Quadratic roots, medium range

Percentage Accurate: 32.1% → 95.2%

Time: 10.5s

Alternatives: 6

Speedup: 29.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 32.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.25, \frac{\frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{20}}}{{b}^{7}}, \frac{-2 \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (fma
   -0.25
   (/ (/ (pow (* c a) 4.0) (/ a 20.0)) (pow b 7.0))
   (- (/ (* -2.0 (* c (* (* c a) (* c a)))) (pow b 5.0)) (/ c b)))
  (* a (/ (* c c) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return fma(-0.25, ((pow((c * a), 4.0) / (a / 20.0)) / pow(b, 7.0)), (((-2.0 * (c * ((c * a) * (c * a)))) / pow(b, 5.0)) - (c / b))) - (a * ((c * c) / pow(b, 3.0)));
}

Julia

function code(a, b, c)
	return Float64(fma(-0.25, Float64(Float64((Float64(c * a) ^ 4.0) / Float64(a / 20.0)) / (b ^ 7.0)), Float64(Float64(Float64(-2.0 * Float64(c * Float64(Float64(c * a) * Float64(c * a)))) / (b ^ 5.0)) - Float64(c / b))) - Float64(a * Float64(Float64(c * c) / (b ^ 3.0))))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.25 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[(a / 20.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * N[(c * N[(N[(c * a), $MachinePrecision] * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 32.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   2. +-commutative 32.7%

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

   3. unsub-neg 32.7%

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

   4. fma-neg 32.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

   5. associate-*l* 32.8%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b}{a \cdot 2} \]

   6. *-commutative 32.8%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b}{a \cdot 2} \]

   7. distribute-rgt-neg-in 32.8%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{a \cdot 2} \]

   8. metadata-eval 32.8%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b}{a \cdot 2} \]

3. Simplified 32.8%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}} \]
4. Step-by-step derivation

   1. fma-udef 32.7%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b}{a \cdot 2} \]

   2. *-commutative 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

   3. metadata-eval 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{\left(-4\right)} \cdot \left(a \cdot c\right)} - b}{a \cdot 2} \]

   4. cancel-sign-sub-inv 32.7%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

   5. associate-*l* 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

   6. *-un-lft-identity 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

   7. prod-diff 32.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b}{a \cdot 2} \]

5. Applied egg-rr 32.7%

   \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b}{a \cdot 2} \]
6. Step-by-step derivation

   1. *-rgt-identity 32.7%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

   2. fma-neg 32.7%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

   3. fma-udef 32.7%

      \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b}{a \cdot 2} \]

   4. *-rgt-identity 32.7%

      \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

   5. *-rgt-identity 32.7%

      \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b}{a \cdot 2} \]

   6. associate--r- 32.7%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -4\right) - \left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right)\right)}} - b}{a \cdot 2} \]

   7. associate--r+ 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right) - a \cdot \left(c \cdot -4\right)\right) - a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

   8. +-inverses 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2} \]

   9. neg-sub0 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

   10. associate-*r* 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -4}\right)} - b}{a \cdot 2} \]

   11. distribute-rgt-neg-in 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot \left(--4\right)}} - b}{a \cdot 2} \]

   12. metadata-eval 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot \color{blue}{4}} - b}{a \cdot 2} \]

   13. *-commutative 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(c \cdot a\right)} \cdot 4} - b}{a \cdot 2} \]

   14. associate-*r* 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2} \]

7. Simplified 32.7%

   \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2} \]

8. Taylor expanded in b around inf 94.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

10. Simplified 94.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{\frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{20}}}{{b}^{7}}, \frac{-2 \cdot \left(c \cdot {\left(c \cdot a\right)}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a} \]
11. Step-by-step derivation

    1. unpow2 94.0%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{20}}}{{b}^{7}}, \frac{-2 \cdot \left(c \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a \]

12. Applied egg-rr 94.0%

    \[\leadsto \mathsf{fma}\left(-0.25, \frac{\frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{20}}}{{b}^{7}}, \frac{-2 \cdot \left(c \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a \]

13. Final simplification 94.0%

    \[\leadsto \mathsf{fma}\left(-0.25, \frac{\frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{20}}}{{b}^{7}}, \frac{-2 \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}} \]

Alternative 2: 93.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{-2 \cdot \left(c \cdot {\left(c \cdot a\right)}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (- (/ (* -2.0 (* c (pow (* c a) 2.0))) (pow b 5.0)) (/ c b))
  (* a (/ (* c c) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (((-2.0 * (c * pow((c * a), 2.0))) / pow(b, 5.0)) - (c / b)) - (a * ((c * c) / pow(b, 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 = ((((-2.0d0) * (c * ((c * a) ** 2.0d0))) / (b ** 5.0d0)) - (c / b)) - (a * ((c * c) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (((-2.0 * (c * Math.pow((c * a), 2.0))) / Math.pow(b, 5.0)) - (c / b)) - (a * ((c * c) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (((-2.0 * (c * math.pow((c * a), 2.0))) / math.pow(b, 5.0)) - (c / b)) - (a * ((c * c) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(-2.0 * Float64(c * (Float64(c * a) ^ 2.0))) / (b ^ 5.0)) - Float64(c / b)) - Float64(a * Float64(Float64(c * c) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((-2.0 * (c * ((c * a) ^ 2.0))) / (b ^ 5.0)) - (c / b)) - (a * ((c * c) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-2.0 * N[(c * N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 32.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   2. +-commutative 32.7%

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

   3. unsub-neg 32.7%

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

   4. fma-neg 32.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

   5. associate-*l* 32.8%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b}{a \cdot 2} \]

   6. *-commutative 32.8%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b}{a \cdot 2} \]

   7. distribute-rgt-neg-in 32.8%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{a \cdot 2} \]

   8. metadata-eval 32.8%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b}{a \cdot 2} \]

3. Simplified 32.8%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}} \]
4. Step-by-step derivation

   1. fma-udef 32.7%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b}{a \cdot 2} \]

   2. *-commutative 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

   3. metadata-eval 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{\left(-4\right)} \cdot \left(a \cdot c\right)} - b}{a \cdot 2} \]

   4. cancel-sign-sub-inv 32.7%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

   5. associate-*l* 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

   6. *-un-lft-identity 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

   7. prod-diff 32.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b}{a \cdot 2} \]

5. Applied egg-rr 32.7%

   \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b}{a \cdot 2} \]
6. Step-by-step derivation

   1. *-rgt-identity 32.7%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

   2. fma-neg 32.7%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

   3. fma-udef 32.7%

      \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b}{a \cdot 2} \]

   4. *-rgt-identity 32.7%

      \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

   5. *-rgt-identity 32.7%

      \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b}{a \cdot 2} \]

   6. associate--r- 32.7%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -4\right) - \left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right)\right)}} - b}{a \cdot 2} \]

   7. associate--r+ 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right) - a \cdot \left(c \cdot -4\right)\right) - a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

   8. +-inverses 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2} \]

   9. neg-sub0 32.7%

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

   10. associate-*r* 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -4}\right)} - b}{a \cdot 2} \]

   11. distribute-rgt-neg-in 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot \left(--4\right)}} - b}{a \cdot 2} \]

   12. metadata-eval 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot \color{blue}{4}} - b}{a \cdot 2} \]

   13. *-commutative 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(c \cdot a\right)} \cdot 4} - b}{a \cdot 2} \]

   14. associate-*r* 32.7%

       \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2} \]

7. Simplified 32.7%

   \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2} \]

8. Taylor expanded in b around inf 92.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]
9. Step-by-step derivation

   1. +-commutative 92.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   2. mul-1-neg 92.3%

      \[\leadsto \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

   3. unsub-neg 92.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

10. Simplified 92.3%

    \[\leadsto \color{blue}{\left(\frac{-2 \cdot \left(c \cdot {\left(c \cdot a\right)}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a} \]

11. Final simplification 92.3%

    \[\leadsto \left(\frac{-2 \cdot \left(c \cdot {\left(c \cdot a\right)}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}} \]

Alternative 3: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(a \cdot 4\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + t_1} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* c (* a 4.0)))) (t_1 (sqrt t_0)))
   (if (<= (/ (- t_1 b) (* a 2.0)) -500.0)
     (* (/ (- t_0 (* b b)) (+ b t_1)) (/ 0.5 a))
     (- (/ (- c) b) (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - (c * (a * 4.0));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -500.0) {
		tmp = ((t_0 - (b * b)) / (b + t_1)) * (0.5 / a);
	} else {
		tmp = (-c / b) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (c * (a * 4.0d0))
    t_1 = sqrt(t_0)
    if (((t_1 - b) / (a * 2.0d0)) <= (-500.0d0)) then
        tmp = ((t_0 - (b * b)) / (b + t_1)) * (0.5d0 / a)
    else
        tmp = (-c / b) - ((c * c) / ((b ** 3.0d0) / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * b) - (c * (a * 4.0));
	double t_1 = Math.sqrt(t_0);
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -500.0) {
		tmp = ((t_0 - (b * b)) / (b + t_1)) * (0.5 / a);
	} else {
		tmp = (-c / b) - ((c * c) / (Math.pow(b, 3.0) / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) - (c * (a * 4.0))
	t_1 = math.sqrt(t_0)
	tmp = 0
	if ((t_1 - b) / (a * 2.0)) <= -500.0:
		tmp = ((t_0 - (b * b)) / (b + t_1)) * (0.5 / a)
	else:
		tmp = (-c / b) - ((c * c) / (math.pow(b, 3.0) / a))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -500.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + t_1)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) - (c * (a * 4.0));
	t_1 = sqrt(t_0);
	tmp = 0.0;
	if (((t_1 - b) / (a * 2.0)) <= -500.0)
		tmp = ((t_0 - (b * b)) / (b + t_1)) * (0.5 / a);
	else
		tmp = (-c / b) - ((c * c) / ((b ^ 3.0) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -500

   1. Initial program 77.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-rgt-identity 77.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 77.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 77.6%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 77.7%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 77.7%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 77.7%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 77.7%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 77.7%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 77.7%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 77.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 77.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 77.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 77.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 77.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 77.6%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b}{a \cdot 2} \]

      2. *-commutative 77.6%

         \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

      3. metadata-eval 77.6%

         \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{\left(-4\right)} \cdot \left(a \cdot c\right)} - b}{a \cdot 2} \]

      4. cancel-sign-sub-inv 77.6%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

      5. associate-*l* 77.6%

         \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      6. *-un-lft-identity 77.6%

         \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. prod-diff 77.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b}{a \cdot 2} \]

   5. Applied egg-rr 77.8%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 77.8%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

      2. fma-neg 77.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

      3. fma-udef 77.4%

         \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b}{a \cdot 2} \]

      4. *-rgt-identity 77.4%

         \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

      5. *-rgt-identity 77.4%

         \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b}{a \cdot 2} \]

      6. associate--r- 77.6%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -4\right) - \left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right)\right)}} - b}{a \cdot 2} \]

      7. associate--r+ 77.6%

         \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right) - a \cdot \left(c \cdot -4\right)\right) - a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

      8. +-inverses 77.6%

         \[\leadsto \frac{\sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2} \]

      9. neg-sub0 77.6%

         \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

      10. associate-*r* 77.6%

          \[\leadsto \frac{\sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -4}\right)} - b}{a \cdot 2} \]

      11. distribute-rgt-neg-in 77.6%

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot \left(--4\right)}} - b}{a \cdot 2} \]

      12. metadata-eval 77.6%

          \[\leadsto \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot \color{blue}{4}} - b}{a \cdot 2} \]

      13. *-commutative 77.6%

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(c \cdot a\right)} \cdot 4} - b}{a \cdot 2} \]

      14. associate-*r* 77.6%

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2} \]

   7. Simplified 77.7%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b\right) \cdot \frac{0.5}{a} \]
   8. Step-by-step derivation

      1. flip-- 77.8%

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b \cdot b}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + b}} \cdot \frac{0.5}{a} \]

      2. add-sqr-sqrt 79.5%

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)} - b \cdot b}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + b} \cdot \frac{0.5}{a} \]

   9. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - b \cdot b}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + b}} \cdot \frac{0.5}{a} \]

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 27.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-rgt-identity 27.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 27.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 27.6%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 27.6%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 27.6%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 27.6%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 27.7%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 27.7%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 27.7%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 27.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 27.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 27.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 27.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 27.7%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 27.7%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]

   4. Taylor expanded in b around inf 91.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 91.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      2. mul-1-neg 91.5%

         \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      3. unsub-neg 91.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. mul-1-neg 91.5%

         \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. distribute-neg-frac 91.5%

         \[\leadsto \color{blue}{\frac{-c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. associate-/l* 91.5%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}} \]

      7. unpow2 91.5%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}} \]

   6. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 4: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\\ \mathbf{if}\;\frac{t_0}{a \cdot 2} \leq -20000000:\\ \;\;\;\;\frac{0.5}{a} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))
   (if (<= (/ t_0 (* a 2.0)) -20000000.0)
     (* (/ 0.5 a) t_0)
     (- (/ (- c) b) (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0)))) - b;
	double tmp;
	if ((t_0 / (a * 2.0)) <= -20000000.0) {
		tmp = (0.5 / a) * t_0;
	} else {
		tmp = (-c / b) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Fortran

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
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0)))) - b
    if ((t_0 / (a * 2.0d0)) <= (-20000000.0d0)) then
        tmp = (0.5d0 / a) * t_0
    else
        tmp = (-c / b) - ((c * c) / ((b ** 3.0d0) / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0)))) - b;
	double tmp;
	if ((t_0 / (a * 2.0)) <= -20000000.0) {
		tmp = (0.5 / a) * t_0;
	} else {
		tmp = (-c / b) - ((c * c) / (Math.pow(b, 3.0) / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0)))) - b
	tmp = 0
	if (t_0 / (a * 2.0)) <= -20000000.0:
		tmp = (0.5 / a) * t_0
	else:
		tmp = (-c / b) - ((c * c) / (math.pow(b, 3.0) / a))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b)
	tmp = 0.0
	if (Float64(t_0 / Float64(a * 2.0)) <= -20000000.0)
		tmp = Float64(Float64(0.5 / a) * t_0);
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0)))) - b;
	tmp = 0.0;
	if ((t_0 / (a * 2.0)) <= -20000000.0)
		tmp = (0.5 / a) * t_0;
	else
		tmp = (-c / b) - ((c * c) / ((b ^ 3.0) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -20000000.0], N[(N[(0.5 / a), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2e7

   1. Initial program 84.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-rgt-identity 84.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 84.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 84.7%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 84.7%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 84.7%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 84.7%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 84.9%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 84.9%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 84.9%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 84.9%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 84.9%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 84.9%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 84.9%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 84.9%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 84.9%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 84.7%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b}{a \cdot 2} \]

      2. *-commutative 84.7%

         \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

      3. metadata-eval 84.7%

         \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{\left(-4\right)} \cdot \left(a \cdot c\right)} - b}{a \cdot 2} \]

      4. cancel-sign-sub-inv 84.7%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

      5. associate-*l* 84.7%

         \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      6. *-un-lft-identity 84.7%

         \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. prod-diff 84.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b}{a \cdot 2} \]

   5. Applied egg-rr 84.9%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 84.8%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

      2. fma-neg 84.5%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

      3. fma-udef 84.5%

         \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b}{a \cdot 2} \]

      4. *-rgt-identity 84.5%

         \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b}{a \cdot 2} \]

      5. *-rgt-identity 84.5%

         \[\leadsto \frac{\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b}{a \cdot 2} \]

      6. associate--r- 84.7%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -4\right) - \left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right)\right)}} - b}{a \cdot 2} \]

      7. associate--r+ 84.7%

         \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right) - a \cdot \left(c \cdot -4\right)\right) - a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

      8. +-inverses 84.7%

         \[\leadsto \frac{\sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2} \]

      9. neg-sub0 84.7%

         \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

      10. associate-*r* 84.7%

          \[\leadsto \frac{\sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -4}\right)} - b}{a \cdot 2} \]

      11. distribute-rgt-neg-in 84.7%

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot \left(--4\right)}} - b}{a \cdot 2} \]

      12. metadata-eval 84.7%

          \[\leadsto \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot \color{blue}{4}} - b}{a \cdot 2} \]

      13. *-commutative 84.7%

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(c \cdot a\right)} \cdot 4} - b}{a \cdot 2} \]

      14. associate-*r* 84.7%

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2} \]

   7. Simplified 84.7%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b\right) \cdot \frac{0.5}{a} \]

   if -2e7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 29.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-rgt-identity 29.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 29.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 29.9%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 29.9%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 29.9%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 29.9%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 30.0%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 30.0%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 30.0%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 30.0%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 30.0%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 30.0%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 30.0%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 30.0%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 30.0%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]

   4. Taylor expanded in b around inf 90.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 90.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      2. mul-1-neg 90.5%

         \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      3. unsub-neg 90.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. mul-1-neg 90.5%

         \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. distribute-neg-frac 90.5%

         \[\leadsto \color{blue}{\frac{-c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. associate-/l* 90.5%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}} \]

      7. unpow2 90.5%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}} \]

   6. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -20000000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 5: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return (-c / b) - ((c * c) / (pow(b, 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 = (-c / b) - ((c * c) / ((b ** 3.0d0) / a))
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-c) / b) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)))
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-rgt-identity 32.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

   2. metadata-eval 32.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

   3. associate-/l* 32.7%

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

   4. associate-*r/ 32.7%

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

   5. +-commutative 32.7%

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

   6. unsub-neg 32.7%

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

   7. fma-neg 32.8%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   8. associate-*l* 32.8%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   9. *-commutative 32.8%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   10. distribute-rgt-neg-in 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   11. metadata-eval 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   12. associate-/r* 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

   13. metadata-eval 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

   14. metadata-eval 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

3. Simplified 32.8%

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]

4. Taylor expanded in b around inf 88.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
5. Step-by-step derivation

   1. +-commutative 88.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   2. mul-1-neg 88.6%

      \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

   3. unsub-neg 88.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   4. mul-1-neg 88.6%

      \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   5. distribute-neg-frac 88.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   6. associate-/l* 88.6%

      \[\leadsto \frac{-c}{b} - \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}} \]

   7. unpow2 88.6%

      \[\leadsto \frac{-c}{b} - \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}} \]

6. Simplified 88.6%

   \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]

7. Final simplification 88.6%

   \[\leadsto \frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

Alternative 6: 80.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (- c) b))

C

double code(double a, double b, double c) {
	return -c / b;
}

Fortran

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
end function

Java

public static double code(double a, double b, double c) {
	return -c / b;
}

Python

def code(a, b, c):
	return -c / b

Julia

function code(a, b, c)
	return Float64(Float64(-c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = -c / b;
end

Wolfram

code[a_, b_, c_] := N[((-c) / b), $MachinePrecision]

Derivation

1. Initial program 32.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-rgt-identity 32.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

   2. metadata-eval 32.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

   3. associate-/l* 32.7%

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

   4. associate-*r/ 32.7%

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

   5. +-commutative 32.7%

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

   6. unsub-neg 32.7%

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

   7. fma-neg 32.8%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   8. associate-*l* 32.8%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   9. *-commutative 32.8%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   10. distribute-rgt-neg-in 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   11. metadata-eval 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   12. associate-/r* 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

   13. metadata-eval 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

   14. metadata-eval 32.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

3. Simplified 32.8%

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]

4. Taylor expanded in b around inf 79.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
5. Step-by-step derivation

   1. mul-1-neg 79.6%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   2. distribute-neg-frac 79.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

6. Simplified 79.6%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]

7. Final simplification 79.6%

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

Reproduce

herbie shell --seed 2023182 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))