Cubic critical, medium range

Average Error: 43.9 → 0.2

Time: 18.4s

Precision: binary64

Cost: 7296

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

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

FPCore

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

↓

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

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) {
	return c / (-b - sqrt(((b * b) + (a * (c * -3.0)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / (-b - sqrt(((b * b) + (a * (c * (-3.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

public static double code(double a, double b, double c) {
	return c / (-b - Math.sqrt(((b * b) + (a * (c * -3.0)))));
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

↓

def code(a, b, c):
	return c / (-b - math.sqrt(((b * b) + (a * (c * -3.0)))))

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)
	return Float64(c / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

↓

function tmp = code(a, b, c)
	tmp = c / (-b - sqrt(((b * b) + (a * (c * -3.0)))));
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_] := N[(c / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.9

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

2. Simplified 43.9

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

   [Start] 43.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   remove-double-neg [<=] 43.9	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
   sub-neg [<=] 43.9	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   div-sub [=>] 44.2	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   neg-mul-1 [=>] 44.2	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-*l/ [<=] 44.0	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   distribute-frac-neg [=>] 44.0	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   fma-neg [=>] 43.1	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
   /-rgt-identity [<=] 43.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   metadata-eval [<=] 43.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   associate-/l* [<=] 43.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   *-commutative [<=] 43.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   neg-mul-1 [<=] 43.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   fma-neg [<=] 44.0	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   neg-mul-1 [=>] 44.0	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

3. Applied egg-rr 43.3

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

4. Simplified 43.3

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

   [Start] 43.3	\[ \frac{-0.3333333333333333}{\frac{a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   associate-/l* [<=] 43.3	\[ \color{blue}{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   *-commutative [=>] 43.3	\[ \frac{\color{blue}{\left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right) \cdot -0.3333333333333333}}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   associate-*l/ [<=] 43.3	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot -0.3333333333333333} \]
   *-commutative [=>] 43.3	\[ \color{blue}{-0.3333333333333333 \cdot \frac{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   associate-/r* [=>] 43.3	\[ -0.3333333333333333 \cdot \color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

5. Taylor expanded in b around 0 0.5

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

6. Applied egg-rr 40.0

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{3 \cdot c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} - 1} \]

7. Simplified 0.2

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

   [Start] 40.0	\[ e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{3 \cdot c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} - 1 \]
   expm1-def [=>] 10.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{3 \cdot c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)\right)} \]
   expm1-log1p [=>] 0.5	\[ \color{blue}{-0.3333333333333333 \cdot \frac{3 \cdot c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   *-commutative [=>] 0.5	\[ \color{blue}{\frac{3 \cdot c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot -0.3333333333333333} \]
   associate-*r/ [<=] 0.4	\[ \color{blue}{\left(3 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} \cdot -0.3333333333333333 \]
   metadata-eval [<=] 0.4	\[ \left(\color{blue}{\frac{-3}{-1}} \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right) \cdot -0.3333333333333333 \]
   times-frac [<=] 0.5	\[ \color{blue}{\frac{-3 \cdot c}{-1 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \cdot -0.3333333333333333 \]
   *-commutative [<=] 0.5	\[ \frac{\color{blue}{c \cdot -3}}{-1 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot -0.3333333333333333 \]
   neg-mul-1 [<=] 0.5	\[ \frac{c \cdot -3}{\color{blue}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \cdot -0.3333333333333333 \]
   associate-/r/ [<=] 0.5	\[ \color{blue}{\frac{c \cdot -3}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{-0.3333333333333333}}} \]
   associate-/l* [=>] 0.4	\[ \color{blue}{\frac{c}{\frac{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{-0.3333333333333333}}{-3}}} \]
   associate-/l/ [=>] 0.2	\[ \frac{c}{\color{blue}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{-3 \cdot -0.3333333333333333}}} \]
   metadata-eval [=>] 0.2	\[ \frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 0.2	\[ \frac{c}{\color{blue}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   distribute-neg-in [=>] 0.2	\[ \frac{c}{\color{blue}{\left(-b\right) + \left(-\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   unsub-neg [=>] 0.2	\[ \frac{c}{\color{blue}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

8. Applied egg-rr 0.2

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

9. Final simplification 0.2

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

Alternatives

Alternative 1
Error	5.6
Cost	832

\[\frac{c}{1.5 \cdot \frac{c \cdot a}{b} + b \cdot -2} \]

Alternative 2
Error	12.0
Cost	320

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

Reproduce

herbie shell --seed 2023187 
(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)))