Quadratic roots, wide range

Percentage Accurate: 17.4% → 97.7%

Time: 13.0s

Alternatives: 8

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: 17.4% 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: 97.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (* (pow c 4.0) (- (* -5.0 (/ a (pow b 7.0))) (/ 2.0 (* 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 * ((a * (pow(c, 4.0) * ((-5.0 * (a / pow(b, 7.0))) - (2.0 / (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 * ((a * ((c ** 4.0d0) * (((-5.0d0) * (a / (b ** 7.0d0))) - (2.0d0 / (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 * ((a * (Math.pow(c, 4.0) * ((-5.0 * (a / Math.pow(b, 7.0))) - (2.0 / (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 * ((a * (math.pow(c, 4.0) * ((-5.0 * (a / math.pow(b, 7.0))) - (2.0 / (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(a * Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64(a / (b ^ 7.0))) - Float64(2.0 / 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 * ((a * ((c ^ 4.0) * ((-5.0 * (a / (b ^ 7.0))) - (2.0 / (c * (b ^ 5.0)))))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(a / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / 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 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in a around 0 96.6%

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

   1. +-commutative 96.6%

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

   2. mul-1-neg 96.6%

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

   3. unsub-neg 96.6%

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

7. Simplified 96.6%

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

8. Taylor expanded in c around inf 96.6%

   \[\leadsto a \cdot \left(a \cdot \color{blue}{\left({c}^{4} \cdot \left(-5 \cdot \frac{a}{{b}^{7}} - 2 \cdot \frac{1}{{b}^{5} \cdot c}\right)\right)} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
9. Step-by-step derivation

   1. associate-*r/ 96.6%

      \[\leadsto a \cdot \left(a \cdot \left({c}^{4} \cdot \left(-5 \cdot \frac{a}{{b}^{7}} - \color{blue}{\frac{2 \cdot 1}{{b}^{5} \cdot c}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

   2. metadata-eval 96.6%

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

   3. *-commutative 96.6%

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

10. Simplified 96.6%

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

Alternative 2: 96.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 95.3%

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

6. Taylor expanded in b around -inf 95.8%

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

   1. associate-*r/ 95.8%

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

8. Simplified 95.8%

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

9. Final simplification 95.8%

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

Alternative 3: 96.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+
    -1.0
    (fma
     -2.0
     (* (pow (* a (- c)) 2.0) (pow b -4.0))
     (* a (* (- c) (pow b -2.0))))))
  b))

C

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

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 + fma(-2.0, Float64((Float64(a * Float64(-c)) ^ 2.0) * (b ^ -4.0)), Float64(a * Float64(Float64(-c) * (b ^ -2.0)))))) / b)
end

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 95.3%

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

6. Taylor expanded in b around inf 95.3%

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

8. Simplified 95.3%

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

   1. associate-*r/ 95.7%

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

   2. +-commutative 95.7%

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

   3. div-inv 95.7%

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

   4. pow-flip 95.7%

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

   5. metadata-eval 95.7%

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

   6. div-inv 95.7%

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

   7. pow-flip 95.7%

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

   8. metadata-eval 95.7%

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

10. Applied egg-rr 95.7%

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

11. Final simplification 95.7%

    \[\leadsto \frac{c \cdot \left(-1 + \mathsf{fma}\left(-2, {\left(a \cdot \left(-c\right)\right)}^{2} \cdot {b}^{-4}, a \cdot \left(\left(-c\right) \cdot {b}^{-2}\right)\right)\right)}{b} \]
12. Add Preprocessing

Alternative 4: 96.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 95.3%

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

6. Taylor expanded in b around inf 95.3%

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

8. Simplified 95.3%

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

9. Taylor expanded in b around inf 95.3%

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

    1. mul-1-neg 95.3%

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

    2. unsub-neg 95.3%

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

    3. unpow2 95.3%

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

    4. unpow2 95.3%

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

    5. swap-sqr 95.3%

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

    6. unpow1 95.3%

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

    7. pow-plus 95.3%

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

    8. *-commutative 95.3%

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

    9. metadata-eval 95.3%

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

    10. *-commutative 95.3%

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

11. Simplified 95.3%

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

12. Final simplification 95.3%

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

Alternative 5: 96.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 95.3%

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

6. Taylor expanded in a around 0 95.3%

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

   1. associate-*r/ 95.3%

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

8. Simplified 95.3%

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

9. Final simplification 95.3%

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

Alternative 6: 95.4% 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 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 95.3%

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

6. Taylor expanded in b around inf 94.7%

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

   1. neg-mul-1 94.7%

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

   2. mul-1-neg 94.7%

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

   3. unsub-neg 94.7%

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

   4. associate-/l* 94.7%

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

   5. unpow2 94.7%

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

   6. unpow2 94.7%

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

   7. times-frac 94.7%

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

   8. sqr-neg 94.7%

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

   9. distribute-frac-neg 94.7%

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

   10. distribute-frac-neg 94.7%

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

   11. unpow2 94.7%

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

   12. distribute-frac-neg 94.7%

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

   13. distribute-neg-frac2 94.7%

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

8. Simplified 94.7%

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

Alternative 7: 95.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 95.3%

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

6. Taylor expanded in b around -inf 94.2%

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

   1. associate-*r/ 94.2%

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

   2. mul-1-neg 94.2%

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

8. Simplified 94.2%

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

   1. unpow2 94.2%

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

10. Applied egg-rr 94.2%

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

11. Final simplification 94.2%

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

Alternative 8: 90.6% 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 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in a around 0 90.9%

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

   1. associate-*r/ 90.9%

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

   2. mul-1-neg 90.9%

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

7. Simplified 90.9%

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

8. Final simplification 90.9%

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

Reproduce

herbie shell --seed 2024137 
(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)))