Quadratic roots, wide range

Percentage Accurate: 17.9% → 97.7%

Time: 10.6s

Alternatives: 7

Speedup: 3.6×

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: 17.9% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{20}{b} \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, \frac{c}{{b}^{3}} \cdot \left(-c\right)\right), a, \frac{-c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (fma
   (fma
    (* -0.25 a)
    (* (/ 20.0 b) (/ (pow c 4.0) (pow b 6.0)))
    (/ (* -2.0 (pow c 3.0)) (pow b 5.0)))
   a
   (* (/ c (pow b 3.0)) (- c)))
  a
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	return fma(fma(fma((-0.25 * a), ((20.0 / b) * (pow(c, 4.0) / pow(b, 6.0))), ((-2.0 * pow(c, 3.0)) / pow(b, 5.0))), a, ((c / pow(b, 3.0)) * -c)), a, (-c / b));
}

Julia

function code(a, b, c)
	return fma(fma(fma(Float64(-0.25 * a), Float64(Float64(20.0 / b) * Float64((c ^ 4.0) / (b ^ 6.0))), Float64(Float64(-2.0 * (c ^ 3.0)) / (b ^ 5.0))), a, Float64(Float64(c / (b ^ 3.0)) * Float64(-c))), a, Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-0.25 * a), $MachinePrecision] * N[(N[(20.0 / b), $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.9%

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

6. Final simplification 97.9%

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

Alternative 2: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return fma((fma(((a * a) * -5.0), pow(c, 4.0), ((b * b) * ((c * c) * fma(-b, b, ((c * a) * -2.0))))) / pow(b, 7.0)), a, (-c / b));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-5 \cdot \left(a \cdot {c}^{4}\right) + -2 \cdot \left({b}^{2} \cdot {c}^{3}\right)}{{b}^{7}}, a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
7. Step-by-step derivation

8. Applied rewrites 97.9%

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

9. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{-5 \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(-2 \cdot \left(a \cdot {c}^{3}\right) + -1 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
10. Step-by-step derivation

11. Applied rewrites 97.9%

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

12. Taylor expanded in c around 0

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left({c}^{2} \cdot \left(-2 \cdot \left(a \cdot c\right) + -1 \cdot {b}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
13. Step-by-step derivation

14. Applied rewrites 97.9%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(\mathsf{fma}\left(-b, b, \left(a \cdot c\right) \cdot -2\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]

15. Final simplification 97.9%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -5, {c}^{4}, \left(b \cdot b\right) \cdot \left(\left(c \cdot c\right) \cdot \mathsf{fma}\left(-b, b, \left(c \cdot a\right) \cdot -2\right)\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
16. Add Preprocessing

Alternative 3: 96.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return fma((fma(((c / pow(b, 5.0)) * a), -2.0, (-1.0 / pow(b, 3.0))) * (c * c)), a, (-c / b));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-5 \cdot \left(a \cdot {c}^{4}\right) + -2 \cdot \left({b}^{2} \cdot {c}^{3}\right)}{{b}^{7}}, a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
7. Step-by-step derivation

8. Applied rewrites 97.9%

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

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
10. Step-by-step derivation

11. Applied rewrites 96.9%

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

12. Final simplification 96.9%

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

Alternative 4: 96.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return fma((fma((-a * b), b, (((a * a) * c) * -2.0)) / pow(b, 5.0)), c, (-1.0 / b)) * c;
}

Julia

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{-2 \cdot \left({a}^{2} \cdot c\right) + -1 \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}}, c, \frac{-1}{b}\right) \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 96.7%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(-a\right) \cdot b, b, \left(\left(a \cdot a\right) \cdot c\right) \cdot -2\right)}{{b}^{5}}, c, \frac{-1}{b}\right) \cdot c \]
9. Add Preprocessing

Alternative 5: 96.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (fma(((c * c) * -2.0), (a * a), ((-b * b) * fma(a, c, (b * b)))) / pow(b, 5.0)) * c;
}

Julia

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 96.2%

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

9. Final simplification 96.2%

   \[\leadsto \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot -2, a \cdot a, \left(\left(-b\right) \cdot b\right) \cdot \mathsf{fma}\left(a, c, b \cdot b\right)\right)}{{b}^{5}} \cdot c \]
10. Add Preprocessing

Alternative 6: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-neg.f64 95.1

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

8. Applied rewrites 95.1%

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

10. Applied rewrites 95.1%

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

Alternative 7: 90.4% accurate, 3.6× 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 18.6%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

   4. lower-neg.f64 89.7

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

5. Applied rewrites 89.7%

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

Reproduce

herbie shell --seed 2024242 
(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)))