Quadratic roots, medium range

Percentage Accurate: 31.7% → 95.3%

Time: 13.0s

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: 31.7% 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: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (-
   (fma
    -2.0
    (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
    (/ (* -5.0 (/ (* (pow c 4.0) (pow a 3.0)) (pow b 6.0))) b))
   (/ c b))
  (/ (* c c) (/ (pow b 3.0) a))))

C

double code(double a, double b, double c) {
	return (fma(-2.0, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), ((-5.0 * ((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 6.0))) / b)) - (c / b)) - ((c * c) / (pow(b, 3.0) / a));
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(-2.0, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), Float64(Float64(-5.0 * Float64(Float64((c ^ 4.0) * (a ^ 3.0)) / (b ^ 6.0))) / b)) - Float64(c / b)) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-5.0 * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

   1. /-rgt-identity 28.5%

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

   2. metadata-eval 28.5%

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

   3. associate-/l* 28.5%

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

   4. associate-*r/ 28.5%

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

   5. +-commutative 28.5%

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

   6. unsub-neg 28.5%

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

   7. fma-neg 28.7%

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

   8. associate-*l* 28.7%

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

   9. *-commutative 28.7%

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

   10. distribute-rgt-neg-in 28.7%

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

   11. metadata-eval 28.7%

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

   12. associate-/r* 28.7%

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

   13. metadata-eval 28.7%

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

   14. metadata-eval 28.7%

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

3. Simplified 28.7%

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

4. Taylor expanded in a around 0 96.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 96.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   2. mul-1-neg 96.6%

      \[\leadsto \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

   3. unsub-neg 96.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

6. Simplified 96.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{\left(-0.25 \cdot {a}^{3}\right) \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]

7. Taylor expanded in c around 0 96.6%

   \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{\color{blue}{-5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

8. Final simplification 96.6%

   \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

Alternative 2: 93.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.5%

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

   1. /-rgt-identity 28.5%

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

   2. metadata-eval 28.5%

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

   3. associate-/l* 28.5%

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

   4. associate-*r/ 28.5%

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

   5. +-commutative 28.5%

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

   6. unsub-neg 28.5%

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

   7. fma-neg 28.7%

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

   8. associate-*l* 28.7%

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

   9. *-commutative 28.7%

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

   10. distribute-rgt-neg-in 28.7%

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

   11. metadata-eval 28.7%

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

   12. associate-/r* 28.7%

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

   13. metadata-eval 28.7%

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

   14. metadata-eval 28.7%

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

3. Simplified 28.7%

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

   1. fma-udef 28.5%

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

   2. *-commutative 28.5%

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

   3. metadata-eval 28.5%

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

   4. cancel-sign-sub-inv 28.5%

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

   5. associate-*l* 28.5%

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

   6. *-un-lft-identity 28.5%

      \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

   7. prod-diff 28.7%

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

5. Applied egg-rr 28.6%

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

   1. *-rgt-identity 28.6%

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

   2. fma-neg 28.6%

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

   3. fma-udef 28.6%

      \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

   4. *-rgt-identity 28.6%

      \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   5. *-rgt-identity 28.6%

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

   6. associate--r- 28.5%

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

   7. associate--r+ 28.5%

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

   8. +-inverses 28.5%

      \[\leadsto \left(\sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a} \]

   9. neg-sub0 28.5%

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

   10. associate-*r* 28.5%

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

   11. distribute-rgt-neg-in 28.5%

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

   12. metadata-eval 28.5%

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

   13. *-commutative 28.5%

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

   14. associate-*r* 28.5%

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

7. Simplified 28.5%

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

8. Taylor expanded in b around inf 95.0%

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

   1. +-commutative 95.0%

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

   2. mul-1-neg 95.0%

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

   3. unsub-neg 95.0%

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

   4. +-commutative 95.0%

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

   5. mul-1-neg 95.0%

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

   6. unsub-neg 95.0%

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

   7. associate-/l* 95.0%

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

   8. unpow2 95.0%

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

   9. associate-/l* 95.0%

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

   10. unpow2 95.0%

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

10. Simplified 95.0%

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

11. Final simplification 95.0%

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

Alternative 3: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(a \cdot 4\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -25000:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + t_1} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

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)) -25000.0)
     (* (/ (- 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)) <= -25000.0) {
		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)) <= (-25000.0d0)) 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)) <= -25000.0) {
		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)) <= -25000.0:
		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)) <= -25000.0)
		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)) <= -25000.0)
		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], -25000.0], 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)) < -25000

   1. Initial program 83.5%

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

      1. /-rgt-identity 83.5%

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

      2. metadata-eval 83.5%

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

      3. associate-/l* 83.5%

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

      4. associate-*r/ 83.7%

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

      5. +-commutative 83.7%

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

      6. unsub-neg 83.7%

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

      7. fma-neg 83.9%

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

      8. associate-*l* 83.9%

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

      9. *-commutative 83.9%

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

      10. distribute-rgt-neg-in 83.9%

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

      11. metadata-eval 83.9%

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

      12. associate-/r* 83.9%

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

      13. metadata-eval 83.9%

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

      14. metadata-eval 83.9%

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

   3. Simplified 83.9%

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

      1. fma-udef 83.7%

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

      2. *-commutative 83.7%

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

      3. metadata-eval 83.7%

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

      4. cancel-sign-sub-inv 83.7%

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

      5. associate-*l* 83.7%

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

      6. *-un-lft-identity 83.7%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      7. prod-diff 83.9%

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

   5. Applied egg-rr 83.6%

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

      1. *-rgt-identity 83.6%

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

      2. fma-neg 83.9%

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

      3. fma-udef 83.9%

         \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

      4. *-rgt-identity 83.9%

         \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      5. *-rgt-identity 83.9%

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

      6. associate--r- 83.7%

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

      7. associate--r+ 83.7%

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

      8. +-inverses 83.7%

         \[\leadsto \left(\sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a} \]

      9. neg-sub0 83.7%

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

      10. associate-*r* 83.7%

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

      11. distribute-rgt-neg-in 83.7%

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

      12. metadata-eval 83.7%

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

      13. *-commutative 83.7%

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

      14. associate-*r* 83.7%

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

   7. Simplified 83.7%

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

      1. flip-- 83.5%

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

      2. add-sqr-sqrt 85.5%

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

      3. *-commutative 85.5%

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

      4. *-commutative 85.5%

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

   9. Applied egg-rr 85.5%

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

   if -25000 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 25.3%

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

      1. /-rgt-identity 25.3%

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

      2. metadata-eval 25.3%

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

      3. associate-/l* 25.3%

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

      4. associate-*r/ 25.3%

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

      5. +-commutative 25.3%

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

      6. unsub-neg 25.3%

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

      7. fma-neg 25.5%

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

      8. associate-*l* 25.5%

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

      9. *-commutative 25.5%

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

      10. distribute-rgt-neg-in 25.5%

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

      11. metadata-eval 25.5%

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

      12. associate-/r* 25.5%

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

      13. metadata-eval 25.5%

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

      14. metadata-eval 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in b around inf 94.1%

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

      1. +-commutative 94.1%

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

      2. mul-1-neg 94.1%

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

      3. unsub-neg 94.1%

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

      4. mul-1-neg 94.1%

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

      5. distribute-neg-frac 94.1%

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

      6. associate-/l* 94.1%

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

      7. unpow2 94.1%

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

   6. Simplified 94.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -25000:\\ \;\;\;\;\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 4: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -25000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(b \cdot b + -8 \cdot \left(c \cdot a\right)\right) - \left(c \cdot a\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -25000.0)
   (* (/ 0.5 a) (- (sqrt (- (+ (* b b) (* -8.0 (* c a))) (* (* c a) -4.0))) b))
   (- (/ (- c) b) (/ (* c c) (/ (pow b 3.0) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -25000.0) {
		tmp = (0.5 / a) * (sqrt((((b * b) + (-8.0 * (c * a))) - ((c * a) * -4.0))) - b);
	} 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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-25000.0d0)) then
        tmp = (0.5d0 / a) * (sqrt((((b * b) + ((-8.0d0) * (c * a))) - ((c * a) * (-4.0d0)))) - b)
    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 tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -25000.0) {
		tmp = (0.5 / a) * (Math.sqrt((((b * b) + (-8.0 * (c * a))) - ((c * a) * -4.0))) - b);
	} else {
		tmp = (-c / b) - ((c * c) / (Math.pow(b, 3.0) / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -25000.0:
		tmp = (0.5 / a) * (math.sqrt((((b * b) + (-8.0 * (c * a))) - ((c * a) * -4.0))) - b)
	else:
		tmp = (-c / b) - ((c * c) / (math.pow(b, 3.0) / a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -25000.0)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(Float64(b * b) + Float64(-8.0 * Float64(c * a))) - Float64(Float64(c * a) * -4.0))) - b));
	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)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -25000.0)
		tmp = (0.5 / a) * (sqrt((((b * b) + (-8.0 * (c * a))) - ((c * a) * -4.0))) - b);
	else
		tmp = (-c / b) - ((c * c) / ((b ^ 3.0) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -25000.0], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(b * b), $MachinePrecision] + N[(-8.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $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)) < -25000

   1. Initial program 83.5%

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

      1. /-rgt-identity 83.5%

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

      2. metadata-eval 83.5%

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

      3. associate-/l* 83.5%

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

      4. associate-*r/ 83.7%

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

      5. +-commutative 83.7%

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

      6. unsub-neg 83.7%

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

      7. fma-neg 83.9%

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

      8. associate-*l* 83.9%

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

      9. *-commutative 83.9%

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

      10. distribute-rgt-neg-in 83.9%

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

      11. metadata-eval 83.9%

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

      12. associate-/r* 83.9%

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

      13. metadata-eval 83.9%

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

      14. metadata-eval 83.9%

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

   3. Simplified 83.9%

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

      1. fma-udef 83.7%

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

      2. *-commutative 83.7%

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

      3. metadata-eval 83.7%

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

      4. cancel-sign-sub-inv 83.7%

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

      5. associate-*l* 83.7%

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

      6. *-un-lft-identity 83.7%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      7. prod-diff 83.9%

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

   5. Applied egg-rr 83.6%

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

      1. +-commutative 83.6%

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

      2. fma-udef 83.6%

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

      3. *-rgt-identity 83.6%

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

      4. *-rgt-identity 83.6%

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

      5. count-2 83.6%

         \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      6. *-commutative 83.6%

         \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(\left(c \cdot -4\right) \cdot a\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      7. *-commutative 83.6%

         \[\leadsto \left(\sqrt{2 \cdot \left(\color{blue}{\left(-4 \cdot c\right)} \cdot a\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      8. associate-*r* 83.6%

         \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(-4 \cdot \left(c \cdot a\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      9. *-rgt-identity 83.6%

         \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]

      10. fma-neg 83.9%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

      11. *-commutative 83.9%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{\left(c \cdot -4\right) \cdot a}\right)} - b\right) \cdot \frac{0.5}{a} \]

      12. *-commutative 83.9%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{\left(-4 \cdot c\right)} \cdot a\right)} - b\right) \cdot \frac{0.5}{a} \]

      13. associate-*r* 83.9%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]

   7. Simplified 83.9%

      \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]
   8. Step-by-step derivation

      1. associate-+r- 84.0%

         \[\leadsto \left(\sqrt{\color{blue}{\left(2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + b \cdot b\right) - -4 \cdot \left(c \cdot a\right)}} - b\right) \cdot \frac{0.5}{a} \]

      2. associate-*r* 84.0%

         \[\leadsto \left(\sqrt{\left(\color{blue}{\left(2 \cdot -4\right) \cdot \left(c \cdot a\right)} + b \cdot b\right) - -4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a} \]

      3. metadata-eval 84.0%

         \[\leadsto \left(\sqrt{\left(\color{blue}{-8} \cdot \left(c \cdot a\right) + b \cdot b\right) - -4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a} \]

      4. *-commutative 84.0%

         \[\leadsto \left(\sqrt{\left(-8 \cdot \left(c \cdot a\right) + b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -4}} - b\right) \cdot \frac{0.5}{a} \]

   9. Applied egg-rr 84.0%

      \[\leadsto \left(\sqrt{\color{blue}{\left(-8 \cdot \left(c \cdot a\right) + b \cdot b\right) - \left(c \cdot a\right) \cdot -4}} - b\right) \cdot \frac{0.5}{a} \]

   if -25000 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 25.3%

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

      1. /-rgt-identity 25.3%

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

      2. metadata-eval 25.3%

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

      3. associate-/l* 25.3%

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

      4. associate-*r/ 25.3%

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

      5. +-commutative 25.3%

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

      6. unsub-neg 25.3%

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

      7. fma-neg 25.5%

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

      8. associate-*l* 25.5%

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

      9. *-commutative 25.5%

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

      10. distribute-rgt-neg-in 25.5%

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

      11. metadata-eval 25.5%

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

      12. associate-/r* 25.5%

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

      13. metadata-eval 25.5%

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

      14. metadata-eval 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in b around inf 94.1%

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

      1. +-commutative 94.1%

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

      2. mul-1-neg 94.1%

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

      3. unsub-neg 94.1%

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

      4. mul-1-neg 94.1%

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

      5. distribute-neg-frac 94.1%

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

      6. associate-/l* 94.1%

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

      7. unpow2 94.1%

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

   6. Simplified 94.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -25000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(b \cdot b + -8 \cdot \left(c \cdot a\right)\right) - \left(c \cdot a\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 5: 90.6% 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 28.5%

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

   1. /-rgt-identity 28.5%

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

   2. metadata-eval 28.5%

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

   3. associate-/l* 28.5%

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

   4. associate-*r/ 28.5%

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

   5. +-commutative 28.5%

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

   6. unsub-neg 28.5%

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

   7. fma-neg 28.7%

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

   8. associate-*l* 28.7%

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

   9. *-commutative 28.7%

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

   10. distribute-rgt-neg-in 28.7%

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

   11. metadata-eval 28.7%

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

   12. associate-/r* 28.7%

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

   13. metadata-eval 28.7%

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

   14. metadata-eval 28.7%

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

3. Simplified 28.7%

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

4. Taylor expanded in b around inf 91.9%

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

   1. +-commutative 91.9%

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

   2. mul-1-neg 91.9%

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

   3. unsub-neg 91.9%

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

   4. mul-1-neg 91.9%

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

   5. distribute-neg-frac 91.9%

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

   6. associate-/l* 91.9%

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

   7. unpow2 91.9%

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

6. Simplified 91.9%

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

7. Final simplification 91.9%

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

Alternative 6: 81.1% 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 28.5%

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

   1. /-rgt-identity 28.5%

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

   2. metadata-eval 28.5%

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

   3. associate-/l* 28.5%

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

   4. associate-*r/ 28.5%

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

   5. +-commutative 28.5%

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

   6. unsub-neg 28.5%

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

   7. fma-neg 28.7%

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

   8. associate-*l* 28.7%

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

   9. *-commutative 28.7%

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

   10. distribute-rgt-neg-in 28.7%

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

   11. metadata-eval 28.7%

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

   12. associate-/r* 28.7%

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

   13. metadata-eval 28.7%

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

   14. metadata-eval 28.7%

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

3. Simplified 28.7%

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

4. Taylor expanded in b around inf 83.4%

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

   1. mul-1-neg 83.4%

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

   2. distribute-neg-frac 83.4%

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

6. Simplified 83.4%

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

7. Final simplification 83.4%

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

Reproduce

herbie shell --seed 2023174 
(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)))