Cubic critical, medium range

Average Error: 43.8 → 0.4

Time: 15.9s

Precision: binary64

Cost: 7744

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/ (* c (* a -3.0)) (* (* a 3.0) (+ b (sqrt (- (* b b) (* c (* 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) {
	return (c * (a * -3.0)) / ((a * 3.0) * (b + sqrt(((b * b) - (c * (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
    code = (c * (a * (-3.0d0))) / ((a * 3.0d0) * (b + sqrt(((b * b) - (c * (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) {
	return (c * (a * -3.0)) / ((a * 3.0) * (b + Math.sqrt(((b * b) - (c * (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):
	return (c * (a * -3.0)) / ((a * 3.0) * (b + math.sqrt(((b * b) - (c * (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)
	return Float64(Float64(c * Float64(a * -3.0)) / Float64(Float64(a * 3.0) * Float64(b + sqrt(Float64(Float64(b * b) - Float64(c * 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)
	tmp = (c * (a * -3.0)) / ((a * 3.0) * (b + sqrt(((b * b) - (c * (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_] := N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 3.0), $MachinePrecision] * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.8

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

2. Applied egg-rr 43.8

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

3. Applied egg-rr 43.2

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

4. Applied egg-rr 0.4

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

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	6.0
Cost	14788

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\\ \mathbf{if}\;\frac{t_0 - b}{a \cdot 3} \leq -650:\\ \;\;\;\;\frac{\frac{b - t_0}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	7552

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

Alternative 3
Error	6.0
Cost	832

\[\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \]

Alternative 4
Error	12.2
Cost	320

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

Alternative 5
Error	12.2
Cost	320

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

Alternative 6
Error	12.1
Cost	320

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

Reproduce

herbie shell --seed 2023031 
(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)))