Average Error: 52.4 → 1.6
Time: 11.7s
Precision: binary64
Cost: 41600
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(-0.16666666666666666, \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right) \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot \frac{1}{b}\right)\right), \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \mathsf{fma}\left(-0.375, a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), \frac{c \cdot -0.5}{b}\right)\right)\right) \]
(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
  (*
   (* (* (* c c) (* (* c c) (pow b -6.0))) 6.328125)
   (* a (* (* a a) (/ 1.0 b))))
  (fma
   -0.5625
   (* (/ (* a a) (pow b 5.0)) (pow c 3.0))
   (fma -0.375 (* a (* (/ c b) (/ c (* b b)))) (/ (* c -0.5) 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, ((((c * c) * ((c * c) * pow(b, -6.0))) * 6.328125) * (a * ((a * a) * (1.0 / b)))), fma(-0.5625, (((a * a) / pow(b, 5.0)) * pow(c, 3.0)), fma(-0.375, (a * ((c / b) * (c / (b * b)))), ((c * -0.5) / 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(Float64(Float64(c * c) * Float64(Float64(c * c) * (b ^ -6.0))) * 6.328125) * Float64(a * Float64(Float64(a * a) * Float64(1.0 / b)))), fma(-0.5625, Float64(Float64(Float64(a * a) / (b ^ 5.0)) * (c ^ 3.0)), fma(-0.375, Float64(a * Float64(Float64(c / b) * Float64(c / Float64(b * b)))), Float64(Float64(c * -0.5) / 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[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] * N[(a * N[(N[(a * a), $MachinePrecision] * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $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, \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right) \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot \frac{1}{b}\right)\right), \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \mathsf{fma}\left(-0.375, a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), \frac{c \cdot -0.5}{b}\right)\right)\right)

Error

Derivation

  1. Initial program 52.4

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

  2. Simplified52.4

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

  3. Taylor expanded in a around 0 1.6

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

  4. Simplified1.6

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

  5. Applied egg-rr1.6

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

  6. Applied egg-rr1.6

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot \color{blue}{\left(a \cdot \left(\left(a \cdot a\right) \cdot \frac{1}{b}\right)\right)}, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \mathsf{fma}\left(-0.375, a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), \frac{c \cdot -0.5}{b}\right)\right)\right) \]

  7. Applied egg-rr1.6

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right)} \cdot 6.328125\right) \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot \frac{1}{b}\right)\right), \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \mathsf{fma}\left(-0.375, a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), \frac{c \cdot -0.5}{b}\right)\right)\right) \]

  8. Final simplification1.6

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right) \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot \frac{1}{b}\right)\right), \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \mathsf{fma}\left(-0.375, a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), \frac{c \cdot -0.5}{b}\right)\right)\right) \]

Alternatives

Alternative 1
Error2.1
Cost20672
\[\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(\left(c \cdot c\right) \cdot a\right) \cdot \left(\frac{-0.375}{{b}^{3}} + \frac{c \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right) \]
Alternative 2
Error3.1
Cost7424
\[-0.375 \cdot \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}} + \frac{c}{b} \cdot -0.5 \]
Alternative 3
Error3.3
Cost7296
\[c \cdot \left(\frac{-0.5}{b} + \frac{c \cdot -0.375}{\frac{{b}^{3}}{a}}\right) \]
Alternative 4
Error6.5
Cost320
\[c \cdot \frac{-0.5}{b} \]
Alternative 5
Error6.5
Cost320
\[\frac{-0.5}{\frac{b}{c}} \]
Alternative 6
Error6.3
Cost320
\[\frac{c \cdot -0.5}{b} \]

Error

Reproduce

herbie shell --seed 2022241 
(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)))