Cubic critical, wide range

Average Error: 52.5 → 1.7

Time: 8.2s

Precision: binary64

Cost: 47296

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

↓

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

FPCore

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

↓

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

C

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

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

↓

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

↓

code[a_, b_, c_] := N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(c * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(c * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5

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

2. Taylor expanded in a around 0 1.5

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

3. Simplified 1.7

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

4. Applied egg-rr 1.7

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

5. Applied egg-rr 1.7

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

6. Taylor expanded in c around 0 1.7

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

7. Simplified 1.7

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

8. Final simplification 1.7

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

Alternatives

Alternative 1
Error	2.0
Cost	33536

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

Alternative 2
Error	2.0
Cost	27328

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

Alternative 3
Error	2.0
Cost	20672

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

Alternative 4
Error	3.0
Cost	7424

\[-0.375 \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot {b}^{-3}\right)\right) + -0.5 \cdot \frac{c}{b} \]

Alternative 5
Error	3.0
Cost	7232

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

Alternative 6
Error	3.4
Cost	1472

\[\frac{\left(a \cdot c\right) \cdot \left(\frac{-1.5}{b} + \frac{-1.125}{\frac{b}{c} \cdot \frac{b \cdot b}{a}}\right)}{a \cdot 3} \]

Alternative 7
Error	6.4
Cost	320

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

Alternative 8
Error	6.4
Cost	320

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

Alternative 9
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2022221 
(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)))