Cubic critical, wide range

Percentage Accurate: 18.0% → 98.6%

Time: 16.5s

Alternatives: 8

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 18.0% 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: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{a \cdot \left(3 \cdot \left(-c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}}{e^{\log \left(3 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1e-21)
   (/
    (/ (* a (* 3.0 (- c))) (+ b (sqrt (fma b b (* (* a c) -3.0)))))
    (exp (log (* 3.0 a))))
   (/ (+ (* c -0.5) (* -0.375 (* a (pow (/ c b) 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1e-21) {
		tmp = ((a * (3.0 * -c)) / (b + sqrt(fma(b, b, ((a * c) * -3.0))))) / exp(log((3.0 * a)));
	} else {
		tmp = ((c * -0.5) + (-0.375 * (a * pow((c / b), 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1e-21)
		tmp = Float64(Float64(Float64(a * Float64(3.0 * Float64(-c))) / Float64(b + sqrt(fma(b, b, Float64(Float64(a * c) * -3.0))))) / exp(log(Float64(3.0 * a))));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1e-21], N[(N[(N[(a * N[(3.0 * (-c)), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[N[Log[N[(3.0 * a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -9.99999999999999908e-22

   1. Initial program 64.1%

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

      1. expm1-log1p-u 63.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)}} \]

      2. expm1-undefine 44.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1}} \]

   4. Applied egg-rr 44.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1}} \]
   5. Step-by-step derivation

      1. expm1-define 63.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)}} \]

   6. Simplified 63.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)}} \]
   7. Step-by-step derivation

      1. add-exp-log 63.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)\right)}}} \]

      2. expm1-log1p-u 63.9%

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

      3. *-commutative 63.9%

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

   8. Applied egg-rr 63.9%

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

      1. flip-+ 64.5%

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

      2. pow2 64.5%

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

      3. add-sqr-sqrt 66.2%

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

      4. pow2 66.2%

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

      5. *-commutative 66.2%

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

      6. *-commutative 66.2%

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

      7. pow2 66.2%

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

      8. *-commutative 66.2%

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

      9. *-commutative 66.2%

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

   10. Applied egg-rr 66.2%

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

       1. unpow2 66.2%

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

       2. sqr-neg 66.2%

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

       3. unpow2 66.2%

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

       4. unpow2 66.2%

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

       5. fmm-def 65.6%

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

       6. associate-*r* 65.6%

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

       7. *-commutative 65.6%

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

       8. distribute-rgt-neg-in 65.6%

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

       9. metadata-eval 65.6%

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

       10. unpow2 65.6%

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

       11. fmm-def 65.6%

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

       12. associate-*r* 65.6%

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

       13. *-commutative 65.6%

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

       14. distribute-rgt-neg-in 65.6%

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

       15. metadata-eval 65.6%

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

   12. Simplified 65.6%

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

   13. Taylor expanded in b around 0 96.2%

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

       1. *-commutative 96.2%

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

       2. associate-*r* 96.3%

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

   15. Simplified 96.3%

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

   if -9.99999999999999908e-22 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 4.6%

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

   3. Simplified 4.6%

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

   5. Taylor expanded in b around inf 100.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   6. Step-by-step derivation

      1. pow2 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{\color{blue}{b \cdot b}}}{b} \]

      2. associate-/l* 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{b \cdot b}\right)}}{b} \]

      3. pow2 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \frac{{c}^{2}}{\color{blue}{{b}^{2}}}\right)}{b} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)}{b} \]

      2. unpow2 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)}{b} \]

      3. times-frac 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)}{b} \]

      4. unpow1 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right)\right)}{b} \]

      5. pow-plus 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}\right)}{b} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right)}{b} \]

   9. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{a \cdot \left(3 \cdot \left(-c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}}{e^{\log \left(3 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{3 \cdot \left(a \cdot c\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}}{-e^{\log \left(3 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1e-21)
   (/
    (/ (* 3.0 (* a c)) (+ b (sqrt (fma b b (* (* a c) -3.0)))))
    (- (exp (log (* 3.0 a)))))
   (/ (+ (* c -0.5) (* -0.375 (* a (pow (/ c b) 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1e-21) {
		tmp = ((3.0 * (a * c)) / (b + sqrt(fma(b, b, ((a * c) * -3.0))))) / -exp(log((3.0 * a)));
	} else {
		tmp = ((c * -0.5) + (-0.375 * (a * pow((c / b), 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1e-21)
		tmp = Float64(Float64(Float64(3.0 * Float64(a * c)) / Float64(b + sqrt(fma(b, b, Float64(Float64(a * c) * -3.0))))) / Float64(-exp(log(Float64(3.0 * a)))));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1e-21], N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Exp[N[Log[N[(3.0 * a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -9.99999999999999908e-22

   1. Initial program 64.1%

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

      1. expm1-log1p-u 63.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)}} \]

      2. expm1-undefine 44.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1}} \]

   4. Applied egg-rr 44.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1}} \]
   5. Step-by-step derivation

      1. expm1-define 63.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)}} \]

   6. Simplified 63.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)}} \]
   7. Step-by-step derivation

      1. add-exp-log 63.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)\right)}}} \]

      2. expm1-log1p-u 63.9%

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

      3. *-commutative 63.9%

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

   8. Applied egg-rr 63.9%

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

      1. flip-+ 64.5%

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

      2. pow2 64.5%

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

      3. add-sqr-sqrt 66.2%

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

      4. pow2 66.2%

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

      5. *-commutative 66.2%

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

      6. *-commutative 66.2%

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

      7. pow2 66.2%

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

      8. *-commutative 66.2%

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

      9. *-commutative 66.2%

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

   10. Applied egg-rr 66.2%

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

       1. unpow2 66.2%

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

       2. sqr-neg 66.2%

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

       3. unpow2 66.2%

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

       4. unpow2 66.2%

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

       5. fmm-def 65.6%

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

       6. associate-*r* 65.6%

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

       7. *-commutative 65.6%

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

       8. distribute-rgt-neg-in 65.6%

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

       9. metadata-eval 65.6%

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

       10. unpow2 65.6%

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

       11. fmm-def 65.6%

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

       12. associate-*r* 65.6%

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

       13. *-commutative 65.6%

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

       14. distribute-rgt-neg-in 65.6%

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

       15. metadata-eval 65.6%

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

   12. Simplified 65.6%

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

   13. Taylor expanded in b around 0 96.2%

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

   if -9.99999999999999908e-22 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 4.6%

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

   3. Simplified 4.6%

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

   5. Taylor expanded in b around inf 100.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   6. Step-by-step derivation

      1. pow2 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{\color{blue}{b \cdot b}}}{b} \]

      2. associate-/l* 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{b \cdot b}\right)}}{b} \]

      3. pow2 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \frac{{c}^{2}}{\color{blue}{{b}^{2}}}\right)}{b} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)}{b} \]

      2. unpow2 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)}{b} \]

      3. times-frac 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)}{b} \]

      4. unpow1 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right)\right)}{b} \]

      5. pow-plus 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}\right)}{b} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right)}{b} \]

   9. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{3 \cdot \left(a \cdot c\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}}{-e^{\log \left(3 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (*
    (pow c 3.0)
    (- (* -0.5625 (/ a (pow b 5.0))) (/ 0.375 (* c (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * (pow(c, 3.0) * ((-0.5625 * (a / pow(b, 5.0))) - (0.375 / (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 = ((-0.5d0) * (c / b)) + (a * ((c ** 3.0d0) * (((-0.5625d0) * (a / (b ** 5.0d0))) - (0.375d0 / (c * (b ** 3.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * (Math.pow(c, 3.0) * ((-0.5625 * (a / Math.pow(b, 5.0))) - (0.375 / (c * Math.pow(b, 3.0))))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * (math.pow(c, 3.0) * ((-0.5625 * (a / math.pow(b, 5.0))) - (0.375 / (c * math.pow(b, 3.0))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64((c ^ 3.0) * Float64(Float64(-0.5625 * Float64(a / (b ^ 5.0))) - Float64(0.375 / Float64(c * (b ^ 3.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((c ^ 3.0) * ((-0.5625 * (a / (b ^ 5.0))) - (0.375 / (c * (b ^ 3.0))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(-0.5625 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Simplified 17.2%

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

5. Taylor expanded in a around 0 96.8%

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

6. Taylor expanded in c around inf 96.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \color{blue}{\left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r/ 96.8%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \color{blue}{\frac{0.375 \cdot 1}{{b}^{3} \cdot c}}\right)\right) \]

   2. metadata-eval 96.8%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \frac{\color{blue}{0.375}}{{b}^{3} \cdot c}\right)\right) \]

   3. *-commutative 96.8%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \frac{0.375}{\color{blue}{c \cdot {b}^{3}}}\right)\right) \]

8. Simplified 96.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \color{blue}{\left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)} \]
9. Add Preprocessing

Alternative 4: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5 + -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (* c -0.5) (* -0.375 (* a (pow (/ c b) 2.0)))) b))

C

double code(double a, double b, double c) {
	return ((c * -0.5) + (-0.375 * (a * pow((c / b), 2.0)))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (-0.5d0)) + ((-0.375d0) * (a * ((c / b) ** 2.0d0)))) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) + (-0.375 * (a * Math.pow((c / b), 2.0)))) / b;
}

Python

def code(a, b, c):
	return ((c * -0.5) + (-0.375 * (a * math.pow((c / b), 2.0)))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) + (-0.375 * (a * ((c / b) ^ 2.0)))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Simplified 17.2%

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

5. Taylor expanded in b around inf 95.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
6. Step-by-step derivation

   1. pow2 95.4%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{\color{blue}{b \cdot b}}}{b} \]

   2. associate-/l* 95.4%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{b \cdot b}\right)}}{b} \]

   3. pow2 95.4%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \frac{{c}^{2}}{\color{blue}{{b}^{2}}}\right)}{b} \]

7. Applied egg-rr 95.4%

   \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b} \]
8. Step-by-step derivation

   1. unpow2 95.4%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)}{b} \]

   2. unpow2 95.4%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)}{b} \]

   3. times-frac 95.4%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)}{b} \]

   4. unpow1 95.4%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right)\right)}{b} \]

   5. pow-plus 95.4%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}\right)}{b} \]

   6. metadata-eval 95.4%

      \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right)}{b} \]

9. Simplified 95.4%

   \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \color{blue}{\left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)}}{b} \]

10. Final simplification 95.4%

    \[\leadsto \frac{c \cdot -0.5 + -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b} \]
11. Add Preprocessing

Alternative 5: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (- (* -0.375 (/ (* a c) (pow b 2.0))) 0.5)) b))

C

double code(double a, double b, double c) {
	return (c * ((-0.375 * ((a * c) / pow(b, 2.0))) - 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 * (((-0.375d0) * ((a * c) / (b ** 2.0d0))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((-0.375 * ((a * c) / Math.pow(b, 2.0))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((-0.375 * ((a * c) / math.pow(b, 2.0))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 2.0))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((-0.375 * ((a * c) / (b ^ 2.0))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Simplified 17.2%

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

5. Taylor expanded in b around inf 95.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

6. Taylor expanded in c around 0 95.4%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}}{b} \]
7. Add Preprocessing

Alternative 6: 95.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{a \cdot c}{b}}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (+ (* b -2.0) (* 1.5 (/ (* a c) b))) c)))

C

double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + (1.5 * ((a * c) / b))) / c);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((b * (-2.0d0)) + (1.5d0 * ((a * c) / b))) / c)
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + (1.5 * ((a * c) / b))) / c);
}

Python

def code(a, b, c):
	return 1.0 / (((b * -2.0) + (1.5 * ((a * c) / b))) / c)

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(1.5 * Float64(Float64(a * c) / b))) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (((b * -2.0) + (1.5 * ((a * c) / b))) / c);
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Simplified 17.2%

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

5. Taylor expanded in b around inf 95.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
6. Step-by-step derivation

   1. clear-num 95.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{b}{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}}} \]

   2. inv-pow 95.0%

      \[\leadsto \color{blue}{{\left(\frac{b}{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}\right)}^{-1}} \]

   3. *-commutative 95.0%

      \[\leadsto {\left(\frac{b}{\color{blue}{c \cdot -0.5} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}\right)}^{-1} \]

   4. fma-define 95.0%

      \[\leadsto {\left(\frac{b}{\color{blue}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}\right)}^{-1} \]

   5. pow2 95.0%

      \[\leadsto {\left(\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \frac{a \cdot {c}^{2}}{\color{blue}{b \cdot b}}\right)}\right)}^{-1} \]

   6. associate-*r/ 95.0%

      \[\leadsto {\left(\frac{b}{\mathsf{fma}\left(c, -0.5, \color{blue}{\frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{b \cdot b}}\right)}\right)}^{-1} \]

   7. pow2 95.0%

      \[\leadsto {\left(\frac{b}{\mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{{b}^{2}}}\right)}\right)}^{-1} \]

7. Applied egg-rr 95.0%

   \[\leadsto \color{blue}{{\left(\frac{b}{\mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 95.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)}}} \]

   2. associate-*r/ 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, \color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}\right)}} \]

   3. associate-*r/ 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right)}} \]

   4. unpow2 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)\right)}} \]

   5. unpow2 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)\right)}} \]

   6. times-frac 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)\right)}} \]

   7. unpow1 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right)\right)\right)}} \]

   8. pow-plus 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}\right)\right)}} \]

   9. metadata-eval 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right)\right)}} \]

9. Simplified 95.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}}} \]

10. Taylor expanded in c around 0 95.2%

    \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]

11. Final simplification 95.2%

    \[\leadsto \frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{a \cdot c}{b}}{c}} \]
12. Add Preprocessing

Alternative 7: 95.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Simplified 17.2%

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

5. Taylor expanded in b around inf 95.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
6. Step-by-step derivation

   1. clear-num 95.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{b}{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}}} \]

   2. inv-pow 95.0%

      \[\leadsto \color{blue}{{\left(\frac{b}{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}\right)}^{-1}} \]

   3. *-commutative 95.0%

      \[\leadsto {\left(\frac{b}{\color{blue}{c \cdot -0.5} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}\right)}^{-1} \]

   4. fma-define 95.0%

      \[\leadsto {\left(\frac{b}{\color{blue}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}\right)}^{-1} \]

   5. pow2 95.0%

      \[\leadsto {\left(\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \frac{a \cdot {c}^{2}}{\color{blue}{b \cdot b}}\right)}\right)}^{-1} \]

   6. associate-*r/ 95.0%

      \[\leadsto {\left(\frac{b}{\mathsf{fma}\left(c, -0.5, \color{blue}{\frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{b \cdot b}}\right)}\right)}^{-1} \]

   7. pow2 95.0%

      \[\leadsto {\left(\frac{b}{\mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{{b}^{2}}}\right)}\right)}^{-1} \]

7. Applied egg-rr 95.0%

   \[\leadsto \color{blue}{{\left(\frac{b}{\mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 95.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)}}} \]

   2. associate-*r/ 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, \color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}\right)}} \]

   3. associate-*r/ 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right)}} \]

   4. unpow2 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)\right)}} \]

   5. unpow2 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)\right)}} \]

   6. times-frac 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)\right)}} \]

   7. unpow1 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right)\right)\right)}} \]

   8. pow-plus 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}\right)\right)}} \]

   9. metadata-eval 95.0%

      \[\leadsto \frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right)\right)}} \]

9. Simplified 95.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(c, -0.5, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}}} \]

10. Taylor expanded in a around 0 95.1%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
11. Add Preprocessing

Alternative 8: 90.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

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

3. Simplified 17.2%

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

5. Taylor expanded in b around inf 90.7%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))