Quadratic roots, narrow range

Percentage Accurate: 55.7% → 90.9%

Time: 18.9s

Alternatives: 9

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: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (+
      (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
      (* -0.25 (* a (/ (* (pow c 4.0) 20.0) (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 * ((pow(c, 4.0) * 20.0) / 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 * (((c ** 4.0d0) * 20.0d0) / (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 * ((Math.pow(c, 4.0) * 20.0) / 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 * ((math.pow(c, 4.0) * 20.0) / 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(a * Float64(Float64((c ^ 4.0) * 20.0) / (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 * (((c ^ 4.0) * 20.0) / (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[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $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 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in a around 0 92.0%

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

6. Taylor expanded in b around 0 92.0%

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

   1. associate-/l* 92.0%

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

   2. distribute-rgt-out 92.0%

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

   3. metadata-eval 92.0%

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

8. Simplified 92.0%

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

9. Final simplification 92.0%

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

Alternative 2: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -260:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{4}}, -2 \cdot \left(c \cdot a + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{2}}\right)\right)}{b}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (fma c -4.0 (/ (pow b 2.0) a)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -260.0)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 2.0))
     (/
      (/
       (fma
        -4.0
        (/ (pow (* c a) 3.0) (pow b 4.0))
        (* -2.0 (+ (* c a) (/ (* (pow c 2.0) (pow a 2.0)) (pow b 2.0)))))
       b)
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = a * fma(c, -4.0, (pow(b, 2.0) / a));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -260.0) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (fma(-4.0, (pow((c * a), 3.0) / pow(b, 4.0)), (-2.0 * ((c * a) + ((pow(c, 2.0) * pow(a, 2.0)) / pow(b, 2.0))))) / b) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * fma(c, -4.0, Float64((b ^ 2.0) / a)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -260.0)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-4.0, Float64((Float64(c * a) ^ 3.0) / (b ^ 4.0)), Float64(-2.0 * Float64(Float64(c * a) + Float64(Float64((c ^ 2.0) * (a ^ 2.0)) / (b ^ 2.0))))) / b) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0 + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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], -260.0], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * N[(N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(c * a), $MachinePrecision] + N[(N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / N[(a * 2.0), $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)) < -260

   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 91.8%

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

      7. distribute-lft-neg-in 91.8%

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

      8. *-commutative 91.8%

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

      9. *-commutative 91.8%

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

      10. distribute-rgt-neg-in 91.8%

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

      11. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in a around inf 91.8%

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

      1. flip-- 91.7%

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

      2. add-sqr-sqrt 92.5%

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

      3. *-commutative 92.5%

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

      4. fma-define 92.5%

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

      5. unpow2 92.5%

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

      6. *-commutative 92.5%

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

      7. fma-define 92.5%

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

   7. Applied egg-rr 92.5%

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

   if -260 < (/.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 54.2%

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

      1. *-commutative 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in b around inf 90.4%

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

      1. fma-define 90.4%

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

      2. cube-prod 90.4%

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

      3. distribute-lft-out 90.4%

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

   7. Simplified 90.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -260:\\ \;\;\;\;\frac{\frac{a \cdot \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right) - {b}^{2}}{b + \sqrt{a \cdot \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{4}}, -2 \cdot \left(c \cdot a + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{2}}\right)\right)}{b}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -260:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{a \cdot \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -260.0)
   (/ 1.0 (/ (* a 2.0) (- (sqrt (* a (fma c -4.0 (/ (pow b 2.0) a)))) b)))
   (*
    c
    (+
     (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -260.0) {
		tmp = 1.0 / ((a * 2.0) / (sqrt((a * fma(c, -4.0, (pow(b, 2.0) / a)))) - b));
	} else {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -260.0)
		tmp = Float64(1.0 / Float64(Float64(a * 2.0) / Float64(sqrt(Float64(a * fma(c, -4.0, Float64((b ^ 2.0) / a)))) - b)));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -260.0], N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(a * N[(c * -4.0 + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -260

   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 91.8%

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

      7. distribute-lft-neg-in 91.8%

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

      8. *-commutative 91.8%

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

      9. *-commutative 91.8%

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

      10. distribute-rgt-neg-in 91.8%

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

      11. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in a around inf 91.8%

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

      1. log1p-expm1-u 3.2%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sqrt{a \cdot \left(-4 \cdot c + \frac{{b}^{2}}{a}\right)} - b}{a \cdot 2}\right)\right)} \]

      2. log1p-undefine 3.2%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{a \cdot \left(-4 \cdot c + \frac{{b}^{2}}{a}\right)} - b}{a \cdot 2}\right)\right)} \]

      3. *-commutative 3.2%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{a \cdot \left(\color{blue}{c \cdot -4} + \frac{{b}^{2}}{a}\right)} - b}{a \cdot 2}\right)\right) \]

      4. fma-define 3.2%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{a \cdot \color{blue}{\mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)}} - b}{a \cdot 2}\right)\right) \]

   7. Applied egg-rr 3.2%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{a \cdot \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)} - b}{a \cdot 2}\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-define 3.2%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sqrt{a \cdot \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)} - b}{a \cdot 2}\right)\right)} \]

      2. log1p-expm1-u 91.8%

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

      3. clear-num 91.8%

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

   9. Applied egg-rr 91.8%

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

   if -260 < (/.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 54.2%

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

      1. *-commutative 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in c around 0 90.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -260:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{a \cdot \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.4:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \left(c \cdot a + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{2}}\right)}{b}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -0.4)
     t_0
     (/
      (/ (* -2.0 (+ (* c a) (/ (* (pow c 2.0) (pow a 2.0)) (pow b 2.0)))) b)
      (* a 2.0)))))

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 <= -0.4) {
		tmp = t_0;
	} else {
		tmp = ((-2.0 * ((c * a) + ((pow(c, 2.0) * pow(a, 2.0)) / pow(b, 2.0)))) / b) / (a * 2.0);
	}
	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 <= (-0.4d0)) then
        tmp = t_0
    else
        tmp = (((-2.0d0) * ((c * a) + (((c ** 2.0d0) * (a ** 2.0d0)) / (b ** 2.0d0)))) / b) / (a * 2.0d0)
    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 <= -0.4) {
		tmp = t_0;
	} else {
		tmp = ((-2.0 * ((c * a) + ((Math.pow(c, 2.0) * Math.pow(a, 2.0)) / Math.pow(b, 2.0)))) / b) / (a * 2.0);
	}
	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 <= -0.4:
		tmp = t_0
	else:
		tmp = ((-2.0 * ((c * a) + ((math.pow(c, 2.0) * math.pow(a, 2.0)) / math.pow(b, 2.0)))) / b) / (a * 2.0)
	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 <= -0.4)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(c * a) + Float64(Float64((c ^ 2.0) * (a ^ 2.0)) / (b ^ 2.0)))) / b) / Float64(a * 2.0));
	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 <= -0.4)
		tmp = t_0;
	else
		tmp = ((-2.0 * ((c * a) + (((c ^ 2.0) * (a ^ 2.0)) / (b ^ 2.0)))) / b) / (a * 2.0);
	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, -0.4], t$95$0, N[(N[(N[(-2.0 * N[(N[(c * a), $MachinePrecision] + N[(N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.40000000000000002

   1. Initial program 81.2%

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

   if -0.40000000000000002 < (/.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 50.5%

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

      1. *-commutative 50.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in b around inf 86.6%

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

      1. distribute-lft-out 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \left(c \cdot a + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{2}}\right)}{b}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ a \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{a}{{b}^{5}} + \frac{-1}{c \cdot {b}^{3}}\right)\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (* (pow c 3.0) (+ (* -2.0 (/ a (pow b 5.0))) (/ -1.0 (* c (pow b 3.0))))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * (pow(c, 3.0) * ((-2.0 * (a / pow(b, 5.0))) + (-1.0 / (c * pow(b, 3.0)))))) - (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * ((c ** 3.0d0) * (((-2.0d0) * (a / (b ** 5.0d0))) + ((-1.0d0) / (c * (b ** 3.0d0)))))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * (Math.pow(c, 3.0) * ((-2.0 * (a / Math.pow(b, 5.0))) + (-1.0 / (c * Math.pow(b, 3.0)))))) - (c / b);
}

Python

def code(a, b, c):
	return (a * (math.pow(c, 3.0) * ((-2.0 * (a / math.pow(b, 5.0))) + (-1.0 / (c * math.pow(b, 3.0)))))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64((c ^ 3.0) * Float64(Float64(-2.0 * Float64(a / (b ^ 5.0))) + Float64(-1.0 / Float64(c * (b ^ 3.0)))))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((c ^ 3.0) * ((-2.0 * (a / (b ^ 5.0))) + (-1.0 / (c * (b ^ 3.0)))))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(-2.0 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in a around 0 88.7%

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

6. Taylor expanded in c around inf 88.7%

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

7. Final simplification 88.7%

   \[\leadsto a \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{a}{{b}^{5}} + \frac{-1}{c \cdot {b}^{3}}\right)\right) - \frac{c}{b} \]
8. Add Preprocessing

Alternative 6: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \frac{-{c}^{2}}{{b}^{3}} - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (- (* a (/ (- (pow c 2.0)) (pow b 3.0))) (/ c b)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in a around 0 82.2%

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

   1. mul-1-neg 82.2%

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

   2. unsub-neg 82.2%

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

   3. mul-1-neg 82.2%

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

   4. distribute-neg-frac2 82.2%

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

   5. associate-/l* 82.2%

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

7. Simplified 82.2%

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

8. Final simplification 82.2%

   \[\leadsto a \cdot \frac{-{c}^{2}}{{b}^{3}} - \frac{c}{b} \]
9. Add Preprocessing

Alternative 7: 81.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in b around inf 82.2%

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

   1. mul-1-neg 82.2%

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

   2. unsub-neg 82.2%

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

   3. mul-1-neg 82.2%

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

7. Simplified 82.2%

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

   1. unpow2 82.2%

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

9. Applied egg-rr 82.2%

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

10. Final simplification 82.2%

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

Alternative 8: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in c around 0 82.1%

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

   1. associate-*r/ 82.1%

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

   2. neg-mul-1 82.1%

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

   3. distribute-rgt-neg-in 82.1%

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

7. Simplified 82.1%

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

8. Final simplification 82.1%

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

Alternative 9: 64.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in b around inf 63.8%

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

   1. associate-*r/ 63.8%

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

   2. mul-1-neg 63.8%

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

7. Simplified 63.8%

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

8. Final simplification 63.8%

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

Reproduce

herbie shell --seed 2024066 
(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)))