Cubic critical, narrow range

Percentage Accurate: 55.8% → 99.1%

Time: 21.9s

Alternatives: 8

Speedup: 23.2×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-3, c \cdot a, \frac{27 \cdot \left(\frac{{b}^{4}}{a} \cdot 0\right)}{c}\right)}{b + \sqrt{\frac{{b}^{6} + {\left(c \cdot a\right)}^{3} \cdot -27}{{b}^{4} + \left(c \cdot a\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (fma -3.0 (* c a) (/ (* 27.0 (* (/ (pow b 4.0) a) 0.0)) c))
   (+
    b
    (sqrt
     (/
      (+ (pow b 6.0) (* (pow (* c a) 3.0) -27.0))
      (+ (pow b 4.0) (* (* c a) (* 3.0 (fma b b (* c (* a 3.0))))))))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (fma(-3.0, (c * a), ((27.0 * ((pow(b, 4.0) / a) * 0.0)) / c)) / (b + sqrt(((pow(b, 6.0) + (pow((c * a), 3.0) * -27.0)) / (pow(b, 4.0) + ((c * a) * (3.0 * fma(b, b, (c * (a * 3.0)))))))))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(-3.0, Float64(c * a), Float64(Float64(27.0 * Float64(Float64((b ^ 4.0) / a) * 0.0)) / c)) / Float64(b + sqrt(Float64(Float64((b ^ 6.0) + Float64((Float64(c * a) ^ 3.0) * -27.0)) / Float64((b ^ 4.0) + Float64(Float64(c * a) * Float64(3.0 * fma(b, b, Float64(c * Float64(a * 3.0)))))))))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-3.0 * N[(c * a), $MachinePrecision] + N[(N[(27.0 * N[(N[(N[Power[b, 4.0], $MachinePrecision] / a), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(N[Power[b, 6.0], $MachinePrecision] + N[(N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * N[(3.0 * N[(b * b + N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

3. Simplified 53.1%

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

5. Applied egg-rr 52.6%

   \[\leadsto \frac{\color{blue}{\frac{\sqrt{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}} - b}{3 \cdot a} \]
6. Step-by-step derivation

   1. sub-neg 52.6%

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} + \left(-b\right)}}{3 \cdot a} \]

   2. flip-+ 52.5%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} \cdot \frac{\sqrt{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} - \left(-b\right) \cdot \left(-b\right)}{\frac{\sqrt{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} - \left(-b\right)}}}{3 \cdot a} \]

7. Applied egg-rr 53.5%

   \[\leadsto \frac{\color{blue}{\frac{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -27}{{b}^{4} + \left(a \cdot c\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -27}{{b}^{4} + \left(a \cdot c\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}} - \left(-b\right)}}}{3 \cdot a} \]

8. Taylor expanded in c around inf 99.1%

   \[\leadsto \frac{\frac{\color{blue}{-3 \cdot \left(a \cdot c\right) + 27 \cdot \frac{-0.012345679012345678 \cdot \frac{{b}^{4}}{a} + 0.012345679012345678 \cdot \frac{{b}^{4}}{a}}{c}}}{\sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -27}{{b}^{4} + \left(a \cdot c\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}} - \left(-b\right)}}{3 \cdot a} \]
9. Step-by-step derivation

   1. fma-def 99.1%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-3, a \cdot c, 27 \cdot \frac{-0.012345679012345678 \cdot \frac{{b}^{4}}{a} + 0.012345679012345678 \cdot \frac{{b}^{4}}{a}}{c}\right)}}{\sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -27}{{b}^{4} + \left(a \cdot c\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}} - \left(-b\right)}}{3 \cdot a} \]

   2. *-commutative 99.1%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-3, \color{blue}{c \cdot a}, 27 \cdot \frac{-0.012345679012345678 \cdot \frac{{b}^{4}}{a} + 0.012345679012345678 \cdot \frac{{b}^{4}}{a}}{c}\right)}{\sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -27}{{b}^{4} + \left(a \cdot c\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}} - \left(-b\right)}}{3 \cdot a} \]

   3. associate-*r/ 99.1%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-3, c \cdot a, \color{blue}{\frac{27 \cdot \left(-0.012345679012345678 \cdot \frac{{b}^{4}}{a} + 0.012345679012345678 \cdot \frac{{b}^{4}}{a}\right)}{c}}\right)}{\sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -27}{{b}^{4} + \left(a \cdot c\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}} - \left(-b\right)}}{3 \cdot a} \]

   4. distribute-rgt-out 99.1%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-3, c \cdot a, \frac{27 \cdot \color{blue}{\left(\frac{{b}^{4}}{a} \cdot \left(-0.012345679012345678 + 0.012345679012345678\right)\right)}}{c}\right)}{\sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -27}{{b}^{4} + \left(a \cdot c\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}} - \left(-b\right)}}{3 \cdot a} \]

   5. metadata-eval 99.1%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-3, c \cdot a, \frac{27 \cdot \left(\frac{{b}^{4}}{a} \cdot \color{blue}{0}\right)}{c}\right)}{\sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -27}{{b}^{4} + \left(a \cdot c\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}} - \left(-b\right)}}{3 \cdot a} \]

10. Simplified 99.1%

    \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-3, c \cdot a, \frac{27 \cdot \left(\frac{{b}^{4}}{a} \cdot 0\right)}{c}\right)}}{\sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -27}{{b}^{4} + \left(a \cdot c\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}} - \left(-b\right)}}{3 \cdot a} \]

11. Final simplification 99.1%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-3, c \cdot a, \frac{27 \cdot \left(\frac{{b}^{4}}{a} \cdot 0\right)}{c}\right)}{b + \sqrt{\frac{{b}^{6} + {\left(c \cdot a\right)}^{3} \cdot -27}{{b}^{4} + \left(c \cdot a\right) \cdot \left(3 \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\right)}}}}{a \cdot 3} \]

Alternative 2: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.4)
   (/ (- (sqrt (fma b b (* c (* -3.0 a)))) b) (* a 3.0))
   (fma
    -0.5
    (/ c b)
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
     (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.4) {
		tmp = (sqrt(fma(b, b, (c * (-3.0 * a)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.5, (c / b), ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-3.0 * a)))) - b) / Float64(a * 3.0));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.4], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $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 3 a) c)))) (*.f64 3 a)) < -0.40000000000000002

   1. Initial program 85.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 85.7%

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

   if -0.40000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 48.4%

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

   5. Applied egg-rr 48.1%

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}} - b}{3 \cdot a} \]

   6. Taylor expanded in c around 0 94.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{{c}^{2} \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + \left(9 \cdot \frac{{a}^{2}}{{b}^{2}} + {\left(-1.5 \cdot \frac{a}{b}\right)}^{2}\right)\right)}{a \cdot b} + \left(-0.16666666666666666 \cdot \frac{{c}^{3} \cdot \left(-3 \cdot \frac{a \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + 9 \cdot \frac{{a}^{2}}{{b}^{2}}\right)}{{b}^{2}} + 1.5 \cdot \frac{a \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + \left(9 \cdot \frac{{a}^{2}}{{b}^{2}} + {\left(-1.5 \cdot \frac{a}{b}\right)}^{2}\right)\right)}{{b}^{2}}\right)}{a \cdot b} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(-9 \cdot \frac{{a}^{2} \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + 9 \cdot \frac{{a}^{2}}{{b}^{2}}\right)}{{b}^{4}} + \left(1.5 \cdot \frac{a \cdot \left(-3 \cdot \frac{a \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + 9 \cdot \frac{{a}^{2}}{{b}^{2}}\right)}{{b}^{2}} + 1.5 \cdot \frac{a \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + \left(9 \cdot \frac{{a}^{2}}{{b}^{2}} + {\left(-1.5 \cdot \frac{a}{b}\right)}^{2}\right)\right)}{{b}^{2}}\right)}{{b}^{2}} + \left(9 \cdot \frac{{a}^{2} \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + 9 \cdot \frac{{a}^{2}}{{b}^{2}}\right)}{{b}^{4}} + {\left(-0.5 \cdot \frac{-9 \cdot \frac{{a}^{2}}{{b}^{2}} + \left(9 \cdot \frac{{a}^{2}}{{b}^{2}} + {\left(-1.5 \cdot \frac{a}{b}\right)}^{2}\right)}{b}\right)}^{2}\right)\right)\right)}{a \cdot b}\right)\right)} \]

   8. Simplified 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{c}^{2}}{a \cdot b} \cdot \left({\left(\frac{a \cdot -1.5}{b}\right)}^{2} + {\left(\frac{a}{b}\right)}^{2} \cdot 0\right), -0.16666666666666666 \cdot \left(\frac{{c}^{3}}{a \cdot b} \cdot \mathsf{fma}\left(-3, \frac{a}{{b}^{2}} \cdot \left({\left(\frac{a}{b}\right)}^{2} \cdot 0\right), \frac{1.5 \cdot a}{\frac{{b}^{2}}{{\left(\frac{a \cdot -1.5}{b}\right)}^{2} + {\left(\frac{a}{b}\right)}^{2} \cdot 0}}\right) + \frac{{c}^{4}}{a} \cdot \frac{\mathsf{fma}\left(-9, \frac{{a}^{2}}{{b}^{4}} \cdot \left({\left(\frac{a}{b}\right)}^{2} \cdot 0\right), \mathsf{fma}\left(1.5, \frac{a}{{b}^{2}} \cdot \mathsf{fma}\left(-3, \frac{a}{{b}^{2}} \cdot \left({\left(\frac{a}{b}\right)}^{2} \cdot 0\right), \frac{1.5 \cdot a}{\frac{{b}^{2}}{{\left(\frac{a \cdot -1.5}{b}\right)}^{2} + {\left(\frac{a}{b}\right)}^{2} \cdot 0}}\right), \mathsf{fma}\left(9, \frac{{a}^{2}}{{b}^{4}} \cdot \left({\left(\frac{a}{b}\right)}^{2} \cdot 0\right), {\left(\frac{-0.5 \cdot \left({\left(\frac{a \cdot -1.5}{b}\right)}^{2} + {\left(\frac{a}{b}\right)}^{2} \cdot 0\right)}{b}\right)}^{2}\right)\right)\right)}{b}\right)\right)\right)} \]

   9. Taylor expanded in c around 0 92.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 3: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.4)
   (/ (- (sqrt (fma b b (* c (* -3.0 a)))) b) (* a 3.0))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+ (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))) (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.4) {
		tmp = (sqrt(fma(b, b, (c * (-3.0 * a)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-3.0 * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.5 * Float64(c / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.4], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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 3 a) c)))) (*.f64 3 a)) < -0.40000000000000002

   1. Initial program 85.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 85.7%

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

   if -0.40000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.4%

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

   2. Taylor expanded in b around inf 92.7%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 4: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.00225:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.00225)
   (/ (- (sqrt (fma b b (* c (* -3.0 a)))) b) (* a 3.0))
   (+ (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.00225) {
		tmp = (sqrt(fma(b, b, (c * (-3.0 * a)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.5 * (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.00225)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-3.0 * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.00225], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / 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 3 a) c)))) (*.f64 3 a)) < -0.00224999999999999983

   1. Initial program 78.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 79.0%

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

   if -0.00224999999999999983 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 43.7%

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

   2. Taylor expanded in b around inf 91.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.00225:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5e-7)
   (/ (- (sqrt (fma b b (* c (* -3.0 a)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5e-7) {
		tmp = (sqrt(fma(b, b, (c * (-3.0 * a)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -5e-7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-3.0 * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / 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 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -5e-7], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4.99999999999999977e-7

   1. Initial program 70.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 70.7%

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

   if -4.99999999999999977e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 30.6%

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

   2. Taylor expanded in b around inf 85.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 85.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

      2. *-commutative 85.0%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   4. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -5e-7) t_0 (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -5e-7) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-5d-7)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -5e-7) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -5e-7:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -5e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -5e-7)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-7], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4.99999999999999977e-7

   1. Initial program 70.6%

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

   if -4.99999999999999977e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 30.6%

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

   2. Taylor expanded in b around inf 85.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 85.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

      2. *-commutative 85.0%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   4. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 64.1% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* c (/ -0.5 b)))

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 * ((-0.5d0) / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * (-0.5 / b);
}

Python

def code(a, b, c):
	return c * (-0.5 / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(-0.5 / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (-0.5 / b);
end

Wolfram

code[a_, b_, c_] := N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

3. Simplified 53.1%

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

4. Taylor expanded in b around inf 35.4%

   \[\leadsto \frac{\color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{3 \cdot a} \]

5. Taylor expanded in b around 0 66.7%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 66.7%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   2. associate-/l* 66.6%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]

7. Simplified 66.6%

   \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]
8. Step-by-step derivation

   1. associate-/r/ 66.6%

      \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot c} \]

9. Applied egg-rr 66.6%

   \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot c} \]

10. Final simplification 66.6%

    \[\leadsto c \cdot \frac{-0.5}{b} \]

Alternative 8: 64.1% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* c -0.5) b))

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 * (-0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * -0.5) / b;
}

Python

def code(a, b, c):
	return (c * -0.5) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * -0.5) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * -0.5) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 53.1%

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

2. Taylor expanded in b around inf 66.7%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Step-by-step derivation

   1. associate-*r/ 66.7%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   2. *-commutative 66.7%

      \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

4. Simplified 66.7%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]

5. Final simplification 66.7%

   \[\leadsto \frac{c \cdot -0.5}{b} \]

Reproduce

herbie shell --seed 2023336 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))