Average Error: 52.7 → 1.5
Time: 6.3s
Precision: binary64
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, -0.5625, \mathsf{fma}\left(\frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, -1.0546875, \mathsf{fma}\left(\frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}, -0.375, -0.5 \cdot \frac{c}{b}\right)\right)\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (fma
  (/ (* (* a a) (pow c 3.0)) (pow b 5.0))
  -0.5625
  (fma
   (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0))
   -1.0546875
   (fma (/ (* c (* a c)) (pow b 3.0)) -0.375 (* -0.5 (/ c 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 fma((((a * a) * pow(c, 3.0)) / pow(b, 5.0)), -0.5625, fma(((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0)), -1.0546875, fma(((c * (a * c)) / pow(b, 3.0)), -0.375, (-0.5 * (c / 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 fma(Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 5.0)), -0.5625, fma(Float64(Float64((a ^ 3.0) * (c ^ 4.0)) / (b ^ 7.0)), -1.0546875, fma(Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)), -0.375, Float64(-0.5 * Float64(c / 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[(N[(N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * -1.0546875 + N[(N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, -0.5625, \mathsf{fma}\left(\frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, -1.0546875, \mathsf{fma}\left(\frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}, -0.375, -0.5 \cdot \frac{c}{b}\right)\right)\right)

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.7

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

  2. Taylor expanded in b around inf 1.5

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

  3. Simplified1.5

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

  4. Final simplification1.5

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

Reproduce

herbie shell --seed 2022140 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))