Quadratic roots, wide range

Percentage Accurate: 18.4% → 99.7%

Time: 11.3s

Alternatives: 6

Speedup: 29.0×

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

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

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: 18.4% 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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot 4\right) \cdot a\\ \frac{\frac{t_0}{a \cdot 2}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(t_0, \mathsf{fma}\left(b, b, t_0\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c 4.0) a)))
   (/
    (/ t_0 (* a 2.0))
    (-
     (- b)
     (pow
      (/
       (fma t_0 (fma b b t_0) (pow b 4.0))
       (fma -64.0 (pow (* c a) 3.0) (pow b 6.0)))
      -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * 4.0) * a;
	return (t_0 / (a * 2.0)) / (-b - pow((fma(t_0, fma(b, b, t_0), pow(b, 4.0)) / fma(-64.0, pow((c * a), 3.0), pow(b, 6.0))), -0.5));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * 4.0) * a)
	return Float64(Float64(t_0 / Float64(a * 2.0)) / Float64(Float64(-b) - (Float64(fma(t_0, fma(b, b, t_0), (b ^ 4.0)) / fma(-64.0, (Float64(c * a) ^ 3.0), (b ^ 6.0))) ^ -0.5)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]}, N[(N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Power[N[(N[(t$95$0 * N[(b * b + t$95$0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / N[(-64.0 * N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision] + N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 16.1%

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

   1. *-commutative 16.1%

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

3. Simplified 16.1%

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

   1. flip3-- 16.6%

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

   2. clear-num 16.4%

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

   3. pow2 16.4%

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

   4. pow2 16.4%

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

   5. pow-prod-up 16.6%

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

   6. metadata-eval 16.6%

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

   7. distribute-rgt-out 16.6%

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

   8. associate-*l* 16.6%

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

   9. +-commutative 16.6%

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

   10. fma-def 16.6%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a \cdot 2} \]

   11. associate-*l* 16.6%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a \cdot 2} \]

5. Applied egg-rr 16.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. flip-+ 16.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}} \cdot \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}}}{\left(-b\right) - \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}}}}}{a \cdot 2} \]

7. Applied egg-rr 17.0%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)} \cdot \left({b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}\right)}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}}{a \cdot 2} \]

8. Taylor expanded in b around 0 99.4%

   \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \frac{\frac{4 \cdot \color{blue}{\left(c \cdot a\right)}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]

   2. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]

10. Simplified 99.4%

    \[\leadsto \frac{\frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. expm1-log1p-u 83.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(4 \cdot c\right) \cdot a}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2}\right)\right)} \]

    2. expm1-udef 20.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left(4 \cdot c\right) \cdot a}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2}\right)} - 1} \]

12. Applied egg-rr 20.5%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c \cdot \left(4 \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - {\left(\frac{\mathsf{fma}\left(c \cdot \left(4 \cdot a\right), \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 83.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c \cdot \left(4 \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - {\left(\frac{\mathsf{fma}\left(c \cdot \left(4 \cdot a\right), \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}\right)}\right)\right)} \]

    2. expm1-log1p 99.4%

       \[\leadsto \color{blue}{\frac{c \cdot \left(4 \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - {\left(\frac{\mathsf{fma}\left(c \cdot \left(4 \cdot a\right), \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}\right)}} \]

    3. associate-/r* 99.8%

       \[\leadsto \color{blue}{\frac{\frac{c \cdot \left(4 \cdot a\right)}{a \cdot 2}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(c \cdot \left(4 \cdot a\right), \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}}} \]

    4. associate-*r* 99.8%

       \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot 4\right) \cdot a}}{a \cdot 2}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(c \cdot \left(4 \cdot a\right), \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}} \]

    5. *-commutative 99.8%

       \[\leadsto \frac{\frac{\left(c \cdot 4\right) \cdot a}{\color{blue}{2 \cdot a}}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(c \cdot \left(4 \cdot a\right), \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}} \]

    6. associate-*r* 99.8%

       \[\leadsto \frac{\frac{\left(c \cdot 4\right) \cdot a}{2 \cdot a}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\color{blue}{\left(c \cdot 4\right) \cdot a}, \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}} \]

    7. associate-*r* 99.8%

       \[\leadsto \frac{\frac{\left(c \cdot 4\right) \cdot a}{2 \cdot a}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(c \cdot 4\right) \cdot a, \mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot 4\right) \cdot a}\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}} \]

14. Simplified 99.8%

    \[\leadsto \color{blue}{\frac{\frac{\left(c \cdot 4\right) \cdot a}{2 \cdot a}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(c \cdot 4\right) \cdot a, \mathsf{fma}\left(b, b, \left(c \cdot 4\right) \cdot a\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}}} \]

15. Final simplification 99.8%

    \[\leadsto \frac{\frac{\left(c \cdot 4\right) \cdot a}{a \cdot 2}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(c \cdot 4\right) \cdot a, \mathsf{fma}\left(b, b, \left(c \cdot 4\right) \cdot a\right), {b}^{4}\right)}{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}} \]

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ \frac{\frac{\left(c \cdot 4\right) \cdot a}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(t_0, \mathsf{fma}\left(b, b, t_0\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))))
   (/
    (/
     (* (* c 4.0) a)
     (-
      (- b)
      (pow
       (/
        (fma t_0 (fma b b t_0) (pow b 4.0))
        (+ (pow b 6.0) (* -64.0 (pow (* c a) 3.0))))
       -0.5)))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	return (((c * 4.0) * a) / (-b - pow((fma(t_0, fma(b, b, t_0), pow(b, 4.0)) / (pow(b, 6.0) + (-64.0 * pow((c * a), 3.0)))), -0.5))) / (a * 2.0);
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	return Float64(Float64(Float64(Float64(c * 4.0) * a) / Float64(Float64(-b) - (Float64(fma(t_0, fma(b, b, t_0), (b ^ 4.0)) / Float64((b ^ 6.0) + Float64(-64.0 * (Float64(c * a) ^ 3.0)))) ^ -0.5))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision] / N[((-b) - N[Power[N[(N[(t$95$0 * N[(b * b + t$95$0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 6.0], $MachinePrecision] + N[(-64.0 * N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 16.1%

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

   1. *-commutative 16.1%

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

3. Simplified 16.1%

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

   1. flip3-- 16.6%

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

   2. clear-num 16.4%

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

   3. pow2 16.4%

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

   4. pow2 16.4%

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

   5. pow-prod-up 16.6%

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

   6. metadata-eval 16.6%

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

   7. distribute-rgt-out 16.6%

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

   8. associate-*l* 16.6%

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

   9. +-commutative 16.6%

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

   10. fma-def 16.6%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a \cdot 2} \]

   11. associate-*l* 16.6%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a \cdot 2} \]

5. Applied egg-rr 16.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. flip-+ 16.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}} \cdot \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}}}{\left(-b\right) - \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}}}}}{a \cdot 2} \]

7. Applied egg-rr 17.0%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)} \cdot \left({b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}\right)}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}}{a \cdot 2} \]

8. Taylor expanded in b around 0 99.4%

   \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \frac{\frac{4 \cdot \color{blue}{\left(c \cdot a\right)}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]

   2. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]

10. Simplified 99.4%

    \[\leadsto \frac{\frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\frac{\left(c \cdot 4\right) \cdot a}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(c \cdot \left(4 \cdot a\right), \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]

Alternative 3: 96.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((-2.0 * (pow(c, 3.0) / (pow(b, 5.0) / pow(a, 2.0)))) - (c / b)) - ((a / pow(b, 3.0)) * pow(c, 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
    code = (((-2.0d0) * ((c ** 3.0d0) / ((b ** 5.0d0) / (a ** 2.0d0)))) - (c / b)) - ((a / (b ** 3.0d0)) * (c ** 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.1%

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

   1. *-commutative 16.1%

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

3. Simplified 16.1%

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

4. Taylor expanded in b around inf 96.8%

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

   1. associate-+r+ 96.8%

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

   2. mul-1-neg 96.8%

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

   3. unsub-neg 96.8%

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

   4. mul-1-neg 96.8%

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

   5. unsub-neg 96.8%

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

   6. *-commutative 96.8%

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

   7. associate-/l* 96.8%

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

   8. associate-/l* 96.8%

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

   9. associate-/r/ 96.8%

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

6. Simplified 96.8%

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

7. Final simplification 96.8%

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

Alternative 4: 95.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-c / b) - ((a / pow(b, 3.0)) * pow(c, 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
    code = (-c / b) - ((a / (b ** 3.0d0)) * (c ** 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.1%

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

   1. *-commutative 16.1%

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

3. Simplified 16.1%

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

4. Taylor expanded in b around inf 95.3%

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

   1. mul-1-neg 95.3%

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

   2. unsub-neg 95.3%

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

   3. mul-1-neg 95.3%

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

   4. distribute-neg-frac 95.3%

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

   5. associate-/l* 95.3%

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

   6. associate-/r/ 95.3%

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

6. Simplified 95.3%

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

7. Final simplification 95.3%

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

Alternative 5: 90.0% 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 16.1%

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

   1. *-commutative 16.1%

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

3. Simplified 16.1%

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

4. Taylor expanded in b around inf 91.4%

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

   1. mul-1-neg 91.4%

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

   2. distribute-neg-frac 91.4%

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

6. Simplified 91.4%

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

7. Final simplification 91.4%

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

Alternative 6: 3.3% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 16.1%

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

   1. *-commutative 16.1%

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

3. Simplified 16.1%

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

   1. flip3-- 16.6%

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

   2. clear-num 16.4%

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

   3. pow2 16.4%

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

   4. pow2 16.4%

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

   5. pow-prod-up 16.6%

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

   6. metadata-eval 16.6%

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

   7. distribute-rgt-out 16.6%

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

   8. associate-*l* 16.6%

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

   9. +-commutative 16.6%

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

   10. fma-def 16.6%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a \cdot 2} \]

   11. associate-*l* 16.6%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a \cdot 2} \]

5. Applied egg-rr 16.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. flip-+ 16.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}} \cdot \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}}}{\left(-b\right) - \sqrt{\frac{1}{\frac{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}}}}}{a \cdot 2} \]

7. Applied egg-rr 17.0%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)} \cdot \left({b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}\right)}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}}{a \cdot 2} \]

8. Taylor expanded in b around 0 99.4%

   \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \frac{\frac{4 \cdot \color{blue}{\left(c \cdot a\right)}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]

   2. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]

10. Simplified 99.4%

    \[\leadsto \frac{\frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(\left(4 \cdot a\right) \cdot c, \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{a \cdot 2} \]

11. Taylor expanded in b around inf 3.3%

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

    1. distribute-rgt-out 3.3%

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

    2. metadata-eval 3.3%

       \[\leadsto \frac{c}{b} \cdot \color{blue}{0} \]

    3. mul0-rgt 3.3%

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

13. Simplified 3.3%

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

14. Final simplification 3.3%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))