Quadratic roots, narrow range

Percentage Accurate: 55.5% → 99.5%

Time: 9.9s

Alternatives: 6

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% 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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{-2 \cdot \frac{c \cdot a}{a}}{b + \sqrt{\frac{\mathsf{fma}\left({\left(c \cdot a\right)}^{2}, -16, {b}^{4}\right)}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* -2.0 (/ (* c a) a))
  (+
   b
   (sqrt
    (/ (fma (pow (* c a) 2.0) -16.0 (pow b 4.0)) (fma b b (* c (* a 4.0))))))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((c * a) / a)) / (b + sqrt((fma(pow((c * a), 2.0), -16.0, pow(b, 4.0)) / fma(b, b, (c * (a * 4.0))))));
}

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64(c * a) / a)) / Float64(b + sqrt(Float64(fma((Float64(c * a) ^ 2.0), -16.0, (b ^ 4.0)) / fma(b, b, Float64(c * Float64(a * 4.0)))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(c * a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision] * -16.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.2%

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

3. Simplified 58.3%

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

   1. *-commutative 58.3%

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

   2. metadata-eval 58.3%

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

   3. distribute-lft-neg-in 58.3%

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

   4. distribute-rgt-neg-in 58.3%

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

   5. *-commutative 58.3%

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

   6. fma-neg 58.2%

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

   7. flip-- 58.0%

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

   8. div-sub 58.0%

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

   9. pow2 58.0%

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

   10. pow2 58.0%

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

   11. pow-prod-up 58.0%

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

   12. metadata-eval 58.0%

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

   13. fma-def 58.2%

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

   14. associate-*l* 58.2%

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

   15. pow2 58.2%

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

   16. associate-*l* 58.2%

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

   17. fma-def 58.2%

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

   18. associate-*l* 58.2%

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

5. Applied egg-rr 58.2%

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

   1. flip-- 58.0%

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

7. Applied egg-rr 58.9%

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

8. Taylor expanded in b around 0 99.3%

   \[\leadsto \frac{\frac{\color{blue}{-4 \cdot \left(c \cdot a\right)}}{\sqrt{\frac{{b}^{4} - 16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} + b}}{a \cdot 2} \]
9. Step-by-step derivation

   1. *-commutative 99.3%

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

10. Simplified 99.3%

    \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot -4}}{\sqrt{\frac{{b}^{4} - 16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} + b}}{a \cdot 2} \]
11. Step-by-step derivation

    1. div-inv 99.3%

       \[\leadsto \color{blue}{\frac{\left(c \cdot a\right) \cdot -4}{\sqrt{\frac{{b}^{4} - 16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} + b} \cdot \frac{1}{a \cdot 2}} \]

12. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{c \cdot \left(a \cdot -4\right)}{b + \sqrt{\frac{{b}^{4} + {\left(c \cdot a\right)}^{2} \cdot -16}{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}}} \cdot \frac{1}{a \cdot 2}} \]
13. Step-by-step derivation

    1. associate-*l/ 99.4%

       \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \frac{1}{a \cdot 2}}{b + \sqrt{\frac{{b}^{4} + {\left(c \cdot a\right)}^{2} \cdot -16}{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}}}} \]

    2. associate-*r/ 99.5%

       \[\leadsto \frac{\color{blue}{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot 1}{a \cdot 2}}}{b + \sqrt{\frac{{b}^{4} + {\left(c \cdot a\right)}^{2} \cdot -16}{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}}} \]

    3. *-rgt-identity 99.5%

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

    4. associate-*r* 99.5%

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

    5. *-commutative 99.5%

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

    6. *-commutative 99.5%

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

    7. times-frac 99.5%

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

    8. metadata-eval 99.5%

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

    9. *-commutative 99.5%

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

    10. associate-*r* 99.5%

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

    11. +-commutative 99.5%

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

    12. fma-def 99.5%

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

    13. associate-*r* 99.5%

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

    14. *-commutative 99.5%

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

    15. associate-*l* 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

    \[\leadsto \frac{-2 \cdot \frac{c \cdot a}{a}}{b + \sqrt{\frac{\mathsf{fma}\left({\left(c \cdot a\right)}^{2}, -16, {b}^{4}\right)}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}} \]

Alternative 2: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0062:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.0062)
   (/ (- (sqrt (- (* b b) (* (* c a) 4.0))) b) (* a 2.0))
   (- (/ (- c) b) (* a (/ c (/ (pow b 3.0) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.0062) {
		tmp = (sqrt(((b * b) - ((c * a) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - (a * (c / (pow(b, 3.0) / 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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-0.0062d0)) then
        tmp = (sqrt(((b * b) - ((c * a) * 4.0d0))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - (a * (c / ((b ** 3.0d0) / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.0062) {
		tmp = (Math.sqrt(((b * b) - ((c * a) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - (a * (c / (Math.pow(b, 3.0) / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.0062:
		tmp = (math.sqrt(((b * b) - ((c * a) * 4.0))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - (a * (c / (math.pow(b, 3.0) / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.0062)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(c * a) * 4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(a * Float64(c / Float64((b ^ 3.0) / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.0062)
		tmp = (sqrt(((b * b) - ((c * a) * 4.0))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - (a * (c / ((b ^ 3.0) / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.0062], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / 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)) < -0.00619999999999999978

   1. Initial program 80.2%

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

   3. Simplified 80.5%

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

      1. *-commutative 80.5%

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

      2. metadata-eval 80.5%

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

      3. distribute-lft-neg-in 80.5%

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

      4. distribute-rgt-neg-in 80.5%

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

      5. *-commutative 80.5%

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

      6. fma-neg 80.2%

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

      7. associate-*l* 80.2%

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

   5. Applied egg-rr 80.2%

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

   if -0.00619999999999999978 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 47.4%

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

   2. Taylor expanded in b around inf 87.2%

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

      1. +-commutative 87.2%

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

      2. mul-1-neg 87.2%

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

      3. unsub-neg 87.2%

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

      4. mul-1-neg 87.2%

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

      5. distribute-neg-frac 87.2%

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

      6. associate-/l* 87.2%

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

      7. associate-/r/ 87.2%

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

      8. unpow2 87.2%

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

      9. associate-/l* 87.2%

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

   4. Simplified 87.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0062:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot a\right) \cdot -4\\ \frac{\frac{t_0}{b + \sqrt{t_0 + b \cdot b}}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c a) -4.0)))
   (/ (/ t_0 (+ b (sqrt (+ t_0 (* b b))))) (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * -4.0;
	return (t_0 / (b + sqrt((t_0 + (b * b))))) / (a * 2.0);
}

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
    t_0 = (c * a) * (-4.0d0)
    code = (t_0 / (b + sqrt((t_0 + (b * b))))) / (a * 2.0d0)
end function

Java

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

Python

def code(a, b, c):
	t_0 = (c * a) * -4.0
	return (t_0 / (b + math.sqrt((t_0 + (b * b))))) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * -4.0)
	return Float64(Float64(t_0 / Float64(b + sqrt(Float64(t_0 + Float64(b * b))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (c * a) * -4.0;
	tmp = (t_0 / (b + sqrt((t_0 + (b * b))))) / (a * 2.0);
end

Wolfram

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

Derivation

1. Initial program 58.2%

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

3. Simplified 58.3%

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

   1. *-commutative 58.3%

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

   2. metadata-eval 58.3%

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

   3. distribute-lft-neg-in 58.3%

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

   4. distribute-rgt-neg-in 58.3%

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

   5. *-commutative 58.3%

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

   6. fma-neg 58.2%

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

   7. flip-- 58.0%

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

   8. div-sub 58.0%

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

   9. pow2 58.0%

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

   10. pow2 58.0%

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

   11. pow-prod-up 58.0%

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

   12. metadata-eval 58.0%

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

   13. fma-def 58.2%

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

   14. associate-*l* 58.2%

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

   15. pow2 58.2%

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

   16. associate-*l* 58.2%

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

   17. fma-def 58.2%

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

   18. associate-*l* 58.2%

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

5. Applied egg-rr 58.2%

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

   1. flip-- 58.0%

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

7. Applied egg-rr 58.9%

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

8. Taylor expanded in b around 0 99.3%

   \[\leadsto \frac{\frac{\color{blue}{-4 \cdot \left(c \cdot a\right)}}{\sqrt{\frac{{b}^{4} - 16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} + b}}{a \cdot 2} \]
9. Step-by-step derivation

   1. *-commutative 99.3%

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

10. Simplified 99.3%

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

11. Taylor expanded in b around 0 99.3%

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

    1. unpow2 99.3%

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

    2. *-commutative 99.3%

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

13. Simplified 99.3%

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

14. Final simplification 99.3%

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

Alternative 4: 81.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in b around inf 78.8%

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

   1. +-commutative 78.8%

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

   2. mul-1-neg 78.8%

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

   3. unsub-neg 78.8%

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

   4. mul-1-neg 78.8%

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

   5. distribute-neg-frac 78.8%

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

   6. associate-/l* 78.8%

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

   7. associate-/r/ 78.8%

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

   8. unpow2 78.8%

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

   9. associate-/l* 78.8%

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

4. Simplified 78.8%

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

5. Final simplification 78.8%

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

Alternative 5: 64.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(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 58.2%

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

2. Taylor expanded in b around inf 61.6%

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

   1. mul-1-neg 61.6%

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

   2. distribute-neg-frac 61.6%

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

4. Simplified 61.6%

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

5. Final simplification 61.6%

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

Alternative 6: 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 58.2%

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

   1. add-sqr-sqrt 58.2%

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

   2. difference-of-squares 58.3%

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

   3. associate-*l* 58.3%

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

   4. sqrt-prod 58.3%

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

   5. metadata-eval 58.3%

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

   6. associate-*l* 58.3%

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

   7. sqrt-prod 58.3%

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

   8. metadata-eval 58.3%

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

3. Applied egg-rr 58.3%

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

   1. *-commutative 58.3%

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

   2. *-commutative 58.3%

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

   3. cancel-sign-sub-inv 58.3%

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

   4. metadata-eval 58.3%

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

   5. *-commutative 58.3%

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

5. Simplified 58.3%

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

6. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{c \cdot a} + 2 \cdot \sqrt{c \cdot a}}{a}} \]
7. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(-2 \cdot \sqrt{c \cdot a} + 2 \cdot \sqrt{c \cdot a}\right)}{a}} \]

   2. distribute-rgt-out 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\sqrt{c \cdot a} \cdot \left(-2 + 2\right)\right)}}{a} \]

   3. *-commutative 3.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\color{blue}{a \cdot c}} \cdot \left(-2 + 2\right)\right)}{a} \]

   4. metadata-eval 3.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{a \cdot c} \cdot \color{blue}{0}\right)}{a} \]

   5. mul0-rgt 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

   6. metadata-eval 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023268 
(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)))