Quadratic roots, narrow range

Percentage Accurate: 55.6% → 91.8%

Time: 13.4s

Alternatives: 10

Speedup: 29.0×

• Rival profile vector overflowed, profile may not be complete

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: 55.6% 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: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (*
     a
     (-
      (*
       a
       (+
        (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
        (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		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 * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], 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[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $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 < 0.087499999999999994

   1. Initial program 87.5%

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

      1. *-commutative 87.5%

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

   3. Simplified 87.7%

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

   if 0.087499999999999994 < 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 53.5%

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

   5. Taylor expanded in a around 0 94.5%

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

   6. Taylor expanded in c around 0 94.5%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \left(c \cdot c\right) \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (*
     a
     (-
      (*
       a
       (+
        (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
        (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
      (* (* c c) (pow b -3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - ((c * c) * pow(b, -3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		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 * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - Float64(Float64(c * c) * (b ^ -3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], 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[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $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 < 0.087499999999999994

   1. Initial program 87.5%

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

      1. *-commutative 87.5%

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

   3. Simplified 87.7%

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

   if 0.087499999999999994 < 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 53.5%

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

   5. Taylor expanded in a around 0 94.5%

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

   6. Taylor expanded in c around 0 94.5%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
   7. Step-by-step derivation

      1. pow1 94.5%

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

      2. mul-1-neg 94.5%

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

      3. div-inv 94.5%

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

      4. pow-flip 94.5%

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

      5. metadata-eval 94.5%

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

   8. Applied egg-rr 94.5%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\color{blue}{{\left(-{c}^{2} \cdot {b}^{-3}\right)}^{1}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
   9. Step-by-step derivation

      1. unpow1 94.5%

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

      2. distribute-rgt-neg-in 94.5%

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

   10. Simplified 94.5%

       \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\color{blue}{{c}^{2} \cdot \left(-{b}^{-3}\right)} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 94.5%

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

   12. Applied egg-rr 94.5%

       \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot \left(-{b}^{-3}\right) + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \left(c \cdot c\right) \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -2 \cdot \left({c}^{3} \cdot \frac{a}{{b}^{4}}\right) - {\left(\frac{c}{-b}\right)}^{2}, -c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (fma
     a
     (- (* -2.0 (* (pow c 3.0) (/ a (pow b 4.0)))) (pow (/ c (- b)) 2.0))
     (- c))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(a, ((-2.0 * (pow(c, 3.0) * (a / pow(b, 4.0)))) - pow((c / -b), 2.0)), -c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, Float64(Float64(-2.0 * Float64((c ^ 3.0) * Float64(a / (b ^ 4.0)))) - (Float64(c / Float64(-b)) ^ 2.0)), Float64(-c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], 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[(N[Power[c, 3.0], $MachinePrecision] * N[(a / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.087499999999999994

   1. Initial program 87.5%

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

      1. *-commutative 87.5%

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

   3. Simplified 87.7%

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

   if 0.087499999999999994 < 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 53.5%

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

   5. Taylor expanded in b around inf 91.6%

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

   6. Taylor expanded in a around 0 91.6%

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

      1. neg-mul-1 91.6%

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

      2. +-commutative 91.6%

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

      3. fma-define 91.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)}}{b} \]

      4. mul-1-neg 91.6%

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

      5. unsub-neg 91.6%

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

      6. *-commutative 91.6%

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

      7. associate-/l* 91.6%

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

      8. unpow2 91.6%

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

      9. unpow2 91.6%

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

      10. times-frac 91.6%

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

      11. sqr-neg 91.6%

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

      12. distribute-frac-neg 91.6%

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

      13. distribute-frac-neg 91.6%

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

      14. unpow2 91.6%

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

   8. Simplified 91.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -2 \cdot \left({c}^{3} \cdot \frac{a}{{b}^{4}}\right) - {\left(\frac{c}{-b}\right)}^{2}, -c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{3} \cdot \left(a \cdot \left(a \cdot \left(-2 \cdot {b}^{-5}\right) - \frac{{b}^{-3}}{c}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (* (pow c 3.0) (* a (- (* a (* -2.0 (pow b -5.0))) (/ (pow b -3.0) c))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (pow(c, 3.0) * (a * ((a * (-2.0 * pow(b, -5.0))) - (pow(b, -3.0) / c)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64((c ^ 3.0) * Float64(a * Float64(Float64(a * Float64(-2.0 * (b ^ -5.0))) - Float64((b ^ -3.0) / c)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], 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[Power[c, 3.0], $MachinePrecision] * N[(a * N[(N[(a * N[(-2.0 * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[b, -3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.087499999999999994

   1. Initial program 87.5%

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

      1. *-commutative 87.5%

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

   3. Simplified 87.7%

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

   if 0.087499999999999994 < 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 53.5%

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

   5. Taylor expanded in a around 0 91.6%

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

   6. Taylor expanded in c around inf 91.6%

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

      1. pow1 91.6%

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

      2. associate-*r* 91.6%

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

      3. fmm-def 91.6%

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

      4. div-inv 91.6%

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

      5. pow-flip 91.6%

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

      6. metadata-eval 91.6%

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

      7. associate-/r* 91.6%

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

      8. pow-flip 91.6%

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

      9. metadata-eval 91.6%

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

   8. Applied egg-rr 91.6%

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

      1. unpow1 91.6%

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

      2. *-commutative 91.6%

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

      3. associate-*l* 91.6%

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

      4. fmm-undef 91.6%

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

      5. associate-*r* 91.6%

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

      6. *-commutative 91.6%

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

      7. associate-*l* 91.6%

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

   10. Simplified 91.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{3} \cdot \left(a \cdot \left(a \cdot \left(-2 \cdot {b}^{-5}\right) - \frac{{b}^{-3}}{c}\right)\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ c (- b)) (* a (/ (* c c) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c / -b) - (a * ((c * c) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64(Float64(c * c) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], 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[(c / (-b)), $MachinePrecision] - N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.087499999999999994

   1. Initial program 87.5%

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

      1. *-commutative 87.5%

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

   3. Simplified 87.7%

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

   if 0.087499999999999994 < 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 53.5%

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

   5. Taylor expanded in a around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

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

      3. mul-1-neg 85.5%

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

      4. distribute-neg-frac2 85.5%

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

      5. associate-/l* 85.5%

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

   7. Simplified 85.5%

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

      1. unpow2 94.5%

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

   9. Applied egg-rr 85.5%

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

Alternative 6: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (- (/ c (- b)) (* a (/ (* c c) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c / -b) - (a * ((c * c) / pow(b, 3.0)));
	}
	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 (b <= 0.0875d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (c / -b) - (a * ((c * c) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c / -b) - (a * ((c * c) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.0875:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (c / -b) - (a * ((c * c) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64(Float64(c * c) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.0875)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (c / -b) - (a * ((c * c) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], 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], N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.087499999999999994

   1. Initial program 87.5%

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

   if 0.087499999999999994 < 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 53.5%

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

   5. Taylor expanded in a around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

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

      3. mul-1-neg 85.5%

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

      4. distribute-neg-frac2 85.5%

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

      5. associate-/l* 85.5%

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

   7. Simplified 85.5%

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

      1. unpow2 94.5%

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

   9. Applied egg-rr 85.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c / -b) - (a * ((c * c) / (b ** 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.7%

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

   1. *-commutative 56.7%

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

3. Simplified 56.9%

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

5. Taylor expanded in a around 0 82.0%

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

   1. mul-1-neg 82.0%

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

   2. unsub-neg 82.0%

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

   3. mul-1-neg 82.0%

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

   4. distribute-neg-frac2 82.0%

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

   5. associate-/l* 82.0%

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

7. Simplified 82.0%

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

   1. unpow2 91.9%

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

9. Applied egg-rr 82.0%

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

Alternative 8: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - \frac{c \cdot a}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (/ (* c a) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((c * a) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-1.0d0) / b) - ((c * a) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((c * a) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - ((c * a) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(c * a) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - ((c * a) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.7%

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

   1. *-commutative 56.7%

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

3. Simplified 56.9%

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

5. Taylor expanded in c around 0 81.8%

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

   1. associate-*r/ 81.8%

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

   2. neg-mul-1 81.8%

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

   3. distribute-rgt-neg-in 81.8%

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

7. Simplified 81.8%

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

8. Taylor expanded in a around 0 81.8%

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

   1. sub-neg 81.8%

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

   2. associate-*r/ 81.8%

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

   3. neg-mul-1 81.8%

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

   4. distribute-rgt-neg-in 81.8%

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

   5. associate-*r/ 81.8%

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

   6. +-commutative 81.8%

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

   7. associate-*r/ 81.8%

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

   8. distribute-rgt-neg-in 81.8%

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

   9. distribute-frac-neg 81.8%

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

   10. unsub-neg 81.8%

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

   11. distribute-neg-frac 81.8%

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

   12. metadata-eval 81.8%

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

10. Simplified 81.8%

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

11. Final simplification 81.8%

    \[\leadsto c \cdot \left(\frac{-1}{b} - \frac{c \cdot a}{{b}^{3}}\right) \]
12. Add Preprocessing

Alternative 9: 64.3% 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 56.7%

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

   1. *-commutative 56.7%

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

3. Simplified 56.9%

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

5. Taylor expanded in b around inf 63.7%

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

   1. associate-*r/ 63.7%

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

   2. mul-1-neg 63.7%

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

7. Simplified 63.7%

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

8. Final simplification 63.7%

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

Alternative 10: 3.2% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 56.7%

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

   1. *-commutative 56.7%

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

3. Simplified 56.9%

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

5. Taylor expanded in b around inf 63.7%

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

   1. associate-*r/ 63.7%

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

   2. mul-1-neg 63.7%

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

7. Simplified 63.7%

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

   1. expm1-log1p-u 56.4%

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

   2. expm1-undefine 45.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-c}{b}\right)} - 1} \]

9. Applied egg-rr 45.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-c}{b}\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 45.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-c}{b}\right)} + \left(-1\right)} \]

    2. metadata-eval 45.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{-c}{b}\right)} + \color{blue}{-1} \]

    3. +-commutative 45.2%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{-c}{b}\right)}} \]

    4. log1p-undefine 45.2%

       \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \frac{-c}{b}\right)}} \]

    5. rem-exp-log 52.5%

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

    6. distribute-frac-neg 52.5%

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

    7. unsub-neg 52.5%

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

11. Simplified 52.5%

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

12. Taylor expanded in c around 0 3.2%

    \[\leadsto -1 + \color{blue}{1} \]

13. Final simplification 3.2%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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)))