Quadratic roots, medium range

Percentage Accurate: 31.2% → 95.7%

Time: 16.3s

Alternatives: 9

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.2% 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: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{c}{b} \cdot \frac{c}{b}, c\right), -0.25 \cdot \frac{\left(20 \cdot {b}^{-6}\right) \cdot {\left(a \cdot c\right)}^{4}}{a}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -2.0
   (* (pow a 2.0) (/ (pow c 3.0) (pow b 4.0)))
   (fma
    -1.0
    (fma a (* (/ c b) (/ c b)) c)
    (* -0.25 (/ (* (* 20.0 (pow b -6.0)) (pow (* a c) 4.0)) a))))
  b))

C

double code(double a, double b, double c) {
	return fma(-2.0, (pow(a, 2.0) * (pow(c, 3.0) / pow(b, 4.0))), fma(-1.0, fma(a, ((c / b) * (c / b)), c), (-0.25 * (((20.0 * pow(b, -6.0)) * pow((a * c), 4.0)) / a)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-2.0, Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 4.0))), fma(-1.0, fma(a, Float64(Float64(c / b) * Float64(c / b)), c), Float64(-0.25 * Float64(Float64(Float64(20.0 * (b ^ -6.0)) * (Float64(a * c) ^ 4.0)) / a)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] + N[(-0.25 * N[(N[(N[(20.0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

3. Simplified 30.4%

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

   1. fma-undefine 30.4%

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

   2. flip-+ 30.4%

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

   3. *-commutative 30.4%

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

   4. associate-*r* 30.4%

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

   5. *-commutative 30.4%

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

   6. *-commutative 30.4%

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

   7. associate-*r* 30.4%

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

   8. *-commutative 30.4%

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

   9. pow2 30.4%

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

   10. *-commutative 30.4%

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

   11. associate-*r* 30.4%

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

   12. *-commutative 30.4%

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

   13. pow2 30.4%

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

   14. pow2 30.4%

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

   15. pow-prod-up 30.3%

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

   16. metadata-eval 30.3%

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

   17. pow2 30.3%

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

6. Applied egg-rr 30.3%

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

7. Taylor expanded in b around inf 94.9%

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

9. Simplified 94.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right), -0.25 \cdot \left(\frac{{c}^{4} \cdot {a}^{4}}{a} \cdot \frac{20}{{b}^{6}}\right)\right)\right)}{b}} \]
10. Step-by-step derivation

    1. associate-*l/ 94.9%

       \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right), -0.25 \cdot \color{blue}{\frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot \frac{20}{{b}^{6}}}{a}}\right)\right)}{b} \]

    2. pow-prod-down 94.9%

       \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right), -0.25 \cdot \frac{\color{blue}{{\left(c \cdot a\right)}^{4}} \cdot \frac{20}{{b}^{6}}}{a}\right)\right)}{b} \]

    3. div-inv 94.9%

       \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right), -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot \color{blue}{\left(20 \cdot \frac{1}{{b}^{6}}\right)}}{a}\right)\right)}{b} \]

    4. pow-flip 94.9%

       \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right), -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot \left(20 \cdot \color{blue}{{b}^{\left(-6\right)}}\right)}{a}\right)\right)}{b} \]

    5. metadata-eval 94.9%

       \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right), -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot \left(20 \cdot {b}^{\color{blue}{-6}}\right)}{a}\right)\right)}{b} \]

11. Applied egg-rr 94.9%

    \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right), -0.25 \cdot \color{blue}{\frac{{\left(c \cdot a\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a}}\right)\right)}{b} \]
12. Step-by-step derivation

    1. unpow2 94.9%

       \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \color{blue}{\frac{c}{-b} \cdot \frac{c}{-b}}, c\right), -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a}\right)\right)}{b} \]

13. Applied egg-rr 94.9%

    \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \color{blue}{\frac{c}{-b} \cdot \frac{c}{-b}}, c\right), -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a}\right)\right)}{b} \]

14. Final simplification 94.9%

    \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{c}{b} \cdot \frac{c}{b}, c\right), -0.25 \cdot \frac{\left(20 \cdot {b}^{-6}\right) \cdot {\left(a \cdot c\right)}^{4}}{a}\right)\right)}{b} \]
15. Add Preprocessing

Alternative 2: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \left(-0.25 \cdot \frac{\left(20 \cdot {b}^{-6}\right) \cdot {\left(a \cdot c\right)}^{4}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -2.0
   (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0))
   (-
    (-
     (* -0.25 (/ (* (* 20.0 (pow b -6.0)) (pow (* a c) 4.0)) a))
     (/ (* a (pow c 2.0)) (pow b 2.0)))
    c))
  b))

C

double code(double a, double b, double c) {
	return fma(-2.0, ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0)), (((-0.25 * (((20.0 * pow(b, -6.0)) * pow((a * c), 4.0)) / a)) - ((a * pow(c, 2.0)) / pow(b, 2.0))) - c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-2.0, Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0)), Float64(Float64(Float64(-0.25 * Float64(Float64(Float64(20.0 * (b ^ -6.0)) * (Float64(a * c) ^ 4.0)) / a)) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) - c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[(20.0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in b around inf 94.9%

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

7. Simplified 94.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]
8. Step-by-step derivation

   1. associate-*l/ 94.9%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \color{blue}{\frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{20}{{b}^{6}}}{a}} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

   2. pow-prod-down 94.9%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \frac{20}{{b}^{6}}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

   3. div-inv 94.9%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \color{blue}{\left(20 \cdot \frac{1}{{b}^{6}}\right)}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

   4. pow-flip 94.9%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot \color{blue}{{b}^{\left(-6\right)}}\right)}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

   5. metadata-eval 94.9%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{\color{blue}{-6}}\right)}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

9. Applied egg-rr 94.9%

   \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a}} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

10. Final simplification 94.9%

    \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \left(-0.25 \cdot \frac{\left(20 \cdot {b}^{-6}\right) \cdot {\left(a \cdot c\right)}^{4}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]
11. Add Preprocessing

Alternative 3: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left({c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{c \cdot {b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     (pow c 4.0)
     (+ (* -5.0 (/ (pow a 2.0) (pow b 7.0))) (* -2.0 (/ a (* c (pow b 5.0))))))
    (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((pow(c, 4.0) * ((-5.0 * (pow(a, 2.0) / pow(b, 7.0))) + (-2.0 * (a / (c * pow(b, 5.0)))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (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 = (a * (((c ** 4.0d0) * (((-5.0d0) * ((a ** 2.0d0) / (b ** 7.0d0))) + ((-2.0d0) * (a / (c * (b ** 5.0d0)))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((Math.pow(c, 4.0) * ((-5.0 * (Math.pow(a, 2.0) / Math.pow(b, 7.0))) + (-2.0 * (a / (c * Math.pow(b, 5.0)))))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((math.pow(c, 4.0) * ((-5.0 * (math.pow(a, 2.0) / math.pow(b, 7.0))) + (-2.0 * (a / (c * math.pow(b, 5.0)))))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64((a ^ 2.0) / (b ^ 7.0))) + Float64(-2.0 * Float64(a / Float64(c * (b ^ 5.0)))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * (((c ^ 4.0) * ((-5.0 * ((a ^ 2.0) / (b ^ 7.0))) + (-2.0 * (a / (c * (b ^ 5.0)))))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(a / N[(c * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in a around 0 94.9%

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

   1. +-commutative 94.9%

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

   2. mul-1-neg 94.9%

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

   3. unsub-neg 94.9%

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

7. Simplified 94.9%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \left(a \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{b} \cdot 20\right)\right) \cdot -0.25\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

8. Taylor expanded in c around inf 94.9%

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

9. Final simplification 94.9%

   \[\leadsto a \cdot \left({c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{c \cdot {b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 4: 94.2% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (fma a (pow (/ c b) 2.0) c)
   (* 2.0 (* (pow a 2.0) (/ (pow c 3.0) (pow b 4.0)))))
  (- b)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in c around 0 93.1%

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

6. Taylor expanded in b around -inf 93.5%

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

   1. mul-1-neg 93.5%

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

   2. distribute-neg-frac2 93.5%

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

8. Simplified 93.5%

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

Alternative 5: 94.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in a around 0 93.4%

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

   1. +-commutative 93.4%

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

   2. mul-1-neg 93.4%

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

   3. unsub-neg 93.4%

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

   4. mul-1-neg 93.4%

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

   5. unsub-neg 93.4%

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

   6. associate-/l* 93.4%

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

7. Simplified 93.4%

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

Alternative 6: 93.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 5.0))) (/ a (pow b 3.0))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / 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 * ((c * (((-2.0d0) * (((a ** 2.0d0) * c) / (b ** 5.0d0))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-2.0 * ((Math.pow(a, 2.0) * c) / Math.pow(b, 5.0))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * ((-2.0 * ((math.pow(a, 2.0) * c) / math.pow(b, 5.0))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-2.0 * (((a ^ 2.0) * c) / (b ^ 5.0))) - (a / (b ^ 3.0)))) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in c around 0 93.2%

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

6. Final simplification 93.2%

   \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right) \]
7. Add Preprocessing

Alternative 7: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.5e-5)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (- (- c) (* a (pow (/ c (- b)) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-5) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.5d-5) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c - (a * ((c / -b) ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-5) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * Math.pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.5e-5:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-c - (a * math.pow((c / -b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.5e-5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.5e-5)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-c - (a * ((c / -b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.5e-5], 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], N[(N[((-c) - N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.50000000000000004e-5

   1. Initial program 83.7%

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

   if 1.50000000000000004e-5 < b

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in c around inf 27.1%

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

   6. Taylor expanded in b around inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. unsub-neg 92.7%

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

      3. mul-1-neg 92.7%

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

      4. associate-/l* 92.7%

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

      5. unpow2 92.7%

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

      6. unpow2 92.7%

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

      7. times-frac 92.7%

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

      8. sqr-neg 92.7%

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

      9. distribute-frac-neg2 92.7%

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

      10. distribute-frac-neg2 92.7%

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

      11. unpow2 92.7%

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

      12. distribute-frac-neg2 92.7%

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

      13. distribute-frac-neg 92.7%

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

   8. Simplified 92.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in c around inf 30.6%

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

6. Taylor expanded in b around inf 90.5%

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

   1. mul-1-neg 90.5%

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

   2. unsub-neg 90.5%

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

   3. mul-1-neg 90.5%

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

   4. associate-/l* 90.5%

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

   5. unpow2 90.5%

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

   6. unpow2 90.5%

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

   7. times-frac 90.5%

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

   8. sqr-neg 90.5%

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

   9. distribute-frac-neg2 90.5%

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

   10. distribute-frac-neg2 90.5%

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

   11. unpow2 90.5%

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

   12. distribute-frac-neg2 90.5%

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

   13. distribute-frac-neg 90.5%

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

8. Simplified 90.5%

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

9. Final simplification 90.5%

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

Alternative 9: 81.5% 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(c / Float64(-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 30.4%

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

   1. *-commutative 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in b around inf 81.7%

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

   1. associate-*r/ 81.7%

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

   2. mul-1-neg 81.7%

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

7. Simplified 81.7%

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

8. Final simplification 81.7%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024088 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))