Cubic critical, wide range

Average Error: 52.8 → 0.4

Time: 16.4s

Precision: binary64

Cost: 7744

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

Math

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

↓

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

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 (* c (* a -3.0))))
   (/ (/ t_0 (+ b (sqrt (+ (* b b) t_0)))) (* a 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) {
	double t_0 = c * (a * -3.0);
	return (t_0 / (b + sqrt(((b * b) + t_0)))) / (a * 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
    real(8) :: t_0
    t_0 = c * (a * (-3.0d0))
    code = (t_0 / (b + sqrt(((b * b) + t_0)))) / (a * 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) {
	double t_0 = c * (a * -3.0);
	return (t_0 / (b + Math.sqrt(((b * b) + t_0)))) / (a * 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):
	t_0 = c * (a * -3.0)
	return (t_0 / (b + math.sqrt(((b * b) + t_0)))) / (a * 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)
	t_0 = Float64(c * Float64(a * -3.0))
	return Float64(Float64(t_0 / Float64(b + sqrt(Float64(Float64(b * b) + t_0)))) / Float64(a * 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)
	t_0 = c * (a * -3.0);
	tmp = (t_0 / (b + sqrt(((b * b) + t_0)))) / (a * 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_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 52.8

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

2. Applied egg-rr 54.4

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

3. Simplified 52.9

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

   [Start] 54.4	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}}{\frac{1}{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]
   associate-/r* [<=] 54.4	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{\left(b \cdot b - \left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right) \cdot \frac{1}{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}{3 \cdot a} \]
   *-commutative [=>] 54.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(b \cdot b - \left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right) \cdot \frac{1}{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]
   *-commutative [=>] 54.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(b \cdot b - \left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right) \cdot \frac{1}{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]
   associate-*r/ [=>] 54.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)}{\color{blue}{\frac{\left(b \cdot b - \left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right) \cdot 1}{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}{3 \cdot a} \]
   associate--r- [=>] 52.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)}{\frac{\color{blue}{\left(\left(b \cdot b - b \cdot b\right) + \left(3 \cdot a\right) \cdot c\right)} \cdot 1}{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]
   *-commutative [=>] 52.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)}{\frac{\left(\left(b \cdot b - b \cdot b\right) + \color{blue}{c \cdot \left(3 \cdot a\right)}\right) \cdot 1}{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]
   *-commutative [=>] 52.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)}{\frac{\left(\left(b \cdot b - b \cdot b\right) + c \cdot \color{blue}{\left(a \cdot 3\right)}\right) \cdot 1}{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]
   *-commutative [=>] 52.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)}{\frac{\left(\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 3\right)\right) \cdot 1}{b - \sqrt{b \cdot b - \color{blue}{c \cdot \left(3 \cdot a\right)}}}}}{3 \cdot a} \]
   *-commutative [=>] 52.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)}{\frac{\left(\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 3\right)\right) \cdot 1}{b - \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 3\right)}}}}}{3 \cdot a} \]

4. Applied egg-rr 52.5

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

5. Applied egg-rr 0.5

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

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.6
Cost	7744

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

Alternative 2
Error	3.0
Cost	7488

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

Alternative 3
Error	3.0
Cost	960

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

Alternative 4
Error	6.2
Cost	320

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

Alternative 5
Error	6.0
Cost	320

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

Reproduce

herbie shell --seed 2023011 
(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)))