Quadratic roots, narrow range

Percentage Accurate: 55.1% → 91.0%

Time: 17.7s

Alternatives: 8

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: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

3. Simplified 54.4%

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

5. Taylor expanded in a around 0 90.7%

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

   1. +-commutative 90.7%

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

   2. mul-1-neg 90.7%

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

   3. unsub-neg 90.7%

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

7. Simplified 90.7%

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

   1. unpow2 90.7%

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

9. Applied egg-rr 90.7%

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

10. Taylor expanded in a around 0 90.7%

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

Alternative 2: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -70:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -70.0)
   (/ (- (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 (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -70.0) {
		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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -70.0)
		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(Float64(-2.0 * Float64(a * (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[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], -70.0], 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[(-2.0 * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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 (/.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)) < -70

   1. Initial program 88.4%

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

   3. Simplified 88.9%

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

   if -70 < (/.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 52.5%

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

      1. *-commutative 52.5%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in a around 0 90.0%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

      4. mul-1-neg 90.0%

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

      5. unsub-neg 90.0%

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

      6. associate-*r/ 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -70:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -70:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -70.0)
   (/ (- (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 (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -70.0) {
		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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -70.0)
		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[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], -70.0], 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 (/.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)) < -70

   1. Initial program 88.4%

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

   3. Simplified 88.9%

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

   if -70 < (/.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 52.5%

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

      1. *-commutative 52.5%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in c around 0 89.7%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -70:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 50:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{-c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 50.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (- (- c) (* a (pow (/ (- c) b) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 50.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * pow((-c / b), 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 50.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(Float64(-c) / b) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 50.0], 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) - N[(a * N[Power[N[((-c) / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 50

   1. Initial program 78.4%

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

   3. Simplified 78.6%

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

   if 50 < b

   1. Initial program 45.2%

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

      1. *-commutative 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in a around 0 90.1%

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

      1. mul-1-neg 90.1%

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

      2. unsub-neg 90.1%

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

      3. associate-*r/ 90.1%

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

      4. mul-1-neg 90.1%

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

      5. associate-/l* 90.1%

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

   7. Simplified 90.1%

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

      1. add-cube-cbrt 88.6%

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

      2. pow3 88.7%

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

      3. *-commutative 88.7%

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

      4. div-inv 88.7%

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

      5. pow-flip 88.7%

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

      6. metadata-eval 88.7%

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

   9. Applied egg-rr 88.7%

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

       1. rem-cube-cbrt 90.1%

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

       2. div-inv 89.9%

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

       3. fma-neg 89.9%

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

       4. associate-*l* 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. fma-undefine 89.9%

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

       2. unsub-neg 89.9%

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

       3. *-commutative 89.9%

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

       4. associate-*l* 89.9%

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

       5. *-commutative 89.9%

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

   13. Simplified 89.9%

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

   14. Taylor expanded in b around inf 90.1%

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

       1. neg-mul-1 90.1%

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

       2. +-commutative 90.1%

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

       3. unsub-neg 90.1%

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

       4. mul-1-neg 90.1%

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

       5. associate-/l* 90.1%

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

       6. unpow2 90.1%

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

       7. unpow2 90.1%

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

       8. times-frac 90.1%

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

       9. sqr-neg 90.1%

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

       10. distribute-frac-neg 90.1%

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

       11. distribute-frac-neg 90.1%

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

       12. unpow2 90.1%

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

       13. distribute-lft-neg-in 90.1%

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

       14. distribute-frac-neg 90.1%

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

       15. distribute-neg-frac2 90.1%

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

   16. Simplified 90.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 50:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{-c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 50:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{-c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 50.0)
   (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
   (/ (- (- c) (* a (pow (/ (- c) b) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 50.0) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * pow((-c / b), 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 50.0)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(Float64(-c) / b) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 50.0], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[((-c) / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 50

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

   3. Simplified 78.5%

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

   if 50 < b

   1. Initial program 45.2%

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

      1. *-commutative 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in a around 0 90.1%

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

      1. mul-1-neg 90.1%

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

      2. unsub-neg 90.1%

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

      3. associate-*r/ 90.1%

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

      4. mul-1-neg 90.1%

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

      5. associate-/l* 90.1%

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

   7. Simplified 90.1%

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

      1. add-cube-cbrt 88.6%

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

      2. pow3 88.7%

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

      3. *-commutative 88.7%

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

      4. div-inv 88.7%

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

      5. pow-flip 88.7%

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

      6. metadata-eval 88.7%

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

   9. Applied egg-rr 88.7%

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

       1. rem-cube-cbrt 90.1%

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

       2. div-inv 89.9%

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

       3. fma-neg 89.9%

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

       4. associate-*l* 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. fma-undefine 89.9%

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

       2. unsub-neg 89.9%

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

       3. *-commutative 89.9%

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

       4. associate-*l* 89.9%

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

       5. *-commutative 89.9%

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

   13. Simplified 89.9%

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

   14. Taylor expanded in b around inf 90.1%

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

       1. neg-mul-1 90.1%

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

       2. +-commutative 90.1%

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

       3. unsub-neg 90.1%

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

       4. mul-1-neg 90.1%

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

       5. associate-/l* 90.1%

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

       6. unpow2 90.1%

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

       7. unpow2 90.1%

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

       8. times-frac 90.1%

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

       9. sqr-neg 90.1%

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

       10. distribute-frac-neg 90.1%

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

       11. distribute-frac-neg 90.1%

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

       12. unpow2 90.1%

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

       13. distribute-lft-neg-in 90.1%

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

       14. distribute-frac-neg 90.1%

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

       15. distribute-neg-frac2 90.1%

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

   16. Simplified 90.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 50:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{-c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 50:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{-c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 50.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (- (- c) (* a (pow (/ (- c) b) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 50.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * pow((-c / b), 2.0))) / 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) :: tmp
    if (b <= 50.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c - (a * ((-c / b) ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 50.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * Math.pow((-c / b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 50.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-c - (a * math.pow((-c / b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 50.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(Float64(-c) / b) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 50.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-c - (a * ((-c / b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 50.0], 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], N[(N[((-c) - N[(a * N[Power[N[((-c) / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 50

   1. Initial program 78.4%

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

   if 50 < b

   1. Initial program 45.2%

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

      1. *-commutative 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in a around 0 90.1%

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

      1. mul-1-neg 90.1%

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

      2. unsub-neg 90.1%

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

      3. associate-*r/ 90.1%

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

      4. mul-1-neg 90.1%

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

      5. associate-/l* 90.1%

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

   7. Simplified 90.1%

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

      1. add-cube-cbrt 88.6%

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

      2. pow3 88.7%

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

      3. *-commutative 88.7%

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

      4. div-inv 88.7%

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

      5. pow-flip 88.7%

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

      6. metadata-eval 88.7%

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

   9. Applied egg-rr 88.7%

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

       1. rem-cube-cbrt 90.1%

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

       2. div-inv 89.9%

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

       3. fma-neg 89.9%

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

       4. associate-*l* 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. fma-undefine 89.9%

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

       2. unsub-neg 89.9%

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

       3. *-commutative 89.9%

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

       4. associate-*l* 89.9%

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

       5. *-commutative 89.9%

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

   13. Simplified 89.9%

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

   14. Taylor expanded in b around inf 90.1%

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

       1. neg-mul-1 90.1%

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

       2. +-commutative 90.1%

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

       3. unsub-neg 90.1%

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

       4. mul-1-neg 90.1%

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

       5. associate-/l* 90.1%

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

       6. unpow2 90.1%

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

       7. unpow2 90.1%

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

       8. times-frac 90.1%

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

       9. sqr-neg 90.1%

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

       10. distribute-frac-neg 90.1%

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

       11. distribute-frac-neg 90.1%

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

       12. unpow2 90.1%

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

       13. distribute-lft-neg-in 90.1%

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

       14. distribute-frac-neg 90.1%

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

       15. distribute-neg-frac2 90.1%

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

   16. Simplified 90.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 50:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{-c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

3. Simplified 54.4%

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

5. Taylor expanded in a around 0 83.1%

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

   1. mul-1-neg 83.1%

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

   2. unsub-neg 83.1%

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

   3. associate-*r/ 83.1%

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

   4. mul-1-neg 83.1%

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

   5. associate-/l* 83.1%

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

7. Simplified 83.1%

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

   1. add-cube-cbrt 82.0%

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

   2. pow3 82.0%

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

   3. *-commutative 82.0%

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

   4. div-inv 82.0%

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

   5. pow-flip 82.0%

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

   6. metadata-eval 82.0%

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

9. Applied egg-rr 82.0%

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

    1. rem-cube-cbrt 83.1%

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

    2. div-inv 83.0%

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

    3. fma-neg 83.0%

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

    4. associate-*l* 83.0%

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

11. Applied egg-rr 83.0%

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

    1. fma-undefine 83.0%

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

    2. unsub-neg 83.0%

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

    3. *-commutative 83.0%

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

    4. associate-*l* 83.0%

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

    5. *-commutative 83.0%

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

13. Simplified 83.0%

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

14. Taylor expanded in b around inf 83.1%

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

    1. neg-mul-1 83.1%

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

    2. +-commutative 83.1%

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

    3. unsub-neg 83.1%

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

    4. mul-1-neg 83.1%

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

    5. associate-/l* 83.1%

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

    6. unpow2 83.1%

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

    7. unpow2 83.1%

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

    8. times-frac 83.1%

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

    9. sqr-neg 83.1%

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

    10. distribute-frac-neg 83.1%

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

    11. distribute-frac-neg 83.1%

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

    12. unpow2 83.1%

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

    13. distribute-lft-neg-in 83.1%

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

    14. distribute-frac-neg 83.1%

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

    15. distribute-neg-frac2 83.1%

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

16. Simplified 83.1%

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

17. Final simplification 83.1%

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

Alternative 8: 64.7% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

3. Simplified 54.4%

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

5. Taylor expanded in b around inf 65.5%

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

   1. associate-*r/ 65.5%

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

   2. mul-1-neg 65.5%

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

7. Simplified 65.5%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024101 
(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)))