Quadratic roots, narrow range

Percentage Accurate: 55.6% → 91.9%

Time: 13.2s

Alternatives: 8

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -90:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -90.0)
     t_0
     (/
      (-
       (*
        a
        (-
         (*
          a
          (+
           (* -5.0 (/ (* a (pow c 4.0)) (pow b 6.0)))
           (* -2.0 (/ (pow c 3.0) (pow b 4.0)))))
         (/ (pow c 2.0) (pow b 2.0))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -90.0) {
		tmp = t_0;
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-2.0 * (pow(c, 3.0) / pow(b, 4.0))))) - (pow(c, 2.0) / pow(b, 2.0)))) - 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-90.0d0)) then
        tmp = t_0
    else
        tmp = ((a * ((a * (((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 6.0d0))) + ((-2.0d0) * ((c ** 3.0d0) / (b ** 4.0d0))))) - ((c ** 2.0d0) / (b ** 2.0d0)))) - 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) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -90.0) {
		tmp = t_0;
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 4.0))))) - (Math.pow(c, 2.0) / Math.pow(b, 2.0)))) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -90.0:
		tmp = t_0
	else:
		tmp = ((a * ((a * ((-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (-2.0 * (math.pow(c, 3.0) / math.pow(b, 4.0))))) - (math.pow(c, 2.0) / math.pow(b, 2.0)))) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -90.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(a * Float64(Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 4.0))))) - Float64((c ^ 2.0) / (b ^ 2.0)))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -90.0)
		tmp = t_0;
	else
		tmp = ((a * ((a * ((-5.0 * ((a * (c ^ 4.0)) / (b ^ 6.0))) + (-2.0 * ((c ^ 3.0) / (b ^ 4.0))))) - ((c ^ 2.0) / (b ^ 2.0)))) - 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -90.0], t$95$0, N[(N[(N[(a * N[(N[(a * N[(N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -90

   1. Initial program 90.0%

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

   if -90 < (/.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 50.3%

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

      1. *-commutative 50.3%

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

      2. +-commutative 50.3%

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

      3. sqr-neg 50.3%

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

      4. unsub-neg 50.3%

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

      5. sqr-neg 50.3%

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

      6. fma-neg 50.3%

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

      7. distribute-lft-neg-in 50.3%

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

      8. *-commutative 50.3%

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

      9. *-commutative 50.3%

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

      10. distribute-rgt-neg-in 50.3%

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

      11. metadata-eval 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in b around inf 93.3%

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

   7. Simplified 93.3%

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

   8. Taylor expanded in a around 0 93.3%

      \[\leadsto \frac{\color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right) - c}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -90:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -90:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{c}{b} \cdot \frac{c}{b}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -90.0)
     t_0
     (/
      (-
       (*
        a
        (-
         (*
          a
          (+
           (* -5.0 (/ (* a (pow c 4.0)) (pow b 6.0)))
           (* -2.0 (/ (pow c 3.0) (pow b 4.0)))))
         (* (/ c b) (/ c b))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -90.0) {
		tmp = t_0;
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-2.0 * (pow(c, 3.0) / pow(b, 4.0))))) - ((c / b) * (c / b)))) - c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-90.0d0)) then
        tmp = t_0
    else
        tmp = ((a * ((a * (((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 6.0d0))) + ((-2.0d0) * ((c ** 3.0d0) / (b ** 4.0d0))))) - ((c / b) * (c / b)))) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -90.0) {
		tmp = t_0;
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 4.0))))) - ((c / b) * (c / b)))) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -90.0:
		tmp = t_0
	else:
		tmp = ((a * ((a * ((-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (-2.0 * (math.pow(c, 3.0) / math.pow(b, 4.0))))) - ((c / b) * (c / b)))) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -90.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(a * Float64(Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 4.0))))) - Float64(Float64(c / b) * Float64(c / b)))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -90.0)
		tmp = t_0;
	else
		tmp = ((a * ((a * ((-5.0 * ((a * (c ^ 4.0)) / (b ^ 6.0))) + (-2.0 * ((c ^ 3.0) / (b ^ 4.0))))) - ((c / b) * (c / b)))) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -90.0], t$95$0, N[(N[(N[(a * N[(N[(a * N[(N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -90

   1. Initial program 90.0%

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

   if -90 < (/.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 50.3%

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

      1. *-commutative 50.3%

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

      2. +-commutative 50.3%

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

      3. sqr-neg 50.3%

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

      4. unsub-neg 50.3%

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

      5. sqr-neg 50.3%

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

      6. fma-neg 50.3%

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

      7. distribute-lft-neg-in 50.3%

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

      8. *-commutative 50.3%

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

      9. *-commutative 50.3%

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

      10. distribute-rgt-neg-in 50.3%

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

      11. metadata-eval 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in b around inf 93.3%

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

   7. Simplified 93.3%

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

   8. Taylor expanded in a around 0 93.3%

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

      1. associate-*r/ 93.3%

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

   10. Applied egg-rr 93.3%

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

       1. associate-*r/ 93.3%

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

       2. mul-1-neg 93.3%

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

       3. unpow2 93.3%

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

       4. unpow2 93.3%

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

       5. times-frac 93.3%

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

       6. sqr-neg 93.3%

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

       7. distribute-frac-neg2 93.3%

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

       8. distribute-frac-neg2 93.3%

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

       9. unpow2 93.3%

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

   12. Simplified 93.3%

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

       1. unpow2 93.3%

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

   14. Applied egg-rr 93.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -90:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{c}{b} \cdot \frac{c}{b}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -52.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{4}}\right) - {\left(\frac{c}{-b}\right)}^{2}, -c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -52.2)
     t_0
     (/
      (fma
       a
       (- (* -2.0 (* a (/ (pow c 3.0) (pow b 4.0)))) (pow (/ c (- b)) 2.0))
       (- c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -52.2) {
		tmp = t_0;
	} else {
		tmp = fma(a, ((-2.0 * (a * (pow(c, 3.0) / pow(b, 4.0)))) - 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(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -52.2)
		tmp = t_0;
	else
		tmp = Float64(fma(a, Float64(Float64(-2.0 * Float64(a * Float64((c ^ 3.0) / (b ^ 4.0)))) - (Float64(c / Float64(-b)) ^ 2.0)), Float64(-c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -52.2], t$95$0, N[(N[(a * N[(N[(-2.0 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.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)) < -52.200000000000003

   1. Initial program 87.8%

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

   if -52.200000000000003 < (/.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 49.8%

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

      1. *-commutative 49.8%

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

      2. +-commutative 49.8%

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

      3. sqr-neg 49.8%

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

      4. unsub-neg 49.8%

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

      5. sqr-neg 49.8%

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

      6. fma-neg 49.8%

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

      7. distribute-lft-neg-in 49.8%

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

      8. *-commutative 49.8%

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

      9. *-commutative 49.8%

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

      10. distribute-rgt-neg-in 49.8%

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

      11. metadata-eval 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in b around inf 93.4%

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

   7. Simplified 93.4%

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

   8. Taylor expanded in a around 0 90.5%

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

      1. fma-neg 90.5%

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

      2. mul-1-neg 90.5%

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

      3. unsub-neg 90.5%

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

      4. associate-/l* 90.5%

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

      5. unpow2 90.5%

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

      6. unpow2 90.5%

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

      7. times-frac 90.5%

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

      8. sqr-neg 90.5%

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

      9. distribute-frac-neg 90.5%

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

      10. distribute-frac-neg 90.5%

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

      11. unpow1 90.5%

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

      12. pow-plus 90.5%

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

      13. distribute-frac-neg 90.5%

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

      14. distribute-neg-frac2 90.5%

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

      15. metadata-eval 90.5%

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

   10. Simplified 90.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -52.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{4}}\right) - {\left(\frac{c}{-b}\right)}^{2}, -c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -52.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -52.2)
     t_0
     (-
      (*
       a
       (- (* -2.0 (* a (/ (pow c 3.0) (pow b 5.0)))) (/ (* c c) (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -52.2) {
		tmp = t_0;
	} else {
		tmp = (a * ((-2.0 * (a * (pow(c, 3.0) / pow(b, 5.0)))) - ((c * 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-52.2d0)) then
        tmp = t_0
    else
        tmp = (a * (((-2.0d0) * (a * ((c ** 3.0d0) / (b ** 5.0d0)))) - ((c * c) / (b ** 3.0d0)))) - (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) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -52.2) {
		tmp = t_0;
	} else {
		tmp = (a * ((-2.0 * (a * (Math.pow(c, 3.0) / Math.pow(b, 5.0)))) - ((c * c) / Math.pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -52.2:
		tmp = t_0
	else:
		tmp = (a * ((-2.0 * (a * (math.pow(c, 3.0) / math.pow(b, 5.0)))) - ((c * c) / math.pow(b, 3.0)))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -52.2)
		tmp = t_0;
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(a * Float64((c ^ 3.0) / (b ^ 5.0)))) - Float64(Float64(c * c) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -52.2)
		tmp = t_0;
	else
		tmp = (a * ((-2.0 * (a * ((c ^ 3.0) / (b ^ 5.0)))) - ((c * c) / (b ^ 3.0)))) - (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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -52.2], t$95$0, N[(N[(a * N[(N[(-2.0 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.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)) < -52.200000000000003

   1. Initial program 87.8%

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

   if -52.200000000000003 < (/.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 49.8%

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

      1. *-commutative 49.8%

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

      2. +-commutative 49.8%

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

      3. sqr-neg 49.8%

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

      4. unsub-neg 49.8%

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

      5. sqr-neg 49.8%

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

      6. fma-neg 49.8%

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

      7. distribute-lft-neg-in 49.8%

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

      8. *-commutative 49.8%

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

      9. *-commutative 49.8%

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

      10. distribute-rgt-neg-in 49.8%

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

      11. metadata-eval 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in b around inf 93.4%

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

   7. Simplified 93.4%

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

   8. Taylor expanded in a around 0 90.4%

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

      1. neg-mul-1 90.4%

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

      2. distribute-frac-neg 90.4%

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

      3. +-commutative 90.4%

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

      4. distribute-frac-neg 90.4%

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

      5. unsub-neg 90.4%

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

      6. mul-1-neg 90.4%

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

      7. unsub-neg 90.4%

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

      8. associate-/l* 90.4%

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

   10. Simplified 90.4%

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

       1. unpow2 90.4%

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

   12. Applied egg-rr 90.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -52.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -52.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 - c \cdot \left(a \cdot \left(\frac{1}{{b}^{2}} - -2 \cdot \frac{a \cdot c}{{b}^{4}}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -52.2)
     t_0
     (/
      (*
       c
       (-
        -1.0
        (* c (* a (- (/ 1.0 (pow b 2.0)) (* -2.0 (/ (* a c) (pow b 4.0))))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -52.2) {
		tmp = t_0;
	} else {
		tmp = (c * (-1.0 - (c * (a * ((1.0 / pow(b, 2.0)) - (-2.0 * ((a * c) / pow(b, 4.0)))))))) / 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) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-52.2d0)) then
        tmp = t_0
    else
        tmp = (c * ((-1.0d0) - (c * (a * ((1.0d0 / (b ** 2.0d0)) - ((-2.0d0) * ((a * c) / (b ** 4.0d0)))))))) / 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) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -52.2) {
		tmp = t_0;
	} else {
		tmp = (c * (-1.0 - (c * (a * ((1.0 / Math.pow(b, 2.0)) - (-2.0 * ((a * c) / Math.pow(b, 4.0)))))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -52.2:
		tmp = t_0
	else:
		tmp = (c * (-1.0 - (c * (a * ((1.0 / math.pow(b, 2.0)) - (-2.0 * ((a * c) / math.pow(b, 4.0)))))))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -52.2)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * Float64(-1.0 - Float64(c * Float64(a * Float64(Float64(1.0 / (b ^ 2.0)) - Float64(-2.0 * Float64(Float64(a * c) / (b ^ 4.0)))))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -52.2)
		tmp = t_0;
	else
		tmp = (c * (-1.0 - (c * (a * ((1.0 / (b ^ 2.0)) - (-2.0 * ((a * c) / (b ^ 4.0)))))))) / 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -52.2], t$95$0, N[(N[(c * N[(-1.0 - N[(c * N[(a * N[(N[(1.0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -52.200000000000003

   1. Initial program 87.8%

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

   if -52.200000000000003 < (/.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 49.8%

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

      1. *-commutative 49.8%

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

      2. +-commutative 49.8%

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

      3. sqr-neg 49.8%

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

      4. unsub-neg 49.8%

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

      5. sqr-neg 49.8%

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

      6. fma-neg 49.8%

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

      7. distribute-lft-neg-in 49.8%

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

      8. *-commutative 49.8%

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

      9. *-commutative 49.8%

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

      10. distribute-rgt-neg-in 49.8%

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

      11. metadata-eval 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in b around inf 93.4%

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

   7. Simplified 93.4%

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

   8. Taylor expanded in c around 0 90.4%

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

   9. Taylor expanded in a around 0 90.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -52.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 - c \cdot \left(a \cdot \left(\frac{1}{{b}^{2}} - -2 \cdot \frac{a \cdot c}{{b}^{4}}\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.04) t_0 (/ (+ c (* a (pow (/ c (- b)) 2.0))) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_0;
	} else {
		tmp = (c + (a * pow((c / -b), 2.0))) / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.04d0)) then
        tmp = t_0
    else
        tmp = (c + (a * ((c / -b) ** 2.0d0))) / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_0;
	} else {
		tmp = (c + (a * Math.pow((c / -b), 2.0))) / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.04:
		tmp = t_0
	else:
		tmp = (c + (a * math.pow((c / -b), 2.0))) / -b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = t_0;
	else
		tmp = Float64(Float64(c + Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.04)
		tmp = t_0;
	else
		tmp = (c + (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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.04], t$95$0, N[(N[(c + N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.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)) < -0.0400000000000000008

   1. Initial program 80.8%

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

   if -0.0400000000000000008 < (/.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 44.8%

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

      1. *-commutative 44.8%

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

      2. +-commutative 44.8%

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

      3. sqr-neg 44.8%

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

      4. unsub-neg 44.8%

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

      5. sqr-neg 44.8%

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

      6. fma-neg 44.7%

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

      7. distribute-lft-neg-in 44.7%

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

      8. *-commutative 44.7%

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

      9. *-commutative 44.7%

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

      10. distribute-rgt-neg-in 44.7%

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

      11. metadata-eval 44.7%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in b around inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. unsub-neg 87.6%

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

      3. mul-1-neg 87.6%

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

      4. associate-/l* 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in a around 0 87.6%

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

      1. associate-/l* 87.6%

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

      2. unpow2 87.6%

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

      3. unpow2 87.6%

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

      4. times-frac 87.6%

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

      5. sqr-neg 87.6%

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

      6. distribute-frac-neg 87.6%

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

      7. distribute-frac-neg 87.6%

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

      8. unpow1 87.6%

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

      9. pow-plus 87.6%

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

      10. distribute-frac-neg 87.6%

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

      11. distribute-neg-frac2 87.6%

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

      12. metadata-eval 87.6%

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

   10. Simplified 87.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.04:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

   1. *-commutative 52.8%

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

   2. +-commutative 52.8%

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

   3. sqr-neg 52.8%

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

   4. unsub-neg 52.8%

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

   5. sqr-neg 52.8%

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

   6. fma-neg 52.8%

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

   7. distribute-lft-neg-in 52.8%

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

   8. *-commutative 52.8%

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

   9. *-commutative 52.8%

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

   10. distribute-rgt-neg-in 52.8%

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

   11. metadata-eval 52.8%

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

3. Simplified 52.8%

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

5. Taylor expanded in b around inf 81.4%

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

   1. mul-1-neg 81.4%

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

   2. unsub-neg 81.4%

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

   3. mul-1-neg 81.4%

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

   4. associate-/l* 81.4%

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

7. Simplified 81.4%

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

8. Taylor expanded in a around 0 81.4%

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

   1. associate-/l* 81.4%

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

   2. unpow2 81.4%

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

   3. unpow2 81.4%

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

   4. times-frac 81.4%

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

   5. sqr-neg 81.4%

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

   6. distribute-frac-neg 81.4%

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

   7. distribute-frac-neg 81.4%

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

   8. unpow1 81.4%

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

   9. pow-plus 81.4%

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

   10. distribute-frac-neg 81.4%

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

   11. distribute-neg-frac2 81.4%

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

   12. metadata-eval 81.4%

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

10. Simplified 81.4%

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

11. Final simplification 81.4%

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

Alternative 8: 64.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

   1. *-commutative 52.8%

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

   2. +-commutative 52.8%

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

   3. sqr-neg 52.8%

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

   4. unsub-neg 52.8%

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

   5. sqr-neg 52.8%

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

   6. fma-neg 52.8%

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

   7. distribute-lft-neg-in 52.8%

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

   8. *-commutative 52.8%

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

   9. *-commutative 52.8%

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

   10. distribute-rgt-neg-in 52.8%

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

   11. metadata-eval 52.8%

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

3. Simplified 52.8%

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

5. Taylor expanded in b around inf 66.1%

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

   1. associate-*r/ 66.1%

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

   2. mul-1-neg 66.1%

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

7. Simplified 66.1%

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

8. Final simplification 66.1%

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

Reproduce

herbie shell --seed 2024139 
(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)))