Cubic critical, wide range

Average Error: 52.5 → 0.5

Time: 19.4s

Precision: binary64

Cost: 39808

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\\ \frac{\frac{-c}{t_0}}{t_0} \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ b (sqrt (fma a (* c -3.0) (* b b)))))))
   (/ (/ (- c) t_0) t_0)))

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) {
	double t_0 = sqrt((b + sqrt(fma(a, (c * -3.0), (b * b)))));
	return (-c / t_0) / t_0;
}

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)
	t_0 = sqrt(Float64(b + sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))))
	return Float64(Float64(Float64(-c) / t_0) / t_0)
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_] := Block[{t$95$0 = N[Sqrt[N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[((-c) / t$95$0), $MachinePrecision] / t$95$0), $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. Simplified 52.5

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

   [Start] 52.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   remove-double-neg [<=] 52.5	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
   sub-neg [<=] 52.5	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   div-sub [=>] 52.6	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   neg-mul-1 [=>] 52.6	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-*l/ [<=] 52.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   distribute-frac-neg [=>] 52.4	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   fma-neg [=>] 51.6	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
   /-rgt-identity [<=] 51.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   metadata-eval [<=] 51.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   associate-/l* [<=] 51.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   *-commutative [<=] 51.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   neg-mul-1 [<=] 51.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   fma-neg [<=] 52.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   neg-mul-1 [=>] 52.4	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

3. Applied egg-rr 52.1

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

4. Taylor expanded in a around 0 0.5

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

5. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\frac{-1 \cdot c}{\sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}{\sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   mul-1-neg [=>] 0.5	\[ \frac{\frac{\color{blue}{-c}}{\sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}{\sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.6
Cost	14400

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

Alternative 2
Error	2.2
Cost	8000

\[\frac{-0.3333333333333333}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c}, -0.5 \cdot \frac{a}{b}\right) - c \cdot \frac{\frac{0.375 \cdot \left(a \cdot a\right)}{b \cdot b}}{b}} \]

Alternative 3
Error	2.0
Cost	8000

\[\frac{-0.3333333333333333}{\mathsf{fma}\left(a, \frac{-0.5}{b}, \frac{0.6666666666666666}{\frac{c}{b}}\right) - c \cdot \frac{\frac{0.375 \cdot \left(a \cdot a\right)}{b \cdot b}}{b}} \]

Alternative 4
Error	2.2
Cost	7936

\[\frac{-0.3333333333333333}{\left(a \cdot \frac{-0.5}{b} + 0.6666666666666666 \cdot \frac{b}{c}\right) - c \cdot \left(0.375 \cdot \frac{a \cdot a}{{b}^{3}}\right)} \]

Alternative 5
Error	3.2
Cost	7104

\[\frac{-0.3333333333333333}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c}, -0.5 \cdot \frac{a}{b}\right)} \]

Alternative 6
Error	3.2
Cost	832

\[\frac{-0.3333333333333333}{-0.5 \cdot \frac{a}{b} + 0.6666666666666666 \cdot \frac{b}{c}} \]

Alternative 7
Error	6.5
Cost	320

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

Alternative 8
Error	6.5
Cost	320

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

Alternative 9
Error	6.3
Cost	320

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

Reproduce

herbie shell --seed 2023034 
(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)))