Cubic critical, narrow range

Average Error: 28.7 → 0.4

Time: 18.7s

Precision: binary64

Cost: 7680

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

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

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 a) (- (/ (- b) a) (/ (sqrt (+ (* b b) (* a (* c -3.0)))) a))))

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

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 / a) / ((-b / a) - (sqrt(((b * b) + (a * (c * (-3.0d0))))) / a))
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 / a) / ((-b / a) - (Math.sqrt(((b * b) + (a * (c * -3.0)))) / a));
}

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 / a) / ((-b / a) - (math.sqrt(((b * b) + (a * (c * -3.0)))) / a))

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(Float64(c / a) / Float64(Float64(Float64(-b) / a) - Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0)))) / a)))
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 / a) / ((-b / a) - (sqrt(((b * b) + (a * (c * -3.0)))) / a));
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[(N[(c / a), $MachinePrecision] / N[(N[((-b) / a), $MachinePrecision] - N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.7

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

2. Simplified 28.7

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

   [Start] 28.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 28.7	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 28.7	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 28.7	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 28.7	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 28.7	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 28.7	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 28.7	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

3. Applied egg-rr 29.3

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

4. Simplified 28.7

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

   [Start] 29.3	\[ \frac{\left(\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 29.3	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-0.3333333333333333}}} \]
   associate-/r/ [=>] 29.3	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333} \]

5. Taylor expanded in b around 0 0.6

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

6. Applied egg-rr 24.3

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

7. Simplified 0.4

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

   [Start] 24.3	\[ e^{\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} - 1 \]
   expm1-def [=>] 9.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)\right)} \]
   expm1-log1p [=>] 0.4	\[ \color{blue}{\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   /-rgt-identity [<=] 0.4	\[ \frac{\color{blue}{\frac{\frac{c}{a} \cdot -1}{1}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 0.4	\[ \frac{\color{blue}{\frac{\frac{c}{a}}{\frac{1}{-1}}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{\frac{c}{a}}{\color{blue}{-1}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l/ [=>] 0.4	\[ \color{blue}{\frac{\frac{c}{a}}{\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -1}} \]
   *-commutative [<=] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{-1 \cdot \left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   neg-mul-1 [<=] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{-\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   neg-sub0 [=>] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{0 - \left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   associate--r+ [=>] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{\left(0 - \frac{b}{a}\right) - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   neg-sub0 [<=] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{\left(-\frac{b}{a}\right)} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   distribute-neg-frac [=>] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{\frac{-b}{a}} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]

8. Applied egg-rr 0.4

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

9. Final simplification 0.4

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

Alternatives

Alternative 1
Error	6.6
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.5:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{1}{\left(\frac{\left(c \cdot a\right) \cdot 0.375}{b \cdot b} \cdot \frac{-1}{b} + 0.6666666666666666 \cdot \frac{b}{c \cdot a}\right) + 0.5 \cdot \frac{-1}{b}}}{a}\\ \end{array} \]

Alternative 2
Error	7.7
Cost	1984

\[-0.3333333333333333 \cdot \frac{\frac{1}{\left(0.6666666666666666 \cdot \frac{b}{c \cdot a} + \frac{c \cdot a}{b \cdot b} \cdot \frac{-0.375}{b}\right) + 0.5 \cdot \frac{-1}{b}}}{a} \]

Alternative 3
Error	11.4
Cost	960

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

Alternative 4
Error	22.7
Cost	320

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

Reproduce

herbie shell --seed 2023083 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))