Quadratic roots, narrow range

Percentage Accurate: 56.1% → 91.7%

Time: 12.0s

Alternatives: 11

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 56.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{{b}^{7}} \cdot \left(20 \cdot {c}^{4}\right), -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - e^{\log \left(\frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= (/ (- t_1 b) (* a 2.0)) -20.0)
     (/ (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) t_1)) (* a 2.0))
     (-
      (-
       (fma
        -0.25
        (* (/ (pow a 3.0) (pow b 7.0)) (* 20.0 (pow c 4.0)))
        (* -2.0 (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))))
       (/ c b))
      (exp (log (/ (* c (* a c)) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -20.0) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, ((pow(a, 3.0) / pow(b, 7.0)) * (20.0 * pow(c, 4.0))), (-2.0 * ((a * a) / (pow(b, 5.0) / pow(c, 3.0))))) - (c / b)) - exp(log(((c * (a * c)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -20.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64(Float64((a ^ 3.0) / (b ^ 7.0)) * Float64(20.0 * (c ^ 4.0))), Float64(-2.0 * Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))))) - Float64(c / b)) - exp(log(Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -20.0], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(20.0 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[Exp[N[Log[N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -20

   1. Initial program 88.1%

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

      1. flip-+ 88.1%

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

      2. pow2 88.1%

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

      3. add-sqr-sqrt 89.7%

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

      4. *-commutative 89.7%

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

      5. *-commutative 89.7%

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

      6. *-commutative 89.7%

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

      7. *-commutative 89.7%

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

   3. Applied egg-rr 89.7%

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

   if -20 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 50.9%

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

      1. neg-sub0 50.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 50.9%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 50.9%

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

      4. neg-mul-1 50.9%

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

      5. associate-*l/ 50.9%

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

      6. *-commutative 50.9%

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

      7. associate-/r* 50.9%

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

      8. /-rgt-identity 50.9%

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

      9. metadata-eval 50.9%

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

   3. Simplified 50.9%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in a around 0 93.2%

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

   6. Simplified 93.2%

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

   7. Taylor expanded in c around 0 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*l/ 93.2%

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

      3. *-commutative 93.2%

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

      4. associate-*r* 93.2%

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

      5. metadata-eval 93.2%

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

      6. distribute-rgt-out 93.2%

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

      7. associate-/l* 93.2%

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

      8. associate-/r/ 93.2%

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

      9. distribute-rgt-out 93.2%

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

      10. metadata-eval 93.2%

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

      11. *-commutative 93.2%

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

   9. Simplified 93.2%

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

       1. add-exp-log 93.2%

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

   11. Applied egg-rr 93.2%

       \[\leadsto \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{{b}^{7}} \cdot \left(20 \cdot {c}^{4}\right), -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \color{blue}{e^{\log \left(\frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(4 \cdot a\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{{b}^{7}} \cdot \left(20 \cdot {c}^{4}\right), -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - e^{\log \left(\frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)}\\ \end{array} \]

Alternative 2: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{{b}^{7}} \cdot \left(20 \cdot {c}^{4}\right), -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= (/ (- t_1 b) (* a 2.0)) -20.0)
     (/ (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) t_1)) (* a 2.0))
     (-
      (-
       (fma
        -0.25
        (* (/ (pow a 3.0) (pow b 7.0)) (* 20.0 (pow c 4.0)))
        (* -2.0 (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))))
       (/ c b))
      (/ (* c (* a c)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -20.0) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, ((pow(a, 3.0) / pow(b, 7.0)) * (20.0 * pow(c, 4.0))), (-2.0 * ((a * a) / (pow(b, 5.0) / pow(c, 3.0))))) - (c / b)) - ((c * (a * c)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -20.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64(Float64((a ^ 3.0) / (b ^ 7.0)) * Float64(20.0 * (c ^ 4.0))), Float64(-2.0 * Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))))) - Float64(c / b)) - Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -20.0], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(20.0 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $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 4 a) c)))) (*.f64 2 a)) < -20

   1. Initial program 88.1%

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

      1. flip-+ 88.1%

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

      2. pow2 88.1%

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

      3. add-sqr-sqrt 89.7%

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

      4. *-commutative 89.7%

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

      5. *-commutative 89.7%

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

      6. *-commutative 89.7%

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

      7. *-commutative 89.7%

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

   3. Applied egg-rr 89.7%

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

   if -20 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 50.9%

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

      1. neg-sub0 50.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 50.9%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 50.9%

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

      4. neg-mul-1 50.9%

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

      5. associate-*l/ 50.9%

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

      6. *-commutative 50.9%

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

      7. associate-/r* 50.9%

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

      8. /-rgt-identity 50.9%

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

      9. metadata-eval 50.9%

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

   3. Simplified 50.9%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in a around 0 93.2%

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

   6. Simplified 93.2%

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

   7. Taylor expanded in c around 0 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*l/ 93.2%

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

      3. *-commutative 93.2%

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

      4. associate-*r* 93.2%

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

      5. metadata-eval 93.2%

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

      6. distribute-rgt-out 93.2%

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

      7. associate-/l* 93.2%

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

      8. associate-/r/ 93.2%

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

      9. distribute-rgt-out 93.2%

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

      10. metadata-eval 93.2%

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

      11. *-commutative 93.2%

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

   9. Simplified 93.2%

      \[\leadsto \left(\mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{3}}{{b}^{7}} \cdot \left(20 \cdot {c}^{4}\right)}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(4 \cdot a\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{{b}^{7}} \cdot \left(20 \cdot {c}^{4}\right), -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 3: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot {c}^{3}\right) \cdot \left(a \cdot \frac{a}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{a \cdot c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= (/ (- t_1 b) (* a 2.0)) -20.0)
     (/ (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) t_1)) (* a 2.0))
     (-
      (- (* (* -2.0 (pow c 3.0)) (* a (/ a (pow b 5.0)))) (/ c b))
      (/ c (/ (pow b 3.0) (* a c)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -20.0) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (((-2.0 * pow(c, 3.0)) * (a * (a / pow(b, 5.0)))) - (c / b)) - (c / (pow(b, 3.0) / (a * c)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    t_1 = sqrt(((b * b) - t_0))
    if (((t_1 - b) / (a * 2.0d0)) <= (-20.0d0)) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0d0)
    else
        tmp = ((((-2.0d0) * (c ** 3.0d0)) * (a * (a / (b ** 5.0d0)))) - (c / b)) - (c / ((b ** 3.0d0) / (a * c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = Math.sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -20.0) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (((-2.0 * Math.pow(c, 3.0)) * (a * (a / Math.pow(b, 5.0)))) - (c / b)) - (c / (Math.pow(b, 3.0) / (a * c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	t_1 = math.sqrt(((b * b) - t_0))
	tmp = 0
	if ((t_1 - b) / (a * 2.0)) <= -20.0:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0)
	else:
		tmp = (((-2.0 * math.pow(c, 3.0)) * (a * (a / math.pow(b, 5.0)))) - (c / b)) - (c / (math.pow(b, 3.0) / (a * c)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -20.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * (c ^ 3.0)) * Float64(a * Float64(a / (b ^ 5.0)))) - Float64(c / b)) - Float64(c / Float64((b ^ 3.0) / Float64(a * c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (4.0 * a);
	t_1 = sqrt(((b * b) - t_0));
	tmp = 0.0;
	if (((t_1 - b) / (a * 2.0)) <= -20.0)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	else
		tmp = (((-2.0 * (c ^ 3.0)) * (a * (a / (b ^ 5.0)))) - (c / b)) - (c / ((b ^ 3.0) / (a * c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -20.0], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] * N[(a * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -20

   1. Initial program 88.1%

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

      1. flip-+ 88.1%

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

      2. pow2 88.1%

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

      3. add-sqr-sqrt 89.7%

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

      4. *-commutative 89.7%

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

      5. *-commutative 89.7%

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

      6. *-commutative 89.7%

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

      7. *-commutative 89.7%

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

   3. Applied egg-rr 89.7%

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

   if -20 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 50.9%

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

      1. log1p-expm1-u 50.9%

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

      2. neg-mul-1 50.9%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)\right) \]

      3. fma-def 50.9%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\right)\right) \]

      4. *-commutative 50.9%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}{2 \cdot a}\right)\right) \]

      5. *-commutative 50.9%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}{2 \cdot a}\right)\right) \]

      6. *-commutative 50.9%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{\color{blue}{a \cdot 2}}\right)\right) \]

   3. Applied egg-rr 50.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{a \cdot 2}\right)\right)} \]

   4. Taylor expanded in b around inf 90.5%

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

      1. +-commutative 90.5%

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

      2. mul-1-neg 90.5%

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

      3. unpow2 90.5%

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

      4. associate-*r* 90.5%

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

      5. unsub-neg 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(4 \cdot a\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot {c}^{3}\right) \cdot \left(a \cdot \frac{a}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{a \cdot c}}\\ \end{array} \]

Alternative 4: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))))
   (if (<= b 11.5)
     (/
      (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) (sqrt (- (* b b) t_0))))
      (* a 2.0))
     (- (/ (- c) b) (/ (* c (* a c)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double tmp;
	if (b <= 11.5) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((c * (a * c)) / pow(b, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if (b <= 11.5d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((c * (a * c)) / (b ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double tmp;
	if (b <= 11.5) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - Math.sqrt(((b * b) - t_0)))) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((c * (a * c)) / Math.pow(b, 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	tmp = 0
	if b <= 11.5:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - math.sqrt(((b * b) - t_0)))) / (a * 2.0)
	else:
		tmp = (-c / b) - ((c * (a * c)) / math.pow(b, 3.0))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	tmp = 0.0
	if (b <= 11.5)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (4.0 * a);
	tmp = 0.0;
	if (b <= 11.5)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 2.0);
	else
		tmp = (-c / b) - ((c * (a * c)) / (b ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 11.5], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 11.5

   1. Initial program 80.7%

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

      1. flip-+ 80.3%

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

      2. pow2 80.3%

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

      3. add-sqr-sqrt 81.7%

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

      4. *-commutative 81.7%

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

      5. *-commutative 81.7%

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

      6. *-commutative 81.7%

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

      7. *-commutative 81.7%

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

   3. Applied egg-rr 81.7%

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

   if 11.5 < b

   1. Initial program 46.0%

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

      1. neg-sub0 46.0%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 46.0%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 46.0%

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

      4. neg-mul-1 46.0%

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

      5. associate-*l/ 46.0%

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

      6. *-commutative 46.0%

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

      7. associate-/r* 46.0%

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

      8. /-rgt-identity 46.0%

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

      9. metadata-eval 46.0%

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

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. associate-*r/ 87.8%

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

      5. neg-mul-1 87.8%

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

      6. unpow2 87.8%

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

      7. associate-*l* 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(4 \cdot a\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 5: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 11.5)
   (* (- (sqrt (fma b b (* (* a c) -4.0))) b) (/ 0.5 a))
   (- (/ (- c) b) (/ (* c (* a c)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 11.5) {
		tmp = (sqrt(fma(b, b, ((a * c) * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = (-c / b) - ((c * (a * c)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 11.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 11.5], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 11.5

   1. Initial program 80.7%

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

      1. /-rgt-identity 80.7%

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

      2. metadata-eval 80.7%

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

      3. associate-/l* 80.7%

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

      4. associate-*r/ 80.7%

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

      5. +-commutative 80.7%

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

      6. unsub-neg 80.7%

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

      7. fma-neg 80.8%

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

      8. associate-*l* 80.8%

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

      9. *-commutative 80.8%

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

      10. distribute-rgt-neg-in 80.8%

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

      11. metadata-eval 80.8%

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

      12. associate-/r* 80.8%

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

      13. metadata-eval 80.8%

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

      14. metadata-eval 80.8%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 80.8%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]

   if 11.5 < b

   1. Initial program 46.0%

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

      1. neg-sub0 46.0%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 46.0%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 46.0%

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

      4. neg-mul-1 46.0%

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

      5. associate-*l/ 46.0%

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

      6. *-commutative 46.0%

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

      7. associate-/r* 46.0%

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

      8. /-rgt-identity 46.0%

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

      9. metadata-eval 46.0%

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

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. associate-*r/ 87.8%

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

      5. neg-mul-1 87.8%

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

      6. unpow2 87.8%

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

      7. associate-*l* 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 6: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 11.5)
   (/ (- (sqrt (fma b b (* (* a c) -4.0))) b) (* a 2.0))
   (- (/ (- c) b) (/ (* c (* a c)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 11.5) {
		tmp = (sqrt(fma(b, b, ((a * c) * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((c * (a * c)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 11.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 11.5], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 11.5

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. +-commutative 80.7%

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

      3. unsub-neg 80.7%

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

      4. fma-neg 80.8%

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

      5. associate-*l* 80.8%

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

      6. *-commutative 80.8%

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

      7. distribute-rgt-neg-in 80.8%

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

      8. metadata-eval 80.8%

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

   3. Simplified 80.8%

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

   if 11.5 < b

   1. Initial program 46.0%

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

      1. neg-sub0 46.0%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 46.0%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 46.0%

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

      4. neg-mul-1 46.0%

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

      5. associate-*l/ 46.0%

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

      6. *-commutative 46.0%

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

      7. associate-/r* 46.0%

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

      8. /-rgt-identity 46.0%

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

      9. metadata-eval 46.0%

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

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. associate-*r/ 87.8%

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

      5. neg-mul-1 87.8%

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

      6. unpow2 87.8%

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

      7. associate-*l* 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 7: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 12:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 12.0)
   (* (/ 0.5 a) (- (sqrt (+ (* b b) (* (* a c) -4.0))) b))
   (- (/ (- c) b) (/ (* c (* a c)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 12.0) {
		tmp = (0.5 / a) * (sqrt(((b * b) + ((a * c) * -4.0))) - b);
	} else {
		tmp = (-c / b) - ((c * (a * c)) / pow(b, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 12.0d0) then
        tmp = (0.5d0 / a) * (sqrt(((b * b) + ((a * c) * (-4.0d0)))) - b)
    else
        tmp = (-c / b) - ((c * (a * c)) / (b ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 12.0) {
		tmp = (0.5 / a) * (Math.sqrt(((b * b) + ((a * c) * -4.0))) - b);
	} else {
		tmp = (-c / b) - ((c * (a * c)) / Math.pow(b, 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 12.0:
		tmp = (0.5 / a) * (math.sqrt(((b * b) + ((a * c) * -4.0))) - b)
	else:
		tmp = (-c / b) - ((c * (a * c)) / math.pow(b, 3.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 12.0)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0))) - b));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 12.0)
		tmp = (0.5 / a) * (sqrt(((b * b) + ((a * c) * -4.0))) - b);
	else
		tmp = (-c / b) - ((c * (a * c)) / (b ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 12.0], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 12

   1. Initial program 80.7%

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

      1. /-rgt-identity 80.7%

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

      2. metadata-eval 80.7%

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

      3. associate-/l* 80.7%

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

      4. associate-*r/ 80.7%

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

      5. +-commutative 80.7%

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

      6. unsub-neg 80.7%

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

      7. fma-neg 80.8%

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

      8. associate-*l* 80.8%

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

      9. *-commutative 80.8%

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

      10. distribute-rgt-neg-in 80.8%

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

      11. metadata-eval 80.8%

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

      12. associate-/r* 80.8%

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

      13. metadata-eval 80.8%

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

      14. metadata-eval 80.8%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 80.8%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 80.7%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b\right) \cdot \frac{0.5}{a} \]

      2. *-commutative 80.7%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 80.7%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + -4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

   if 12 < b

   1. Initial program 46.0%

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

      1. neg-sub0 46.0%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 46.0%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 46.0%

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

      4. neg-mul-1 46.0%

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

      5. associate-*l/ 46.0%

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

      6. *-commutative 46.0%

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

      7. associate-/r* 46.0%

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

      8. /-rgt-identity 46.0%

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

      9. metadata-eval 46.0%

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

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. associate-*r/ 87.8%

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

      5. neg-mul-1 87.8%

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

      6. unpow2 87.8%

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

      7. associate-*l* 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 12:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 8: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 11.5)
   (/ (- (sqrt (+ (* b b) (* (* a c) -4.0))) b) (* a 2.0))
   (- (/ (- c) b) (/ (* c (* a c)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 11.5) {
		tmp = (sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((c * (a * c)) / pow(b, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 11.5d0) then
        tmp = (sqrt(((b * b) + ((a * c) * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((c * (a * c)) / (b ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 11.5) {
		tmp = (Math.sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((c * (a * c)) / Math.pow(b, 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 11.5:
		tmp = (math.sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((c * (a * c)) / math.pow(b, 3.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 11.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 11.5)
		tmp = (sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((c * (a * c)) / (b ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 11.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 11.5

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. +-commutative 80.7%

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

      3. unsub-neg 80.7%

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

      4. fma-neg 80.8%

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

      5. associate-*l* 80.8%

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

      6. *-commutative 80.8%

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

      7. distribute-rgt-neg-in 80.8%

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

      8. metadata-eval 80.8%

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

   3. Simplified 80.8%

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

      1. fma-udef 80.7%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b\right) \cdot \frac{0.5}{a} \]

      2. *-commutative 80.7%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 80.7%

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

   if 11.5 < b

   1. Initial program 46.0%

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

      1. neg-sub0 46.0%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 46.0%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 46.0%

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

      4. neg-mul-1 46.0%

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

      5. associate-*l/ 46.0%

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

      6. *-commutative 46.0%

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

      7. associate-/r* 46.0%

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

      8. /-rgt-identity 46.0%

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

      9. metadata-eval 46.0%

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

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. associate-*r/ 87.8%

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

      5. neg-mul-1 87.8%

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

      6. unpow2 87.8%

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

      7. associate-*l* 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 9: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-c / b) - ((c * (a * c)) / 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 / b) - ((c * (a * c)) / (b ** 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

   1. neg-sub0 54.1%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 54.1%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 54.1%

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

   4. neg-mul-1 54.1%

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

   5. associate-*l/ 54.1%

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

   6. *-commutative 54.1%

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

   7. associate-/r* 54.1%

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

   8. /-rgt-identity 54.1%

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

   9. metadata-eval 54.1%

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

3. Simplified 54.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 81.5%

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

   1. +-commutative 81.5%

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

   2. mul-1-neg 81.5%

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

   3. unsub-neg 81.5%

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

   4. associate-*r/ 81.5%

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

   5. neg-mul-1 81.5%

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

   6. unpow2 81.5%

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

   7. associate-*l* 81.5%

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

6. Simplified 81.5%

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

7. Final simplification 81.5%

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

Alternative 10: 63.8% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

   1. neg-sub0 54.1%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 54.1%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 54.1%

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

   4. neg-mul-1 54.1%

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

   5. associate-*l/ 54.1%

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

   6. *-commutative 54.1%

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

   7. associate-/r* 54.1%

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

   8. /-rgt-identity 54.1%

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

   9. metadata-eval 54.1%

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

3. Simplified 54.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 65.0%

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

   1. associate-*r/ 65.0%

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

   2. neg-mul-1 65.0%

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

6. Simplified 65.0%

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

7. Final simplification 65.0%

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

Alternative 11: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

   1. log1p-expm1-u 47.3%

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

   2. neg-mul-1 47.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)\right) \]

   3. fma-def 47.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\right)\right) \]

   4. *-commutative 47.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}{2 \cdot a}\right)\right) \]

   5. *-commutative 47.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}{2 \cdot a}\right)\right) \]

   6. *-commutative 47.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{\color{blue}{a \cdot 2}}\right)\right) \]

3. Applied egg-rr 47.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{a \cdot 2}\right)\right)} \]

4. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
5. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b + -1 \cdot b\right)}{a}} \]

   2. distribute-rgt1-in 3.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

6. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

7. Final simplification 3.2%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023199 
(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)))