Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.7%

Time: 11.1s

Alternatives: 9

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: 17.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot a\right) \cdot c\\ \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \mathsf{fma}\left(-5, \frac{t\_0 \cdot t\_0}{{b}^{6}}, \frac{\left(-c\right) \cdot c}{b \cdot b}\right), -c\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c a) c)))
   (/
    (fma
     (* (* -2.0 a) a)
     (* (/ (* c c) (* b b)) (/ c (* b b)))
     (fma
      a
      (fma -5.0 (/ (* t_0 t_0) (pow b 6.0)) (/ (* (- c) c) (* b b)))
      (- c)))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * c;
	return fma(((-2.0 * a) * a), (((c * c) / (b * b)) * (c / (b * b))), fma(a, fma(-5.0, ((t_0 * t_0) / pow(b, 6.0)), ((-c * c) / (b * b))), -c)) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * c)
	return Float64(fma(Float64(Float64(-2.0 * a) * a), Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(c / Float64(b * b))), fma(a, fma(-5.0, Float64(Float64(t_0 * t_0) / (b ^ 6.0)), Float64(Float64(Float64(-c) * c) / Float64(b * b))), Float64(-c))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * c), $MachinePrecision]}, N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(-5.0 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[((-c) * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 21.3%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 96.9%

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

7. Applied rewrites 96.9%

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

8. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, a \cdot \left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} - \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]
9. Step-by-step derivation

10. Applied rewrites 96.9%

    \[\leadsto \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \mathsf{fma}\left(-5, \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}}, \frac{c \cdot c}{\left(-b\right) \cdot b}\right), -c\right)\right)}{b} \]
11. Step-by-step derivation

12. Applied rewrites 96.9%

    \[\leadsto \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \mathsf{fma}\left(-5, \frac{\left(\left(c \cdot a\right) \cdot c\right) \cdot \left(\left(c \cdot a\right) \cdot c\right)}{{b}^{6}}, \frac{c \cdot c}{\left(-b\right) \cdot b}\right), -c\right)\right)}{b} \]

13. Final simplification 96.9%

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

Alternative 2: 96.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 96.9%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 96.0%

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

Alternative 3: 96.6% accurate, 0.3× speedup

Math

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

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(-a), Float64(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[((-a) * N[(b * b), $MachinePrecision] + 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 21.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 95.7%

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

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 95.7%

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

Alternative 4: 96.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 21.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 95.7%

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

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 95.2%

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

Alternative 5: 95.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

3. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. associate-*r/ N/A

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

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-sub N/A

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

   8. unsub-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-out N/A

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

   11. associate-/l* N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 93.8%

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

6. Final simplification 93.8%

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

Alternative 6: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 96.9%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 93.8%

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

Alternative 7: 95.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. distribute-neg-frac N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. associate-*l/ N/A

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

   8. associate-*r/ N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 93.5%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 93.5%

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

Alternative 8: 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 21.3%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 88.0

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

5. Applied rewrites 88.0%

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

Alternative 9: 3.3% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 21.3%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 21.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lift-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. distribute-neg-frac2 N/A

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

   18. lower-/.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. distribute-lft-neg-in N/A

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

   21. metadata-eval N/A

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

6. Applied rewrites 22.1%

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

7. Taylor expanded in a around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*l/ N/A

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

   4. mul0-rgt 3.3

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

9. Applied rewrites 3.3%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(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)))