Cubic critical, wide range

Percentage Accurate: 17.6% → 97.6%

Time: 12.4s

Alternatives: 9

Speedup: 2.9×

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

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

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

Wolfram

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

Initial Program: 17.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{6.328125}{{b}^{6}} \cdot \left({a}^{4} \cdot {c}^{4}\right), \mathsf{fma}\left(\frac{-0.375 \cdot \left(c \cdot c\right)}{b}, \frac{a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (* (* a a) -0.5625)
   (/ (pow c 3.0) (pow b 4.0))
   (fma
    (/ -0.16666666666666666 a)
    (* (/ 6.328125 (pow b 6.0)) (* (pow a 4.0) (pow c 4.0)))
    (fma (/ (* -0.375 (* c c)) b) (/ a b) (* -0.5 c))))
  b))

C

double code(double a, double b, double c) {
	return fma(((a * a) * -0.5625), (pow(c, 3.0) / pow(b, 4.0)), fma((-0.16666666666666666 / a), ((6.328125 / pow(b, 6.0)) * (pow(a, 4.0) * pow(c, 4.0))), fma(((-0.375 * (c * c)) / b), (a / b), (-0.5 * c)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(a * a) * -0.5625), Float64((c ^ 3.0) / (b ^ 4.0)), fma(Float64(-0.16666666666666666 / a), Float64(Float64(6.328125 / (b ^ 6.0)) * Float64((a ^ 4.0) * (c ^ 4.0))), fma(Float64(Float64(-0.375 * Float64(c * c)) / b), Float64(a / b), Float64(-0.5 * c)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 / a), $MachinePrecision] * N[(N[(6.328125 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \left({c}^{4} \cdot {a}^{4}\right) \cdot \frac{6.328125}{{b}^{6}}, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]

6. Final simplification 98.3%

   \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{6.328125}{{b}^{6}} \cdot \left({a}^{4} \cdot {c}^{4}\right), \mathsf{fma}\left(\frac{-0.375 \cdot \left(c \cdot c\right)}{b}, \frac{a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \]
7. Add Preprocessing

Alternative 2: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{6.328125}{b} \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (fma
   (fma
    (* -0.16666666666666666 a)
    (* (/ 6.328125 b) (/ (pow c 4.0) (pow b 6.0)))
    (/ (* (pow c 3.0) -0.5625) (pow b 5.0)))
   a
   (/ (* -0.375 (* c c)) (pow b 3.0)))
  a
  (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	return fma(fma(fma((-0.16666666666666666 * a), ((6.328125 / b) * (pow(c, 4.0) / pow(b, 6.0))), ((pow(c, 3.0) * -0.5625) / pow(b, 5.0))), a, ((-0.375 * (c * c)) / pow(b, 3.0))), a, ((c / b) * -0.5));
}

Julia

function code(a, b, c)
	return fma(fma(fma(Float64(-0.16666666666666666 * a), Float64(Float64(6.328125 / b) * Float64((c ^ 4.0) / (b ^ 6.0))), Float64(Float64((c ^ 3.0) * -0.5625) / (b ^ 5.0))), a, Float64(Float64(-0.375 * Float64(c * c)) / (b ^ 3.0))), a, Float64(Float64(c / b) * -0.5))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-0.16666666666666666 * a), $MachinePrecision] * N[(N[(6.328125 / b), $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Final simplification 98.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{6.328125}{b} \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right) \]
7. Add Preprocessing

Alternative 3: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(c \cdot a, -1.0546875, \left(b \cdot b\right) \cdot -0.5625\right) \cdot \left(a \cdot a\right)}{{b}^{6}}, c, \frac{a}{b \cdot b} \cdot -0.375\right), c, -0.5\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   (fma
    (fma
     (/ (* (fma (* c a) -1.0546875 (* (* b b) -0.5625)) (* a a)) (pow b 6.0))
     c
     (* (/ a (* b b)) -0.375))
    c
    -0.5)
   c)
  b))

C

double code(double a, double b, double c) {
	return (fma(fma(((fma((c * a), -1.0546875, ((b * b) * -0.5625)) * (a * a)) / pow(b, 6.0)), c, ((a / (b * b)) * -0.375)), c, -0.5) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(fma(Float64(Float64(fma(Float64(c * a), -1.0546875, Float64(Float64(b * b) * -0.5625)) * Float64(a * a)) / (b ^ 6.0)), c, Float64(Float64(a / Float64(b * b)) * -0.375)), c, -0.5) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * -1.0546875 + N[(N[(b * b), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * c + N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \left({c}^{4} \cdot {a}^{4}\right) \cdot \frac{6.328125}{{b}^{6}}, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]

6. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - \frac{1}{2}\right)}{b} \]
7. Step-by-step derivation

8. Applied rewrites 98.2%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{6}}\right), c, \frac{a}{b \cdot b} \cdot -0.375\right), c, -0.5\right) \cdot c}{b} \]

9. Taylor expanded in b around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{3} \cdot c\right) + \frac{-9}{16} \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{b}^{6}}, c, \frac{a}{b \cdot b} \cdot \frac{-3}{8}\right), c, \frac{-1}{2}\right) \cdot c}{b} \]
10. Step-by-step derivation

11. Applied rewrites 98.2%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot -0.5625\right) \cdot b, b, \left({a}^{3} \cdot c\right) \cdot -1.0546875\right)}{{b}^{6}}, c, \frac{a}{b \cdot b} \cdot -0.375\right), c, -0.5\right) \cdot c}{b} \]

12. Taylor expanded in a around 0

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{a}^{2} \cdot \left(\frac{-135}{128} \cdot \left(a \cdot c\right) + \frac{-9}{16} \cdot {b}^{2}\right)}{{b}^{6}}, c, \frac{a}{b \cdot b} \cdot \frac{-3}{8}\right), c, \frac{-1}{2}\right) \cdot c}{b} \]
13. Step-by-step derivation

14. Applied rewrites 98.2%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(c \cdot a, -1.0546875, \left(b \cdot b\right) \cdot -0.5625\right) \cdot \left(a \cdot a\right)}{{b}^{6}}, c, \frac{a}{b \cdot b} \cdot -0.375\right), c, -0.5\right) \cdot c}{b} \]
15. Add Preprocessing

Alternative 4: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{-0.375 \cdot \left(c \cdot c\right)}{b}, \frac{a}{b}, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{4}}, -0.5625, -0.5\right) \cdot c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (/ (* -0.375 (* c c)) b)
   (/ a b)
   (* (fma (/ (* (* c c) (* a a)) (pow b 4.0)) -0.5625 -0.5) c))
  b))

C

double code(double a, double b, double c) {
	return fma(((-0.375 * (c * c)) / b), (a / b), (fma((((c * c) * (a * a)) / pow(b, 4.0)), -0.5625, -0.5) * c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(-0.375 * Float64(c * c)) / b), Float64(a / b), Float64(fma(Float64(Float64(Float64(c * c) * Float64(a * a)) / (b ^ 4.0)), -0.5625, -0.5) * c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + N[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + -0.5), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

5. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

6. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b}, \frac{a}{b}, c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - \frac{1}{2}\right)\right)}{b} \]
7. Step-by-step derivation

8. Applied rewrites 97.6%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{4}}, -0.5625, -0.5\right) \cdot c\right)}{b} \]

9. Final simplification 97.6%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.375 \cdot \left(c \cdot c\right)}{b}, \frac{a}{b}, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{4}}, -0.5625, -0.5\right) \cdot c\right)}{b} \]
10. Add Preprocessing

Alternative 5: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot -0.5625}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b}, c, -0.5\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   (fma
    (/ (fma (/ (* (* a a) -0.5625) b) (/ c b) (* -0.375 a)) (* b b))
    c
    -0.5)
   c)
  b))

C

double code(double a, double b, double c) {
	return (fma((fma((((a * a) * -0.5625) / b), (c / b), (-0.375 * a)) / (b * b)), c, -0.5) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(fma(Float64(Float64(Float64(a * a) * -0.5625) / b), Float64(c / b), Float64(-0.375 * a)) / Float64(b * b)), c, -0.5) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

5. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

6. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right)}{b} \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{a}{b}, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}}\right), c, -0.5\right) \cdot c}{b} \]

9. Taylor expanded in b around inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{2}}, c, \frac{-1}{2}\right) \cdot c}{b} \]
10. Step-by-step derivation

11. Applied rewrites 97.5%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot -0.5625}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b}, c, -0.5\right) \cdot c}{b} \]
12. Add Preprocessing

Alternative 6: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{-0.375 \cdot \left(c \cdot c\right)}{b}, \frac{a}{b}, -0.5 \cdot c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma (/ (* -0.375 (* c c)) b) (/ a b) (* -0.5 c)) b))

C

double code(double a, double b, double c) {
	return fma(((-0.375 * (c * c)) / b), (a / b), (-0.5 * c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(-0.375 * Float64(c * c)) / b), Float64(a / b), Float64(-0.5 * c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}} + \frac{-1}{2} \cdot c}{b} \]

   4. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{b \cdot b}} + \frac{-1}{2} \cdot c}{b} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \color{blue}{\left({c}^{2} \cdot a\right)}}{b \cdot b} + \frac{-1}{2} \cdot c}{b} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{-3}{8} \cdot {c}^{2}\right) \cdot a}}{b \cdot b} + \frac{-1}{2} \cdot c}{b} \]

   7. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{b} \cdot \frac{a}{b}} + \frac{-1}{2} \cdot c}{b} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{-3}{8} \cdot {c}^{2}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}}{b} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{b}}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{{c}^{2} \cdot \frac{-3}{8}}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{{c}^{2} \cdot \frac{-3}{8}}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

   12. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\left(c \cdot c\right)} \cdot \frac{-3}{8}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\left(c \cdot c\right)} \cdot \frac{-3}{8}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b}, \color{blue}{\frac{a}{b}}, \frac{-1}{2} \cdot c\right)}{b} \]

   15. lower-*.f64 96.0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, \color{blue}{-0.5 \cdot c}\right)}{b} \]

5. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, -0.5 \cdot c\right)}{b}} \]

6. Final simplification 96.0%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.375 \cdot \left(c \cdot c\right)}{b}, \frac{a}{b}, -0.5 \cdot c\right)}{b} \]
7. Add Preprocessing

Alternative 7: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{a}{b \cdot b} \cdot -0.375, c, -0.5\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (fma (* (/ a (* b b)) -0.375) c -0.5) c) b))

C

double code(double a, double b, double c) {
	return (fma(((a / (b * b)) * -0.375), c, -0.5) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(Float64(a / Float64(b * b)) * -0.375), c, -0.5) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] * c + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

5. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

6. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right)}{b} \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{a}{b}, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}}\right), c, -0.5\right) \cdot c}{b} \]

9. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}}, c, \frac{-1}{2}\right) \cdot c}{b} \]
10. Step-by-step derivation

11. Applied rewrites 96.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b \cdot b} \cdot -0.375, c, -0.5\right) \cdot c}{b} \]
12. Add Preprocessing

Alternative 8: 90.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -0.5))

C

double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

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) * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -0.5;
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 89.9

      \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

5. Applied rewrites 89.9%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Add Preprocessing

Alternative 9: 90.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{b} \cdot c \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ -0.5 b) c))

C

double code(double a, double b, double c) {
	return (-0.5 / b) * c;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) / b) * c
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 / b) * c;
}

Python

def code(a, b, c):
	return (-0.5 / b) * c

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 / b) * c)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 / b) * c;
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 / b), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 89.9

      \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

5. Applied rewrites 89.9%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 89.6%

   \[\leadsto c \cdot \color{blue}{\frac{-0.5}{b}} \]

8. Final simplification 89.6%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))