Cubic critical, wide range

Percentage Accurate: 17.8% → 97.5%

Time: 7.4s

Alternatives: 10

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: 17.8% 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: 97.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.7%

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

7. Applied rewrites 95.7%

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

8. Final simplification 95.7%

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

Alternative 2: 97.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.7%

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

7. Applied rewrites 95.7%

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

8. Taylor expanded in c around 0

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

10. Applied rewrites 95.7%

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

11. Taylor expanded in a around 0

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

13. Applied rewrites 95.7%

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

14. Final simplification 95.7%

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

Alternative 3: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.7%

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

7. Applied rewrites 95.7%

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

8. Taylor expanded in c around 0

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

10. Applied rewrites 95.7%

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

11. Taylor expanded in a around 0

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

13. Applied rewrites 94.7%

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

14. Final simplification 94.7%

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

Alternative 4: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.7%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 94.6%

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

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 94.6%

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

12. Final simplification 94.6%

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

Alternative 5: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.375 \cdot a}{b}, \frac{c}{b}, 0.5\right)}{b}, a, -0.6666666666666666 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.3333333333333333
  (fma
   (/ (fma (/ (* 0.375 a) b) (/ c b) 0.5) b)
   a
   (* -0.6666666666666666 (/ b c)))))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / fma((fma(((0.375 * a) / b), (c / b), 0.5) / b), a, (-0.6666666666666666 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / fma(Float64(fma(Float64(Float64(0.375 * a) / b), Float64(c / b), 0.5) / b), a, Float64(-0.6666666666666666 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(N[(N[(N[(N[(0.375 * a), $MachinePrecision] / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 0.5), $MachinePrecision] / b), $MachinePrecision] * a + N[(-0.6666666666666666 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 23.0

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}} \]

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 23.0

       \[\leadsto \frac{0.3333333333333333}{\frac{a}{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}} \]

4. Applied rewrites 23.0%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{a}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}}} \]

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 94.5%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.375 \cdot \frac{c}{{b}^{3}}, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}} \]

8. Taylor expanded in b around inf

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

10. Applied rewrites 94.5%

    \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.375 \cdot a}{b}, \frac{c}{b}, 0.5\right)}{b}, a, \frac{b}{c} \cdot -0.6666666666666666\right)} \]

11. Final simplification 94.5%

    \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.375 \cdot a}{b}, \frac{c}{b}, 0.5\right)}{b}, a, -0.6666666666666666 \cdot \frac{b}{c}\right)} \]
12. Add Preprocessing

Alternative 6: 95.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 92.5

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

5. Applied rewrites 92.5%

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

Alternative 7: 95.2% 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 23.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.7%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 94.6%

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

9. Taylor expanded in a 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 92.5%

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

Alternative 8: 94.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.7%

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

7. Applied rewrites 95.7%

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

8. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

10. Applied rewrites 92.2%

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

11. Final simplification 92.2%

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

Alternative 9: 90.4% 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 23.0%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 86.6

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

5. Applied rewrites 86.6%

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

6. Final simplification 86.6%

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

Alternative 10: 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 23.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 23.0

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}} \]

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 23.0

       \[\leadsto \frac{0.3333333333333333}{\frac{a}{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}} \]

4. Applied rewrites 23.0%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{a}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 22.7

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-b\right)} \]

6. Applied rewrites 22.7%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + \frac{0.3333333333333333}{a} \cdot \left(-b\right)} \]

7. Taylor expanded in a around 0

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

   1. distribute-rgt-out N/A

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

   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 2024332 
(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)))