?

Average Accuracy: 31.5% → 99.7%
Time: 14.3s
Precision: binary64
Cost: 7296

?

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\frac{c}{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}} \]
(FPCore (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)))))))
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)))));
}

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

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)))));
}

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)))))

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

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

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]

\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)}}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 31.5%

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

  2. Simplified31.5%

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

    [Start]31.5

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

    *-lft-identity [<=]31.5

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

    metadata-eval [<=]31.5

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

    times-frac [<=]31.5

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

    neg-mul-1 [<=]31.5

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

    distribute-rgt-neg-in [=>]31.5

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

    times-frac [=>]31.5

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

    *-commutative [=>]31.5

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

  3. Applied egg-rr32.4%

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

  4. Simplified32.4%

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

    [Start]32.4

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

    times-frac [=>]32.4

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

  5. Taylor expanded in b around 0 99.1%

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

  6. Applied egg-rr37.0%

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

  7. Simplified99.7%

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

    [Start]37.0

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

    expm1-def [=>]84.2

    \[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 \cdot \frac{c \cdot a}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)\right)} \]

    expm1-log1p [=>]99.6

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

    associate-*r/ [=>]99.6

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

    *-commutative [<=]99.6

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

    associate-/r* [<=]99.4

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

    *-commutative [=>]99.4

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

    neg-mul-1 [<=]99.4

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

    distribute-frac-neg [=>]99.4

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

    mul-1-neg [<=]99.4

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

    metadata-eval [<=]99.4

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

    times-frac [<=]99.4

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

    *-commutative [=>]99.4

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

    times-frac [=>]99.3

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

    associate-/r* [=>]99.3

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

    times-frac [<=]99.4

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

  8. Applied egg-rr99.7%

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

  9. Final simplification99.7%

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

Alternatives

Alternative 1
Accuracy90.9%
Cost832
\[\frac{c}{1.5 \cdot \frac{c \cdot a}{b} + b \cdot -2} \]
Alternative 2
Accuracy80.9%
Cost320
\[c \cdot \frac{-0.5}{b} \]
Alternative 3
Accuracy80.9%
Cost320
\[\frac{-0.5}{\frac{b}{c}} \]
Alternative 4
Accuracy81.2%
Cost320
\[\frac{c \cdot -0.5}{b} \]

Error

Reproduce?

herbie shell --seed 2023129 
(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)))