Quadratic roots, narrow range

Percentage Accurate: 56.0% → 90.8%

Time: 28.0s

Alternatives: 11

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: 56.0% 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: 90.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * (((c ** 4.0d0) * (((-5.0d0) * ((a ** 2.0d0) / (b ** 7.0d0))) + ((-2.0d0) * (a / (c * (b ** 5.0d0)))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((Math.pow(c, 4.0) * ((-5.0 * (Math.pow(a, 2.0) / Math.pow(b, 7.0))) + (-2.0 * (a / (c * Math.pow(b, 5.0)))))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((math.pow(c, 4.0) * ((-5.0 * (math.pow(a, 2.0) / math.pow(b, 7.0))) + (-2.0 * (a / (c * math.pow(b, 5.0)))))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (a * (((c ^ 4.0) * ((-5.0 * ((a ^ 2.0) / (b ^ 7.0))) + (-2.0 * (a / (c * (b ^ 5.0)))))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in a around 0 90.3%

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

6. Taylor expanded in c around 0 90.3%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\left(20 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 90.3%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\left(\frac{a \cdot {c}^{4}}{{b}^{7}} \cdot 20\right)}\right)\right) \]

   2. associate-*l/ 90.3%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\frac{\left(a \cdot {c}^{4}\right) \cdot 20}{{b}^{7}}}\right)\right) \]

   3. associate-*r* 90.3%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{a \cdot \left({c}^{4} \cdot 20\right)}}{{b}^{7}}\right)\right) \]

   4. metadata-eval 90.3%

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

   5. distribute-rgt-out 90.3%

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

   6. associate-/l* 90.3%

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

   7. distribute-rgt-out 90.3%

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

   8. metadata-eval 90.3%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(a \cdot \frac{{c}^{4} \cdot \color{blue}{20}}{{b}^{7}}\right)\right)\right) \]

   9. *-commutative 90.3%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(a \cdot \frac{\color{blue}{20 \cdot {c}^{4}}}{{b}^{7}}\right)\right)\right) \]

8. Simplified 90.3%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\left(a \cdot \frac{20 \cdot {c}^{4}}{{b}^{7}}\right)}\right)\right) \]

9. Taylor expanded in c around inf 90.3%

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

10. Final simplification 90.3%

    \[\leadsto a \cdot \left({c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{c \cdot {b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
11. Add Preprocessing

Alternative 2: 87.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0035:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({a}^{2} \cdot -2\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.0035)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (-
     (- (* (* (pow a 2.0) -2.0) (/ (pow c 3.0) (pow b 4.0))) c)
     (/ (* a (pow c 2.0)) (pow b 2.0)))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.0035) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((((pow(a, 2.0) * -2.0) * (pow(c, 3.0) / pow(b, 4.0))) - c) - ((a * pow(c, 2.0)) / pow(b, 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.0035)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64((a ^ 2.0) * -2.0) * Float64((c ^ 3.0) / (b ^ 4.0))) - c) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) / 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], -0.0035], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Power[a, 2.0], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.00350000000000000007

   1. Initial program 79.0%

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

      1. *-commutative 79.0%

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

   3. Simplified 79.0%

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

   if -0.00350000000000000007 < (/.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 45.0%

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

      1. *-commutative 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in b around inf 94.1%

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

      1. associate-+r+ 94.1%

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

      2. mul-1-neg 94.1%

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

      3. unsub-neg 94.1%

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

      4. mul-1-neg 94.1%

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

      5. unsub-neg 94.1%

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

      6. associate-/l* 94.1%

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

      7. associate-*r* 94.1%

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

   7. Simplified 94.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0035:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({a}^{2} \cdot -2\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0035:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{3} \cdot \left(\frac{{a}^{2} \cdot -2}{{b}^{5}} - \frac{\frac{a}{c}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.0035)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (*
     (pow c 3.0)
     (- (/ (* (pow a 2.0) -2.0) (pow b 5.0)) (/ (/ a c) (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)) <= -0.0035) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (pow(c, 3.0) * (((pow(a, 2.0) * -2.0) / pow(b, 5.0)) - ((a / c) / 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)) <= -0.0035)
		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(Float64(Float64((a ^ 2.0) * -2.0) / (b ^ 5.0)) - Float64(Float64(a / c) / (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], -0.0035], 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[(N[(N[(N[Power[a, 2.0], $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[(a / 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.00350000000000000007

   1. Initial program 79.0%

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

      1. *-commutative 79.0%

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

   3. Simplified 79.0%

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

   if -0.00350000000000000007 < (/.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 45.0%

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

      1. *-commutative 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. unsub-neg 94.0%

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

      3. associate-*r/ 94.0%

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

      4. *-commutative 94.0%

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

      5. associate-/r* 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0035:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{3} \cdot \left(\frac{{a}^{2} \cdot -2}{{b}^{5}} - \frac{\frac{a}{c}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0035:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.0035)
   (/ (- (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)) <= -0.0035) {
		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)) <= -0.0035)
		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], -0.0035], 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)) < -0.00350000000000000007

   1. Initial program 79.0%

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

      1. *-commutative 79.0%

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

   3. Simplified 79.0%

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

   if -0.00350000000000000007 < (/.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 45.0%

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

      1. *-commutative 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in c around 0 93.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0035:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0004:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.0004)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (- (- c) (/ (* a (pow c 2.0)) (pow b 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.0004) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - ((a * pow(c, 2.0)) / pow(b, 2.0))) / 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)) <= -0.0004)
		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(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) / 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], -0.0004], 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[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $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)) < -4.00000000000000019e-4

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

   3. Simplified 78.3%

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

   if -4.00000000000000019e-4 < (/.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 42.3%

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

      1. *-commutative 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in b around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

      3. mul-1-neg 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0004:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.0004:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{c}^{2}}{{b}^{3}} \cdot \left(-a\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -0.0004)
     t_0
     (- (* (/ (pow c 2.0) (pow b 3.0)) (- a)) (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0004) {
		tmp = t_0;
	} else {
		tmp = ((pow(c, 2.0) / pow(b, 3.0)) * -a) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.0004d0)) then
        tmp = t_0
    else
        tmp = (((c ** 2.0d0) / (b ** 3.0d0)) * -a) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0004) {
		tmp = t_0;
	} else {
		tmp = ((Math.pow(c, 2.0) / Math.pow(b, 3.0)) * -a) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.0004:
		tmp = t_0
	else:
		tmp = ((math.pow(c, 2.0) / math.pow(b, 3.0)) * -a) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.0004)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64((c ^ 2.0) / (b ^ 3.0)) * Float64(-a)) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.0004)
		tmp = t_0;
	else
		tmp = (((c ^ 2.0) / (b ^ 3.0)) * -a) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0004], t$95$0, N[(N[(N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * (-a)), $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)) < -4.00000000000000019e-4

   1. Initial program 78.3%

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

   if -4.00000000000000019e-4 < (/.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 42.3%

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

      1. *-commutative 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in a around 0 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

      3. mul-1-neg 90.8%

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

      4. distribute-neg-frac2 90.8%

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

      5. associate-/l* 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0004:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{c}^{2}}{{b}^{3}} \cdot \left(-a\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.0004:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -0.0004)
     t_0
     (/ (- (- c) (/ (* a (pow c 2.0)) (pow b 2.0))) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0004) {
		tmp = t_0;
	} else {
		tmp = (-c - ((a * pow(c, 2.0)) / pow(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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.0004d0)) then
        tmp = t_0
    else
        tmp = (-c - ((a * (c ** 2.0d0)) / (b ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0004) {
		tmp = t_0;
	} else {
		tmp = (-c - ((a * Math.pow(c, 2.0)) / Math.pow(b, 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.0004:
		tmp = t_0
	else:
		tmp = (-c - ((a * math.pow(c, 2.0)) / math.pow(b, 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.0004)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-c) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.0004)
		tmp = t_0;
	else
		tmp = (-c - ((a * (c ^ 2.0)) / (b ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0004], t$95$0, N[(N[((-c) - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $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)) < -4.00000000000000019e-4

   1. Initial program 78.3%

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

   if -4.00000000000000019e-4 < (/.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 42.3%

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

      1. *-commutative 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in b around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

      3. mul-1-neg 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0004:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.0004:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 - a \cdot \frac{c}{{b}^{2}}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -0.0004) t_0 (/ (* c (- -1.0 (* a (/ c (pow b 2.0))))) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0004) {
		tmp = t_0;
	} else {
		tmp = (c * (-1.0 - (a * (c / pow(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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.0004d0)) then
        tmp = t_0
    else
        tmp = (c * ((-1.0d0) - (a * (c / (b ** 2.0d0))))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0004) {
		tmp = t_0;
	} else {
		tmp = (c * (-1.0 - (a * (c / Math.pow(b, 2.0))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.0004:
		tmp = t_0
	else:
		tmp = (c * (-1.0 - (a * (c / math.pow(b, 2.0))))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.0004)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * Float64(-1.0 - Float64(a * Float64(c / (b ^ 2.0))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.0004)
		tmp = t_0;
	else
		tmp = (c * (-1.0 - (a * (c / (b ^ 2.0))))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0004], t$95$0, N[(N[(c * N[(-1.0 - N[(a * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $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)) < -4.00000000000000019e-4

   1. Initial program 78.3%

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

   if -4.00000000000000019e-4 < (/.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 42.3%

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

      1. *-commutative 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in b around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

      3. mul-1-neg 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in c around 0 90.7%

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

      1. sub-neg 90.7%

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

      2. metadata-eval 90.7%

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

      3. +-commutative 90.7%

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

      4. mul-1-neg 90.7%

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

      5. unsub-neg 90.7%

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

      6. associate-/l* 90.7%

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

   10. Simplified 90.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0004:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 - a \cdot \frac{c}{{b}^{2}}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.3% 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 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around inf 80.2%

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

   1. mul-1-neg 80.2%

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

   2. unsub-neg 80.2%

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

   3. mul-1-neg 80.2%

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

7. Simplified 80.2%

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

8. Taylor expanded in c around 0 80.0%

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

   1. sub-neg 80.0%

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

   2. distribute-neg-frac 80.0%

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

   3. metadata-eval 80.0%

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

   4. +-commutative 80.0%

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

   5. mul-1-neg 80.0%

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

   6. unsub-neg 80.0%

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

10. Simplified 80.0%

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

11. Final simplification 80.0%

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

Alternative 10: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 - a \cdot \frac{c}{{b}^{2}}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (- -1.0 (* a (/ c (pow b 2.0))))) b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 - (a * (c / pow(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 * ((-1.0d0) - (a * (c / (b ** 2.0d0))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 - (a * (c / Math.pow(b, 2.0))))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 - (a * (c / math.pow(b, 2.0))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 - Float64(a * Float64(c / (b ^ 2.0))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 - (a * (c / (b ^ 2.0))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-1.0 - N[(a * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around inf 80.2%

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

   1. mul-1-neg 80.2%

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

   2. unsub-neg 80.2%

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

   3. mul-1-neg 80.2%

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

7. Simplified 80.2%

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

8. Taylor expanded in c around 0 80.1%

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

   1. sub-neg 80.1%

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

   2. metadata-eval 80.1%

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

   3. +-commutative 80.1%

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

   4. mul-1-neg 80.1%

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

   5. unsub-neg 80.1%

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

   6. associate-/l* 80.1%

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

10. Simplified 80.1%

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

11. Final simplification 80.1%

    \[\leadsto \frac{c \cdot \left(-1 - a \cdot \frac{c}{{b}^{2}}\right)}{b} \]
12. Add Preprocessing

Alternative 11: 63.9% 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 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around inf 63.6%

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

   1. associate-*r/ 63.6%

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

   2. mul-1-neg 63.6%

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

7. Simplified 63.6%

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

8. Final simplification 63.6%

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

Reproduce

herbie shell --seed 2024095 
(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)))