?

Average Accuracy: 31.8% → 95.2%
Time: 15.4s
Precision: binary64
Cost: 47680

?

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.375, a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right), -0.5 \cdot \frac{c}{b}\right)\right)\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (fma
  -0.16666666666666666
  (* (/ (pow a 3.0) b) (* (* (* c c) (* (* c c) (pow b -6.0))) 6.328125))
  (fma
   -0.5625
   (* (* (* c c) (* c (pow b -5.0))) (* a a))
   (fma -0.375 (* a (* c (/ c (pow b 3.0)))) (* -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(-0.16666666666666666, ((pow(a, 3.0) / b) * (((c * c) * ((c * c) * pow(b, -6.0))) * 6.328125)), fma(-0.5625, (((c * c) * (c * pow(b, -5.0))) * (a * a)), fma(-0.375, (a * (c * (c / pow(b, 3.0)))), (-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(-0.16666666666666666, Float64(Float64((a ^ 3.0) / b) * Float64(Float64(Float64(c * c) * Float64(Float64(c * c) * (b ^ -6.0))) * 6.328125)), fma(-0.5625, Float64(Float64(Float64(c * c) * Float64(c * (b ^ -5.0))) * Float64(a * a)), fma(-0.375, Float64(a * Float64(c * Float64(c / (b ^ 3.0)))), 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[(-0.16666666666666666 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(c * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.375, a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right), -0.5 \cdot \frac{c}{b}\right)\right)\right)

Error?

Derivation?

  1. Initial program 29.0%

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

  2. Simplified29.2%

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

  3. Taylor expanded in a around 0 95.7%

    \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

  4. Simplified95.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.375, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot a, -0.5 \cdot \frac{c}{b}\right)\right)\right)} \]

  5. Applied egg-rr95.7%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right)} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.375, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot a, -0.5 \cdot \frac{c}{b}\right)\right)\right) \]

  6. Applied egg-rr95.7%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right)} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.375, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot a, -0.5 \cdot \frac{c}{b}\right)\right)\right) \]

  7. Final simplification95.7%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.375, a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right), -0.5 \cdot \frac{c}{b}\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy93.3%
Cost33536
\[\mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \mathsf{fma}\left(-0.375, c \cdot \frac{c}{\frac{{b}^{3}}{a}}, c \cdot \frac{-0.5}{b}\right)\right) \]
Alternative 2
Accuracy93.6%
Cost33536
\[\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\right) \]
Alternative 3
Accuracy93.6%
Cost33536
\[\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}, -0.5625 \cdot \left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right) \]
Alternative 4
Accuracy90.4%
Cost7488
\[\mathsf{fma}\left(-0.375, \frac{\frac{c \cdot \left(a \cdot c\right)}{b \cdot b}}{b}, -0.5 \cdot \frac{c}{b}\right) \]
Alternative 5
Accuracy90.2%
Cost7296
\[c \cdot \left(\frac{-0.5}{b} + \left(c \cdot {b}^{-3}\right) \cdot \left(a \cdot -0.375\right)\right) \]
Alternative 6
Accuracy80.7%
Cost320
\[c \cdot \frac{-0.5}{b} \]
Alternative 7
Accuracy80.9%
Cost320
\[\frac{c \cdot -0.5}{b} \]

Error

Reproduce?

herbie shell --seed 2023157 -o generate:proofs
(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)))