Average Error: 28.5 → 0.3
Time: 7.7s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\frac{-c}{\mathsf{fma}\left(\sqrt{\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{3}}}}, \sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}, b\right)} \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/
  (- c)
  (fma
   (sqrt (sqrt (cbrt (pow (fma c (* a -3.0) (* b b)) 3.0))))
   (sqrt (sqrt (fma a (* c -3.0) (* b b))))
   b)))
double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}
↓
double code(double a, double b, double c) {
	return -c / fma(sqrt(sqrt(cbrt(pow(fma(c, (a * -3.0), (b * b)), 3.0)))), sqrt(sqrt(fma(a, (c * -3.0), (b * b)))), b);
}

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
↓
function code(a, b, c)
	return Float64(Float64(-c) / fma(sqrt(sqrt(cbrt((fma(c, Float64(a * -3.0), Float64(b * b)) ^ 3.0)))), sqrt(sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))), b))
end

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]
↓
code[a_, b_, c_] := N[((-c) / N[(N[Sqrt[N[Sqrt[N[Power[N[Power[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.5

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

  2. Simplified28.5

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

  3. Applied flip--_binary6428.5

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

  4. Applied associate-*l/_binary6428.5

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

  5. Simplified0.6

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -3, 0\right) \cdot \frac{0.3333333333333333}{a}}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b} \]

  6. Taylor expanded in a around 0 0.3

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

  7. Simplified0.3

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

  8. Applied add-sqr-sqrt_binary640.4

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

  9. Applied fma-def_binary640.3

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

  10. Applied add-cbrt-cube_binary640.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2022129 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))