 0.3333333333333333))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -75000.0)
     (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* t_0 (pow t_0 2.0)))
     (fma -0.5 (/ c b) (/ (* (* c c) (* a -0.375)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = cbrt((a / 0.3333333333333333));
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -75000.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (t_0 * pow(t_0, 2.0));
	} else {
		tmp = fma(-0.5, (c / b), (((c * c) * (a * -0.375)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = cbrt(Float64(a / 0.3333333333333333))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -75000.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(t_0 * (t_0 ^ 2.0)));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * Float64(a * -0.375)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a / 0.3333333333333333), $MachinePrecision], 1/3], $MachinePrecision]}, 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], -75000.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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)) < -75000

   1. Initial program 82.2%

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

      1. neg-sub0 82.2%

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

      2. sqr-neg 82.2%

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

      3. associate-+l- 82.2%

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

      4. sub0-neg 82.2%

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

      5. neg-mul-1 82.2%

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

   3. Simplified 82.4%

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

      1. add-cube-cbrt 82.8%

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

      2. div-inv 82.7%

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

      3. metadata-eval 82.7%

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

      4. *-commutative 82.7%

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

      5. div-inv 82.7%

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

      6. metadata-eval 82.7%

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

      7. *-commutative 82.7%

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

      8. pow2 82.7%

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

      9. *-commutative 82.7%

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

      10. metadata-eval 82.7%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\sqrt[3]{a \cdot \color{blue}{\frac{1}{0.3333333333333333}}}\right)}^{2} \cdot \sqrt[3]{\frac{a}{0.3333333333333333}}} \]

      11. div-inv 82.8%

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

   5. Applied egg-rr 82.8%

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

   if -75000 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.3%

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

   2. Taylor expanded in b around inf 91.8%

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

      1. fma-def 91.8%

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

      2. associate-*r/ 91.8%

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

      3. associate-*r* 91.8%

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

      4. unpow2 91.8%

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

   4. Simplified 91.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -75000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\sqrt[3]{\frac{a}{0.3333333333333333}} \cdot {\left(\sqrt[3]{\frac{a}{0.3333333333333333}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 4: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -75000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\sqrt[3]{\frac{1}{\frac{0.3333333333333333}{a}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -75000.0)
   (/
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (pow (cbrt (/ 1.0 (/ 0.3333333333333333 a))) 3.0))
   (fma -0.5 (/ c b) (/ (* (* c c) (* a -0.375)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -75000.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / pow(cbrt((1.0 / (0.3333333333333333 / a))), 3.0);
	} else {
		tmp = fma(-0.5, (c / b), (((c * c) * (a * -0.375)) / pow(b, 3.0)));
	}
	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)) <= -75000.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / (cbrt(Float64(1.0 / Float64(0.3333333333333333 / a))) ^ 3.0));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * Float64(a * -0.375)) / (b ^ 3.0)));
	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], -75000.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[Power[N[(1.0 / N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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)) < -75000

   1. Initial program 82.2%

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

      1. neg-sub0 82.2%

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

      2. sqr-neg 82.2%

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

      3. associate-+l- 82.2%

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

      4. sub0-neg 82.2%

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

      5. neg-mul-1 82.2%

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

   3. Simplified 82.4%

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

      1. div-inv 82.4%

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

      2. metadata-eval 82.4%

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

      3. *-commutative 82.4%

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

      4. add-cube-cbrt 82.7%

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

      5. pow3 82.6%

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

      6. *-commutative 82.6%

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

      7. metadata-eval 82.6%

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

      8. div-inv 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. clear-num 82.7%

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

      2. inv-pow 82.7%

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

   7. Applied egg-rr 82.7%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\sqrt[3]{\color{blue}{{\left(\frac{0.3333333333333333}{a}\right)}^{-1}}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. unpow-1 82.7%

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

   9. Simplified 82.7%

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

   if -75000 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.3%

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

   2. Taylor expanded in b around inf 91.8%

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

      1. fma-def 91.8%

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

      2. associate-*r/ 91.8%

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

      3. associate-*r* 91.8%

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

      4. unpow2 91.8%

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

   4. Simplified 91.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -75000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\sqrt[3]{\frac{1}{\frac{0.3333333333333333}{a}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -75000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\sqrt[3]{\frac{a}{0.3333333333333333}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -75000.0)
   (/
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (pow (cbrt (/ a 0.3333333333333333)) 3.0))
   (fma -0.5 (/ c b) (/ (* (* c c) (* a -0.375)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -75000.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / pow(cbrt((a / 0.3333333333333333)), 3.0);
	} else {
		tmp = fma(-0.5, (c / b), (((c * c) * (a * -0.375)) / pow(b, 3.0)));
	}
	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)) <= -75000.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / (cbrt(Float64(a / 0.3333333333333333)) ^ 3.0));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * Float64(a * -0.375)) / (b ^ 3.0)));
	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], -75000.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[Power[N[(a / 0.3333333333333333), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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)) < -75000

   1. Initial program 82.2%

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

      1. neg-sub0 82.2%

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

      2. sqr-neg 82.2%

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

      3. associate-+l- 82.2%

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

      4. sub0-neg 82.2%

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

      5. neg-mul-1 82.2%

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

   3. Simplified 82.4%

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

      1. div-inv 82.4%

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

      2. metadata-eval 82.4%

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

      3. *-commutative 82.4%

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

      4. add-cube-cbrt 82.7%

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

      5. pow3 82.6%

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

      6. *-commutative 82.6%

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

      7. metadata-eval 82.6%

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

      8. div-inv 82.7%

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

   5. Applied egg-rr 82.7%

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

   if -75000 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.3%

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

   2. Taylor expanded in b around inf 91.8%

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

      1. fma-def 91.8%

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

      2. associate-*r/ 91.8%

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

      3. associate-*r* 91.8%

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

      4. unpow2 91.8%

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

   4. Simplified 91.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -75000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\sqrt[3]{\frac{a}{0.3333333333333333}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 6: 94.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

2. Taylor expanded in b around inf 94.3%

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

   1. fma-def 94.3%

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

   2. cube-prod 94.3%

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

   3. fma-def 94.4%

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

   4. associate-/l* 94.3%

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

   5. fma-def 94.3%

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

4. Simplified 94.3%

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

5. Taylor expanded in a around 0 93.2%

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

   1. fma-def 93.2%

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

   2. associate-/l* 93.2%

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

   3. associate-/r/ 93.2%

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

   4. unpow2 93.2%

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

   5. associate-*r/ 93.2%

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

   6. associate-*l/ 92.9%

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

   7. +-commutative 92.9%

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

   8. fma-def 92.9%

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

   9. associate-/l* 92.9%

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

   10. associate-/r/ 92.9%

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

   11. unpow2 92.9%

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

   12. *-commutative 92.9%

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

7. Simplified 92.9%

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

8. Final simplification 92.9%

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

Alternative 7: 94.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return fma(-0.5625, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(a / Float64((b ^ 3.0) / Float64(c * c))))))
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[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

2. Taylor expanded in b around inf 93.2%

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

   1. fma-def 93.2%

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

   2. *-commutative 93.2%

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

   3. unpow2 93.2%

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

   4. fma-def 93.2%

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

   5. associate-/l* 93.2%

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

   6. unpow2 93.2%

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

4. Simplified 93.2%

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

5. Final simplification 93.2%

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

Alternative 8: 91.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -75000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -75000.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ a 0.3333333333333333))
   (fma -0.5 (/ c b) (/ (* (* c c) (* a -0.375)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -75000.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a / 0.3333333333333333);
	} else {
		tmp = fma(-0.5, (c / b), (((c * c) * (a * -0.375)) / pow(b, 3.0)));
	}
	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)) <= -75000.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a / 0.3333333333333333));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * Float64(a * -0.375)) / (b ^ 3.0)));
	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], -75000.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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)) < -75000

   1. Initial program 82.2%

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

      1. neg-sub0 82.2%

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

      2. sqr-neg 82.2%

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

      3. associate-+l- 82.2%

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

      4. sub0-neg 82.2%

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

      5. neg-mul-1 82.2%

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

   3. Simplified 82.4%

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

   if -75000 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.3%

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

   2. Taylor expanded in b around inf 91.8%

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

      1. fma-def 91.8%

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

      2. associate-*r/ 91.8%

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

      3. associate-*r* 91.8%

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

      4. unpow2 91.8%

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

   4. Simplified 91.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -75000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot \left(a \cdot -0.375\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 9: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -6e-5)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma b b (* c (* a -3.0)))) b))
   (* -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)) <= -6e-5) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma(b, b, (c * (a * -3.0)))) - b);
	} 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)) <= -6e-5)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b));
	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], -6e-5], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $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)) < -6.00000000000000015e-5

   1. Initial program 73.6%

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

      1. neg-sub0 73.6%

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

      2. sqr-neg 73.6%

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

      3. associate-+l- 73.6%

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

      4. sub0-neg 73.6%

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

      5. neg-mul-1 73.6%

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

   3. Simplified 73.8%

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

      1. add-exp-log 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. add-exp-log 73.8%

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

      2. div-sub 72.9%

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

      3. div-inv 72.3%

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

      4. metadata-eval 72.3%

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

      5. div-inv 73.0%

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

      6. metadata-eval 73.0%

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

   7. Applied egg-rr 73.0%

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

      1. div-sub 73.8%

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

      2. *-lft-identity 73.8%

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

      3. associate-*l/ 73.7%

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

      4. /-rgt-identity 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-/r* 73.7%

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

      7. metadata-eval 73.7%

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

      8. /-rgt-identity 73.7%

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

      9. associate-*r* 73.7%

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

      10. rem-exp-log 0.0%

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

      11. *-commutative 0.0%

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

      12. associate-*l* 0.0%

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

      13. rem-exp-log 73.7%

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

   9. Simplified 73.7%

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

   if -6.00000000000000015e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 20.0%

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

   2. Taylor expanded in b around inf 89.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -6e-5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ a 0.3333333333333333))
   (* -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)) <= -6e-5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a / 0.3333333333333333);
	} 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)) <= -6e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a / 0.3333333333333333));
	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], -6e-5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $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)) < -6.00000000000000015e-5

   1. Initial program 73.6%

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

      1. neg-sub0 73.6%

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

      2. sqr-neg 73.6%

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

      3. associate-+l- 73.6%

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

      4. sub0-neg 73.6%

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

      5. neg-mul-1 73.6%

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

   3. Simplified 73.8%

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

   if -6.00000000000000015e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 20.0%

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

   2. Taylor expanded in b around inf 89.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -6e-5)
   (/ (- (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)) <= -6e-5) {
		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)) <= -6e-5)
		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], -6e-5], 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)) < -6.00000000000000015e-5

   1. Initial program 73.6%

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

      1. neg-sub0 73.6%

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

      2. sqr-neg 73.6%

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

      3. associate-+l- 73.6%

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

      4. sub0-neg 73.6%

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

   3. Simplified 73.8%

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

   if -6.00000000000000015e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 20.0%

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

   2. Taylor expanded in b around inf 89.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -6 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))))
   (if (<= t_0 -6e-5) t_0 (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -6e-5) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    if (t_0 <= (-6d-5)) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -6e-5) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -6e-5:
		tmp = t_0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -6e-5)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -6e-5)
		tmp = t_0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -6e-5], t$95$0, 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)) < -6.00000000000000015e-5

   1. Initial program 73.6%

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

   if -6.00000000000000015e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 20.0%

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

   2. Taylor expanded in b around inf 89.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13: 81.9% 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 33.0%

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

2. Taylor expanded in b around inf 79.7%

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

3. Final simplification 79.7%

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

Reproduce

herbie shell --seed 2023293 
(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)))