Quadratic roots, medium range

Percentage Accurate: 31.4% → 95.5%

Time: 14.0s

Alternatives: 10

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-0.25 * ((a * 20.0) * (pow(c, 4.0) / pow(b, 7.0)))))) - (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 * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-0.25d0) * ((a * 20.0d0) * ((c ** 4.0d0) / (b ** 7.0d0)))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-0.25 * ((a * 20.0) * (Math.pow(c, 4.0) / Math.pow(b, 7.0)))))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((a * ((-2.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-0.25 * ((a * 20.0) * (math.pow(c, 4.0) / math.pow(b, 7.0)))))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((a * ((-2.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-0.25 * ((a * 20.0) * ((c ^ 4.0) / (b ^ 7.0)))))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(a * 20.0), $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 7.0], $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 32.5%

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

   1. *-commutative 32.5%

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

3. Simplified 32.5%

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

5. Taylor expanded in a around 0 94.8%

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

6. Taylor expanded in c around 0 94.8%

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

   1. associate-/l* 94.8%

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

   2. associate-*r* 94.8%

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

8. Simplified 94.8%

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

9. Final simplification 94.8%

   \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(\left(a \cdot 20\right) \cdot \frac{{c}^{4}}{{b}^{7}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 2: 94.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -3000000:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{3} \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} - \frac{\frac{a}{c}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -3000000.0)
   (/ (- (sqrt (* c (+ (* a -4.0) (/ (pow b 2.0) c)))) b) (* a 2.0))
   (-
    (*
     (pow c 3.0)
     (- (* -2.0 (/ (pow a 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)) <= -3000000.0) {
		tmp = (sqrt((c * ((a * -4.0) + (pow(b, 2.0) / c)))) - b) / (a * 2.0);
	} else {
		tmp = (pow(c, 3.0) * ((-2.0 * (pow(a, 2.0) / pow(b, 5.0))) - ((a / c) / pow(b, 3.0)))) - (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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-3000000.0d0)) then
        tmp = (sqrt((c * ((a * (-4.0d0)) + ((b ** 2.0d0) / c)))) - b) / (a * 2.0d0)
    else
        tmp = ((c ** 3.0d0) * (((-2.0d0) * ((a ** 2.0d0) / (b ** 5.0d0))) - ((a / c) / (b ** 3.0d0)))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -3000000.0) {
		tmp = (Math.sqrt((c * ((a * -4.0) + (Math.pow(b, 2.0) / c)))) - b) / (a * 2.0);
	} else {
		tmp = (Math.pow(c, 3.0) * ((-2.0 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))) - ((a / c) / Math.pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -3000000.0:
		tmp = (math.sqrt((c * ((a * -4.0) + (math.pow(b, 2.0) / c)))) - b) / (a * 2.0)
	else:
		tmp = (math.pow(c, 3.0) * ((-2.0 * (math.pow(a, 2.0) / math.pow(b, 5.0))) - ((a / c) / math.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)) <= -3000000.0)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(a * -4.0) + Float64((b ^ 2.0) / c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64((c ^ 3.0) * Float64(Float64(-2.0 * Float64((a ^ 2.0) / (b ^ 5.0))) - Float64(Float64(a / c) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -3000000.0)
		tmp = (sqrt((c * ((a * -4.0) + ((b ^ 2.0) / c)))) - b) / (a * 2.0);
	else
		tmp = ((c ^ 3.0) * ((-2.0 * ((a ^ 2.0) / (b ^ 5.0))) - ((a / c) / (b ^ 3.0)))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -3000000.0], N[(N[(N[Sqrt[N[(c * N[(N[(a * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $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)) < -3e6

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. +-commutative 91.8%

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

      3. sqr-neg 91.8%

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

      4. unsub-neg 91.8%

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

      5. sqr-neg 91.8%

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

      6. fma-neg 91.9%

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

      7. distribute-lft-neg-in 91.9%

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

      8. *-commutative 91.9%

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

      9. *-commutative 91.9%

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

      10. distribute-rgt-neg-in 91.9%

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

      11. metadata-eval 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in c around inf 91.9%

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

   if -3e6 < (/.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 30.1%

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

      1. *-commutative 30.1%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in a around 0 94.8%

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

   6. Taylor expanded in c around inf 94.8%

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

      1. mul-1-neg 94.8%

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

      2. unsub-neg 94.8%

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

      3. *-commutative 94.8%

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

      4. associate-/r* 94.8%

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

   8. Simplified 94.8%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -3000000:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{3} \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} - \frac{\frac{a}{c}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -3000000:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-2 \cdot {\left(c \cdot a\right)}^{2} - {b}^{2} \cdot \left(c \cdot a\right)}{{b}^{5}} + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -3000000.0)
   (/ (- (sqrt (* c (+ (* a -4.0) (/ (pow b 2.0) c)))) b) (* a 2.0))
   (*
    c
    (+
     (/ (- (* -2.0 (pow (* c a) 2.0)) (* (pow b 2.0) (* c a))) (pow b 5.0))
     (/ -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)) <= -3000000.0) {
		tmp = (sqrt((c * ((a * -4.0) + (pow(b, 2.0) / c)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((((-2.0 * pow((c * a), 2.0)) - (pow(b, 2.0) * (c * a))) / pow(b, 5.0)) + (-1.0 / 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 (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-3000000.0d0)) then
        tmp = (sqrt((c * ((a * (-4.0d0)) + ((b ** 2.0d0) / c)))) - b) / (a * 2.0d0)
    else
        tmp = c * (((((-2.0d0) * ((c * a) ** 2.0d0)) - ((b ** 2.0d0) * (c * a))) / (b ** 5.0d0)) + ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -3000000.0) {
		tmp = (Math.sqrt((c * ((a * -4.0) + (Math.pow(b, 2.0) / c)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((((-2.0 * Math.pow((c * a), 2.0)) - (Math.pow(b, 2.0) * (c * a))) / Math.pow(b, 5.0)) + (-1.0 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -3000000.0:
		tmp = (math.sqrt((c * ((a * -4.0) + (math.pow(b, 2.0) / c)))) - b) / (a * 2.0)
	else:
		tmp = c * ((((-2.0 * math.pow((c * a), 2.0)) - (math.pow(b, 2.0) * (c * a))) / math.pow(b, 5.0)) + (-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)) <= -3000000.0)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(a * -4.0) + Float64((b ^ 2.0) / c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(Float64(Float64(-2.0 * (Float64(c * a) ^ 2.0)) - Float64((b ^ 2.0) * Float64(c * a))) / (b ^ 5.0)) + Float64(-1.0 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -3000000.0)
		tmp = (sqrt((c * ((a * -4.0) + ((b ^ 2.0) / c)))) - b) / (a * 2.0);
	else
		tmp = c * ((((-2.0 * ((c * a) ^ 2.0)) - ((b ^ 2.0) * (c * a))) / (b ^ 5.0)) + (-1.0 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -3000000.0], N[(N[(N[Sqrt[N[(c * N[(N[(a * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(N[(N[(-2.0 * N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[b, 2.0], $MachinePrecision] * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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)) < -3e6

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. +-commutative 91.8%

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

      3. sqr-neg 91.8%

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

      4. unsub-neg 91.8%

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

      5. sqr-neg 91.8%

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

      6. fma-neg 91.9%

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

      7. distribute-lft-neg-in 91.9%

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

      8. *-commutative 91.9%

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

      9. *-commutative 91.9%

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

      10. distribute-rgt-neg-in 91.9%

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

      11. metadata-eval 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in c around inf 91.9%

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

   if -3e6 < (/.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 30.1%

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

      1. *-commutative 30.1%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in c around 0 94.6%

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

   6. Taylor expanded in b around 0 94.6%

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

      1. fma-define 94.6%

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

      2. unpow2 94.6%

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

      3. unpow2 94.6%

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

      4. swap-sqr 94.6%

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

      5. unpow2 94.6%

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

      6. mul-1-neg 94.6%

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

      7. fma-neg 94.6%

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

      8. *-commutative 94.6%

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

      9. associate-*r* 94.6%

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

   8. Simplified 94.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -3000000:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-2 \cdot {\left(c \cdot a\right)}^{2} - {b}^{2} \cdot \left(c \cdot a\right)}{{b}^{5}} + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -2000000:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -2000000.0)
   (/ (- (sqrt (* c (+ (* a -4.0) (/ (pow b 2.0) c)))) b) (* a 2.0))
   (/ (fma a (pow (/ c b) 2.0) 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)) <= -2000000.0) {
		tmp = (sqrt((c * ((a * -4.0) + (pow(b, 2.0) / c)))) - b) / (a * 2.0);
	} else {
		tmp = fma(a, pow((c / b), 2.0), 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)) <= -2000000.0)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(a * -4.0) + Float64((b ^ 2.0) / c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-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], -2000000.0], N[(N[(N[Sqrt[N[(c * N[(N[(a * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + c), $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)) < -2e6

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

      2. +-commutative 90.7%

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

      3. sqr-neg 90.7%

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

      4. unsub-neg 90.7%

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

      5. sqr-neg 90.7%

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

      6. fma-neg 90.8%

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

      7. distribute-lft-neg-in 90.8%

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

      8. *-commutative 90.8%

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

      9. *-commutative 90.8%

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

      10. distribute-rgt-neg-in 90.8%

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

      11. metadata-eval 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in c around inf 90.8%

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

   if -2e6 < (/.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 29.9%

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

      1. *-commutative 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in c around 0 91.4%

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

      1. associate-*r/ 91.4%

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

      2. neg-mul-1 91.4%

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

      3. distribute-rgt-neg-in 91.4%

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

   7. Simplified 91.4%

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

   8. Taylor expanded in a around inf 91.3%

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

      1. pow1 91.3%

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

      2. associate-*r* 91.3%

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

      3. mul-1-neg 91.3%

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

      4. div-inv 91.3%

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

      5. pow-flip 91.3%

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

      6. metadata-eval 91.3%

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

   10. Applied egg-rr 91.3%

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

       1. unpow1 91.3%

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

       2. *-commutative 91.3%

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

       3. distribute-lft-neg-in 91.3%

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

   12. Simplified 91.3%

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

   13. Taylor expanded in a around 0 91.7%

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

       1. neg-mul-1 91.7%

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

       2. mul-1-neg 91.7%

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

       3. unsub-neg 91.7%

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

       4. distribute-neg-frac 91.7%

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

       5. unpow3 91.7%

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

       6. unpow2 91.7%

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

       7. associate-/r* 91.7%

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

       8. div-sub 91.7%

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

       9. unsub-neg 91.7%

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

       10. distribute-neg-out 91.7%

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

       11. mul-1-neg 91.7%

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

   15. Simplified 91.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -2000000:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -2000000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -2000000.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (fma a (pow (/ c b) 2.0) 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)) <= -2000000.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(a, pow((c / b), 2.0), 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)) <= -2000000.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-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], -2000000.0], 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[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + c), $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)) < -2e6

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

      2. +-commutative 90.7%

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

      3. sqr-neg 90.7%

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

      4. unsub-neg 90.7%

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

      5. sqr-neg 90.7%

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

      6. fma-neg 90.8%

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

      7. distribute-lft-neg-in 90.8%

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

      8. *-commutative 90.8%

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

      9. *-commutative 90.8%

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

      10. distribute-rgt-neg-in 90.8%

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

      11. metadata-eval 90.8%

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

   3. Simplified 90.8%

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

   if -2e6 < (/.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 29.9%

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

      1. *-commutative 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in c around 0 91.4%

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

      1. associate-*r/ 91.4%

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

      2. neg-mul-1 91.4%

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

      3. distribute-rgt-neg-in 91.4%

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

   7. Simplified 91.4%

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

   8. Taylor expanded in a around inf 91.3%

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

      1. pow1 91.3%

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

      2. associate-*r* 91.3%

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

      3. mul-1-neg 91.3%

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

      4. div-inv 91.3%

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

      5. pow-flip 91.3%

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

      6. metadata-eval 91.3%

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

   10. Applied egg-rr 91.3%

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

       1. unpow1 91.3%

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

       2. *-commutative 91.3%

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

       3. distribute-lft-neg-in 91.3%

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

   12. Simplified 91.3%

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

   13. Taylor expanded in a around 0 91.7%

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

       1. neg-mul-1 91.7%

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

       2. mul-1-neg 91.7%

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

       3. unsub-neg 91.7%

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

       4. distribute-neg-frac 91.7%

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

       5. unpow3 91.7%

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

       6. unpow2 91.7%

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

       7. associate-/r* 91.7%

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

       8. div-sub 91.7%

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

       9. unsub-neg 91.7%

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

       10. distribute-neg-out 91.7%

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

       11. mul-1-neg 91.7%

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

   15. Simplified 91.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -2000000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -2000000.0) t_0 (/ (fma a (pow (/ c b) 2.0) 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 <= -2000000.0) {
		tmp = t_0;
	} else {
		tmp = fma(a, pow((c / b), 2.0), 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 <= -2000000.0)
		tmp = t_0;
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-b));
	end
	return 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, -2000000.0], t$95$0, N[(N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + c), $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)) < -2e6

   1. Initial program 90.7%

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

   if -2e6 < (/.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 29.9%

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

      1. *-commutative 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in c around 0 91.4%

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

      1. associate-*r/ 91.4%

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

      2. neg-mul-1 91.4%

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

      3. distribute-rgt-neg-in 91.4%

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

   7. Simplified 91.4%

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

   8. Taylor expanded in a around inf 91.3%

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

      1. pow1 91.3%

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

      2. associate-*r* 91.3%

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

      3. mul-1-neg 91.3%

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

      4. div-inv 91.3%

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

      5. pow-flip 91.3%

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

      6. metadata-eval 91.3%

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

   10. Applied egg-rr 91.3%

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

       1. unpow1 91.3%

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

       2. *-commutative 91.3%

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

       3. distribute-lft-neg-in 91.3%

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

   12. Simplified 91.3%

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

   13. Taylor expanded in a around 0 91.7%

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

       1. neg-mul-1 91.7%

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

       2. mul-1-neg 91.7%

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

       3. unsub-neg 91.7%

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

       4. distribute-neg-frac 91.7%

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

       5. unpow3 91.7%

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

       6. unpow2 91.7%

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

       7. associate-/r* 91.7%

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

       8. div-sub 91.7%

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

       9. unsub-neg 91.7%

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

       10. distribute-neg-out 91.7%

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

       11. mul-1-neg 91.7%

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

   15. Simplified 91.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -2000000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \left(c \cdot a\right) \cdot {b}^{-3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -2000000.0) t_0 (* c (- (/ -1.0 b) (* (* c a) (pow b -3.0)))))))

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 <= -2000000.0) {
		tmp = t_0;
	} else {
		tmp = c * ((-1.0 / b) - ((c * a) * pow(b, -3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-2000000.0d0)) then
        tmp = t_0
    else
        tmp = c * (((-1.0d0) / b) - ((c * a) * (b ** (-3.0d0))))
    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 <= -2000000.0) {
		tmp = t_0;
	} else {
		tmp = c * ((-1.0 / b) - ((c * a) * Math.pow(b, -3.0)));
	}
	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 <= -2000000.0:
		tmp = t_0
	else:
		tmp = c * ((-1.0 / b) - ((c * a) * math.pow(b, -3.0)))
	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 <= -2000000.0)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(c * a) * (b ^ -3.0))));
	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 <= -2000000.0)
		tmp = t_0;
	else
		tmp = c * ((-1.0 / b) - ((c * a) * (b ^ -3.0)));
	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, -2000000.0], t$95$0, N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $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)) < -2e6

   1. Initial program 90.7%

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

   if -2e6 < (/.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 29.9%

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

      1. *-commutative 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in c around 0 91.4%

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

      1. associate-*r/ 91.4%

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

      2. neg-mul-1 91.4%

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

      3. distribute-rgt-neg-in 91.4%

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

   7. Simplified 91.4%

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

      1. div-inv 91.4%

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

      2. *-un-lft-identity 91.4%

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

      3. prod-diff 91.4%

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

      4. pow-flip 91.4%

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

      5. metadata-eval 91.4%

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

   9. Applied egg-rr 91.4%

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

       1. +-commutative 91.4%

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

       2. fma-undefine 91.4%

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

       3. *-rgt-identity 91.4%

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

       4. associate-+r+ 91.4%

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

   11. Simplified 91.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -2000000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \left(c \cdot a\right) \cdot {b}^{-3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.5% accurate, 1.0× speedup

Math

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

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 32.5%

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

   1. *-commutative 32.5%

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

3. Simplified 32.5%

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

5. Taylor expanded in c around 0 89.4%

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

   1. associate-*r/ 89.4%

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

   2. neg-mul-1 89.4%

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

   3. distribute-rgt-neg-in 89.4%

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

7. Simplified 89.4%

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

   1. div-inv 89.4%

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

   2. *-un-lft-identity 89.4%

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

   3. prod-diff 89.4%

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

   4. pow-flip 89.4%

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

   5. metadata-eval 89.4%

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

9. Applied egg-rr 89.4%

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

    1. +-commutative 89.4%

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

    2. fma-undefine 89.4%

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

    3. *-rgt-identity 89.4%

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

    4. associate-+r+ 89.4%

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

11. Simplified 89.4%

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

12. Final simplification 89.4%

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

Alternative 9: 81.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.5%

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

   1. *-commutative 32.5%

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

3. Simplified 32.5%

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

5. Taylor expanded in b around inf 80.4%

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

   1. associate-*r/ 80.4%

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

   2. mul-1-neg 80.4%

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

7. Simplified 80.4%

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

8. Final simplification 80.4%

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

Alternative 10: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 32.5%

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

   1. *-commutative 32.5%

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

3. Simplified 32.5%

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

5. Taylor expanded in c around 0 89.4%

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

   1. associate-*r/ 89.4%

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

   2. neg-mul-1 89.4%

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

   3. distribute-rgt-neg-in 89.4%

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

7. Simplified 89.4%

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

8. Taylor expanded in a around 0 80.2%

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

   1. expm1-log1p-u 70.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)\right)} \]

   2. expm1-undefine 30.3%

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

   3. associate-*r/ 30.3%

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

10. Applied egg-rr 30.3%

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

    1. sub-neg 30.3%

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

    2. metadata-eval 30.3%

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

    3. +-commutative 30.3%

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

    4. log1p-undefine 30.3%

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

    5. rem-exp-log 39.7%

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

    6. *-commutative 39.7%

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

    7. associate-*r/ 39.7%

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

    8. mul-1-neg 39.7%

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

    9. unsub-neg 39.7%

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

12. Simplified 39.7%

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

13. Taylor expanded in c around 0 3.2%

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

14. Final simplification 3.2%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))