Quadratic roots, narrow range

Percentage Accurate: 55.7% → 91.6%

Time: 28.3s

Alternatives: 10

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.7% 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: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0052:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \left(a \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{b} \cdot 20\right)\right) \cdot -0.25\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0052)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (*
     a
     (-
      (*
       a
       (fma
        -2.0
        (/ (pow c 3.0) (pow b 5.0))
        (* (* a (* (/ (/ (pow c 4.0) (pow b 6.0)) b) 20.0)) -0.25)))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0052) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), ((a * (((pow(c, 4.0) / pow(b, 6.0)) / b) * 20.0)) * -0.25))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0052)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(Float64(a * Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) / b) * 20.0)) * -0.25))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0052], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0051999999999999998

   1. Initial program 91.8%

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

   3. Simplified 92.0%

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

   if 0.0051999999999999998 < b

   1. Initial program 53.4%

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

      1. *-commutative 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in a around 0 92.6%

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

      1. +-commutative 92.6%

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

      2. mul-1-neg 92.6%

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

      3. unsub-neg 92.6%

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

   7. Simplified 92.6%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \left(a \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{b} \cdot 20\right)\right) \cdot -0.25\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0052:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \left(a \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{b} \cdot 20\right)\right) \cdot -0.25\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.005:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{c \cdot {b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.005)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (*
     a
     (-
      (*
       (pow c 4.0)
       (+
        (* -5.0 (/ (pow a 2.0) (pow b 7.0)))
        (* -2.0 (/ a (* c (pow b 5.0))))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.005) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((pow(c, 4.0) * ((-5.0 * (pow(a, 2.0) / pow(b, 7.0))) + (-2.0 * (a / (c * pow(b, 5.0)))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.005)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64((a ^ 2.0) / (b ^ 7.0))) + Float64(-2.0 * Float64(a / Float64(c * (b ^ 5.0)))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.005], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(a / N[(c * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0050000000000000001

   1. Initial program 91.8%

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

   3. Simplified 92.0%

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

   if 0.0050000000000000001 < b

   1. Initial program 53.4%

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

      1. *-commutative 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in a around 0 92.6%

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

      1. +-commutative 92.6%

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

      2. mul-1-neg 92.6%

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

      3. unsub-neg 92.6%

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

   7. Simplified 92.6%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \left(a \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{b} \cdot 20\right)\right) \cdot -0.25\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

   8. Taylor expanded in c around inf 92.6%

      \[\leadsto a \cdot \left(\color{blue}{{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5} \cdot c}\right)} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.005:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{c \cdot {b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}\right) - c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.7)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (-
     (- (* -2.0 (* (pow a 2.0) (/ (pow c 3.0) (pow b 4.0)))) c)
     (* a (/ (pow c 2.0) (pow b 2.0))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.7) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (pow(a, 2.0) * (pow(c, 3.0) / pow(b, 4.0)))) - c) - (a * (pow(c, 2.0) / pow(b, 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 4.0)))) - c) - Float64(a * Float64((c ^ 2.0) / (b ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.7], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.69999999999999996

   1. Initial program 86.9%

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

   3. Simplified 87.0%

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

   if 0.69999999999999996 < b

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

   3. Simplified 51.2%

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

      1. div-sub 50.4%

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

      2. *-un-lft-identity 50.4%

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

      3. *-commutative 50.4%

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

      4. times-frac 50.4%

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

      5. metadata-eval 50.4%

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

      6. pow2 50.4%

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

      7. *-un-lft-identity 50.4%

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

      8. *-commutative 50.4%

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

      9. times-frac 50.4%

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

      10. metadata-eval 50.4%

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

   6. Applied egg-rr 50.4%

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

   7. Taylor expanded in b around inf 90.9%

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

   9. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{\left(-2 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}\right) - c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}\right) - c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.7)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (*
     a
     (-
      (* -2.0 (* a (/ (pow c 3.0) (pow b 5.0))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.7) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((-2.0 * (a * (pow(c, 3.0) / pow(b, 5.0)))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(a * Float64((c ^ 3.0) / (b ^ 5.0)))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.7], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-2.0 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.69999999999999996

   1. Initial program 86.9%

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

   3. Simplified 87.0%

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

   if 0.69999999999999996 < b

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in a around 0 90.9%

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

      1. +-commutative 90.9%

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

      2. mul-1-neg 90.9%

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

      3. unsub-neg 90.9%

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

      4. mul-1-neg 90.9%

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

      5. unsub-neg 90.9%

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

      6. associate-/l* 90.9%

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

   7. Simplified 90.9%

      \[\leadsto \color{blue}{a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{c}{b}\right)}^{2} \cdot \left(-a\right)}{b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.01)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (* (pow (/ c b) 2.0) (- a)) b) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.01) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((pow((c / b), 2.0) * -a) / b) - (c / b);
	}
	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)) <= -0.01)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64((Float64(c / b) ^ 2.0) * Float64(-a)) / b) - Float64(c / b));
	end
	return 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], -0.01], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] * (-a)), $MachinePrecision] / b), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0100000000000000002

   1. Initial program 80.2%

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

   3. Simplified 80.3%

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

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 47.1%

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

      1. *-commutative 47.1%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in b around inf 88.0%

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

      1. mul-1-neg 88.0%

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

      2. unsub-neg 88.0%

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

      3. mul-1-neg 88.0%

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

   7. Simplified 88.0%

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

      1. log1p-expm1-u 88.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\right)\right)} \]

      2. associate-/l* 88.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}}{b}\right)\right) \]

   9. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}}{b}\right)\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 88.0%

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

       2. div-sub 88.0%

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

       3. add-sqr-sqrt 88.0%

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

       4. pow2 88.0%

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

       5. sqrt-div 88.0%

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

       6. sqrt-pow1 88.0%

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

       7. metadata-eval 88.0%

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

       8. pow1 88.0%

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

       9. sqrt-pow1 88.0%

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

       10. metadata-eval 88.0%

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

       11. pow1 88.0%

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

   11. Applied egg-rr 88.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{c}{b}\right)}^{2} \cdot \left(-a\right)}{b} - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.7)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (*
    c
    (+
     (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.7) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.7], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.69999999999999996

   1. Initial program 86.9%

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

   3. Simplified 87.0%

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

   if 0.69999999999999996 < b

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in c around 0 90.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{c}{b}\right)}^{2} \cdot \left(-a\right)}{b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -0.01) t_0 (- (/ (* (pow (/ c b) 2.0) (- a)) b) (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_0;
	} else {
		tmp = ((pow((c / b), 2.0) * -a) / b) - (c / b);
	}
	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) / (a * 2.0d0)
    if (t_0 <= (-0.01d0)) then
        tmp = t_0
    else
        tmp = ((((c / b) ** 2.0d0) * -a) / b) - (c / b)
    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) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_0;
	} else {
		tmp = ((Math.pow((c / b), 2.0) * -a) / b) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.01:
		tmp = t_0
	else:
		tmp = ((math.pow((c / b), 2.0) * -a) / b) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64((Float64(c / b) ^ 2.0) * Float64(-a)) / b) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = t_0;
	else
		tmp = ((((c / b) ^ 2.0) * -a) / b) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -0.01], t$95$0, N[(N[(N[(N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] * (-a)), $MachinePrecision] / b), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0100000000000000002

   1. Initial program 80.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 47.1%

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

      1. *-commutative 47.1%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in b around inf 88.0%

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

      1. mul-1-neg 88.0%

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

      2. unsub-neg 88.0%

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

      3. mul-1-neg 88.0%

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

   7. Simplified 88.0%

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

      1. log1p-expm1-u 88.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\right)\right)} \]

      2. associate-/l* 88.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}}{b}\right)\right) \]

   9. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}}{b}\right)\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 88.0%

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

       2. div-sub 88.0%

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

       3. add-sqr-sqrt 88.0%

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

       4. pow2 88.0%

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

       5. sqrt-div 88.0%

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

       6. sqrt-pow1 88.0%

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

       7. metadata-eval 88.0%

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

       8. pow1 88.0%

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

       9. sqrt-pow1 88.0%

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

       10. metadata-eval 88.0%

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

       11. pow1 88.0%

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

   11. Applied egg-rr 88.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{c}{b}\right)}^{2} \cdot \left(-a\right)}{b} - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

3. Simplified 56.4%

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

5. Taylor expanded in b around inf 80.5%

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

   1. mul-1-neg 80.5%

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

   2. unsub-neg 80.5%

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

   3. mul-1-neg 80.5%

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

7. Simplified 80.5%

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

   1. log1p-expm1-u 77.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\right)\right)} \]

   2. associate-/l* 77.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}}{b}\right)\right) \]

9. Applied egg-rr 77.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}}{b}\right)\right)} \]
10. Step-by-step derivation

    1. log1p-expm1-u 80.5%

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

    2. div-sub 80.5%

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

    3. add-sqr-sqrt 80.5%

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

    4. pow2 80.5%

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

    5. sqrt-div 80.5%

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

    6. sqrt-pow1 80.5%

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

    7. metadata-eval 80.5%

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

    8. pow1 80.5%

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

    9. sqrt-pow1 80.5%

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

    10. metadata-eval 80.5%

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

    11. pow1 80.5%

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

11. Applied egg-rr 80.5%

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

12. Final simplification 80.5%

    \[\leadsto \frac{{\left(\frac{c}{b}\right)}^{2} \cdot \left(-a\right)}{b} - \frac{c}{b} \]
13. Add Preprocessing

Alternative 9: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

3. Simplified 56.4%

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

5. Taylor expanded in b around inf 80.5%

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

   1. mul-1-neg 80.5%

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

   2. unsub-neg 80.5%

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

   3. mul-1-neg 80.5%

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

7. Simplified 80.5%

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

   1. log1p-expm1-u 77.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\right)\right)} \]

   2. associate-/l* 77.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}}{b}\right)\right) \]

9. Applied egg-rr 77.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}}{b}\right)\right)} \]
10. Step-by-step derivation

    1. log1p-expm1-u 80.5%

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

    2. add-sqr-sqrt 80.5%

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

    3. pow2 80.5%

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

    4. sqrt-div 80.5%

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

    5. sqrt-pow1 80.5%

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

    6. metadata-eval 80.5%

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

    7. pow1 80.5%

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

    8. sqrt-pow1 80.5%

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

    9. metadata-eval 80.5%

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

    10. pow1 80.5%

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

11. Applied egg-rr 80.5%

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

12. Final simplification 80.5%

    \[\leadsto \frac{{\left(\frac{c}{b}\right)}^{2} \cdot \left(-a\right) - c}{b} \]
13. Add Preprocessing

Alternative 10: 64.1% 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(c / Float64(-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 56.4%

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

   1. *-commutative 56.4%

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

3. Simplified 56.4%

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

5. Taylor expanded in b around inf 63.8%

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

   1. associate-*r/ 63.8%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. mul-1-neg 63.8%

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

7. Simplified 63.8%

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

8. Final simplification 63.8%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))