Cubic critical, medium range

Percentage Accurate: 31.0% → 95.6%

Time: 7.4s

Alternatives: 12

Speedup: 3.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- 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)
use fmin_fmax_functions
    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]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	((- b) + (sqrt(((b * b) - (((3) * a) * c))))) / ((3) * a)
END code

Initial Program: 31.0% accurate, 1.0× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- 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)
use fmin_fmax_functions
    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]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	((- b) + (sqrt(((b * b) - (((3) * a) * c))))) / ((3) * a)
END code

Alternative 1: 95.6% accurate, 0.2× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/
 (fma
  (/ c (* b b))
  (* (* -0.375 a) c)
  (fma
   -0.5
   c
   (fma
    (* -1.0546875 (pow (* c a) 4.0))
    (/ (pow b -6.0) a)
    (* (* (* (* a a) c) c) (* c (* (pow b -4.0) -0.5625))))))
 b))

C

double code(double a, double b, double c) {
	return fma((c / (b * b)), ((-0.375 * a) * c), fma(-0.5, c, fma((-1.0546875 * pow((c * a), 4.0)), (pow(b, -6.0) / a), ((((a * a) * c) * c) * (c * (pow(b, -4.0) * -0.5625)))))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(c / Float64(b * b)), Float64(Float64(-0.375 * a) * c), fma(-0.5, c, fma(Float64(-1.0546875 * (Float64(c * a) ^ 4.0)), Float64((b ^ -6.0) / a), Float64(Float64(Float64(Float64(a * a) * c) * c) * Float64(c * Float64((b ^ -4.0) * -0.5625)))))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.375 * a), $MachinePrecision] * c), $MachinePrecision] + N[(-0.5 * c + N[(N[(-1.0546875 * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -6.0], $MachinePrecision] / a), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * N[(c * N[(N[Power[b, -4.0], $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	(((c / (b * b)) * (((-375e-3) * a) * c)) + (((-5e-1) * c) + ((((-10546875e-7) * ((c * a) ^ (4))) * ((b ^ (-6)) / a)) + ((((a * a) * c) * c) * (c * ((b ^ (-4)) * (-5625e-4))))))) / b
END code

Derivation

1. Initial program 31.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

6. Applied rewrites 95.6%

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

8. Applied rewrites 95.6%

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

Alternative 2: 95.5% accurate, 0.2× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/
 (fma
  c
  (fma (* -0.375 a) (/ c (* b b)) -0.5)
  (fma
   (* (pow b -4.0) c)
   (* -0.5625 (* (* (* a a) c) c))
   (* (* (/ (pow b -6.0) a) -1.0546875) (pow (* c a) 4.0))))
 b))

C

double code(double a, double b, double c) {
	return fma(c, fma((-0.375 * a), (c / (b * b)), -0.5), fma((pow(b, -4.0) * c), (-0.5625 * (((a * a) * c) * c)), (((pow(b, -6.0) / a) * -1.0546875) * pow((c * a), 4.0)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(c, fma(Float64(-0.375 * a), Float64(c / Float64(b * b)), -0.5), fma(Float64((b ^ -4.0) * c), Float64(-0.5625 * Float64(Float64(Float64(a * a) * c) * c)), Float64(Float64(Float64((b ^ -6.0) / a) * -1.0546875) * (Float64(c * a) ^ 4.0)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(-0.375 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * N[(-0.5625 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[b, -6.0], $MachinePrecision] / a), $MachinePrecision] * -1.0546875), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	((c * ((((-375e-3) * a) * (c / (b * b))) + (-5e-1))) + ((((b ^ (-4)) * c) * ((-5625e-4) * (((a * a) * c) * c))) + ((((b ^ (-6)) / a) * (-10546875e-7)) * ((c * a) ^ (4))))) / b
END code

Derivation

1. Initial program 31.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

6. Applied rewrites 95.6%

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

8. Applied rewrites 95.6%

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

10. Applied rewrites 95.5%

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

Alternative 3: 95.5% accurate, 0.2× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/
 (fma
  (pow (* c a) 4.0)
  (* -1.0546875 (/ (pow b -6.0) a))
  (fma
   (* (* (* (pow b -4.0) -0.5625) c) (* a a))
   (* c c)
   (* c (fma (* -0.375 a) (/ c (* b b)) -0.5))))
 b))

C

double code(double a, double b, double c) {
	return fma(pow((c * a), 4.0), (-1.0546875 * (pow(b, -6.0) / a)), fma((((pow(b, -4.0) * -0.5625) * c) * (a * a)), (c * c), (c * fma((-0.375 * a), (c / (b * b)), -0.5)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma((Float64(c * a) ^ 4.0), Float64(-1.0546875 * Float64((b ^ -6.0) / a)), fma(Float64(Float64(Float64((b ^ -4.0) * -0.5625) * c) * Float64(a * a)), Float64(c * c), Float64(c * fma(Float64(-0.375 * a), Float64(c / Float64(b * b)), -0.5)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(-1.0546875 * N[(N[Power[b, -6.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * -0.5625), $MachinePrecision] * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(c * N[(N[(-0.375 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	((((c * a) ^ (4)) * ((-10546875e-7) * ((b ^ (-6)) / a))) + ((((((b ^ (-4)) * (-5625e-4)) * c) * (a * a)) * (c * c)) + (c * ((((-375e-3) * a) * (c / (b * b))) + (-5e-1))))) / b
END code

Derivation

1. Initial program 31.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

6. Applied rewrites 95.6%

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

8. Applied rewrites 95.6%

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

10. Applied rewrites 95.5%

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

Alternative 4: 94.0% accurate, 0.3× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\frac{\mathsf{fma}\left(-0.5, c, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{4}}, -0.375 \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{b} \]

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/
 (fma
  -0.5
  c
  (*
   a
   (fma
    -0.5625
    (/ (* a (pow c 3.0)) (pow b 4.0))
    (* -0.375 (/ (pow c 2.0) (pow b 2.0))))))
 b))

C

double code(double a, double b, double c) {
	return fma(-0.5, c, (a * fma(-0.5625, ((a * pow(c, 3.0)) / pow(b, 4.0)), (-0.375 * (pow(c, 2.0) / pow(b, 2.0)))))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-0.5, c, Float64(a * fma(-0.5625, Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0)), Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 2.0)))))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * c + N[(a * N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	(((-5e-1) * c) + (a * (((-5625e-4) * ((a * (c ^ (3))) / (b ^ (4)))) + ((-375e-3) * ((c ^ (2)) / (b ^ (2))))))) / b
END code

Derivation

1. Initial program 31.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

5. Taylor expanded in a around 0

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

7. Applied rewrites 94.0%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{4}}, -0.375 \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{b} \]
8. Add Preprocessing

Alternative 5: 94.0% accurate, 0.3× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\frac{\mathsf{fma}\left(\frac{c}{b \cdot b}, \left(-0.375 \cdot a\right) \cdot c, \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, -0.5 \cdot c\right)\right)}{b} \]

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/
 (fma
  (/ c (* b b))
  (* (* -0.375 a) c)
  (fma
   -0.5625
   (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0))
   (* -0.5 c)))
 b))

C

double code(double a, double b, double c) {
	return fma((c / (b * b)), ((-0.375 * a) * c), fma(-0.5625, ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0)), (-0.5 * c))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(c / Float64(b * b)), Float64(Float64(-0.375 * a) * c), fma(-0.5625, Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0)), Float64(-0.5 * c))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.375 * a), $MachinePrecision] * c), $MachinePrecision] + N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	(((c / (b * b)) * (((-375e-3) * a) * c)) + (((-5625e-4) * (((a ^ (2)) * (c ^ (3))) / (b ^ (4)))) + ((-5e-1) * c))) / b
END code

Derivation

1. Initial program 31.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

6. Applied rewrites 95.6%

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

8. Applied rewrites 95.6%

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

9. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{c}{b \cdot b}, \left(-0.375 \cdot a\right) \cdot c, \frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \frac{-1}{2} \cdot c\right)}{b} \]
10. Step-by-step derivation

11. Applied rewrites 94.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{c}{b \cdot b}, \left(-0.375 \cdot a\right) \cdot c, \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, -0.5 \cdot c\right)\right)}{b} \]
12. Add Preprocessing

Alternative 6: 94.0% accurate, 0.3× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot c}{{b}^{4}}, -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b} \]

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/
 (*
  c
  (-
   (*
    c
    (fma
     -0.5625
     (/ (* (pow a 2.0) c) (pow b 4.0))
     (* -0.375 (/ a (pow b 2.0)))))
   0.5))
 b))

C

double code(double a, double b, double c) {
	return (c * ((c * fma(-0.5625, ((pow(a, 2.0) * c) / pow(b, 4.0)), (-0.375 * (a / pow(b, 2.0))))) - 0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * fma(-0.5625, Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0)), Float64(-0.375 * Float64(a / (b ^ 2.0))))) - 0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	(c * ((c * (((-5625e-4) * (((a ^ (2)) * c) / (b ^ (4)))) + ((-375e-3) * (a / (b ^ (2)))))) - (5e-1))) / b
END code

Derivation

1. Initial program 31.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

5. Taylor expanded in c around 0

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

7. Applied rewrites 94.0%

   \[\leadsto \frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot c}{{b}^{4}}, -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b} \]
8. Add Preprocessing

Alternative 7: 90.9% accurate, 1.0× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (fma (/ c (* b b)) (* (* -0.375 a) c) (* -0.5 c)) b))

C

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

Julia

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

Wolfram

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

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	(((c / (b * b)) * (((-375e-3) * a) * c)) + ((-5e-1) * c)) / b
END code

Derivation

1. Initial program 31.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

6. Applied rewrites 95.6%

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

8. Applied rewrites 95.6%

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

9. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{c}{b \cdot b}, \left(-0.375 \cdot a\right) \cdot c, \frac{-1}{2} \cdot c\right)}{b} \]
10. Step-by-step derivation

11. Applied rewrites 90.9%

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

Alternative 8: 84.3% accurate, 0.5× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))
     -3e-9)
  (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -3e-9) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -3e-9)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -3e-9], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	LET tmp = IF ((((- b) + (sqrt(((b * b) - (((3) * a) * c))))) / ((3) * a)) <= (-2999999999999999980049621235082650538839033060867222957313060760498046875e-81)) THEN (((- b) + (sqrt(((b * b) + (((-3) * a) * c))))) / ((3) * a)) ELSE ((-5e-1) * (c / b)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -3e-9

   1. Initial program 31.0%

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

   3. Applied rewrites 31.0%

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

   if -3e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 81.6%

      \[\leadsto -0.5 \cdot \frac{c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.3% accurate, 0.5× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -3 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))
     -3e-9)
  (/ (* 0.3333333333333333 (- (sqrt (fma (* c -3.0) a (* b b))) b)) a)
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -3e-9) {
		tmp = (0.3333333333333333 * (sqrt(fma((c * -3.0), a, (b * b))) - b)) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -3e-9)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b)) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -3e-9], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	LET tmp = IF ((((- b) + (sqrt(((b * b) - (((3) * a) * c))))) / ((3) * a)) <= (-2999999999999999980049621235082650538839033060867222957313060760498046875e-81)) THEN (((333333333333333314829616256247390992939472198486328125e-54) * ((sqrt((((c * (-3)) * a) + (b * b)))) - b)) / a) ELSE ((-5e-1) * (c / b)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -3e-9

   1. Initial program 31.0%

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

   3. Applied rewrites 31.0%

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

   if -3e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 81.6%

      \[\leadsto -0.5 \cdot \frac{c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 84.3% accurate, 0.5× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))
     -3e-9)
  (* (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) a) 0.3333333333333333)
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -3e-9) {
		tmp = ((sqrt(fma((c * -3.0), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -3e-9)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -3e-9], N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	LET tmp = IF ((((- b) + (sqrt(((b * b) - (((3) * a) * c))))) / ((3) * a)) <= (-2999999999999999980049621235082650538839033060867222957313060760498046875e-81)) THEN ((((sqrt((((c * (-3)) * a) + (b * b)))) - b) / a) * (333333333333333314829616256247390992939472198486328125e-54)) ELSE ((-5e-1) * (c / b)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -3e-9

   1. Initial program 31.0%

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

   3. Applied rewrites 31.0%

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

   if -3e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 81.6%

      \[\leadsto -0.5 \cdot \frac{c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 84.3% accurate, 0.5× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -3 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))
     -3e-9)
  (* (/ 0.3333333333333333 a) (- (sqrt (fma (* c -3.0) a (* b b))) b))
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -3e-9) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((c * -3.0), a, (b * b))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -3e-9)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -3e-9], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	LET tmp = IF ((((- b) + (sqrt(((b * b) - (((3) * a) * c))))) / ((3) * a)) <= (-2999999999999999980049621235082650538839033060867222957313060760498046875e-81)) THEN (((333333333333333314829616256247390992939472198486328125e-54) / a) * ((sqrt((((c * (-3)) * a) + (b * b)))) - b)) ELSE ((-5e-1) * (c / b)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -3e-9

   1. Initial program 31.0%

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

   3. Applied rewrites 31.0%

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

   if -3e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 81.6%

      \[\leadsto -0.5 \cdot \frac{c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 81.6% accurate, 3.2× speedup

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[-0.5 \cdot \frac{c}{b} \]

FPCore

(FPCore (a b c)
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a)
               (< a 9007199254740992.0))
          (and (< 1.1102230246251565e-16 b)
               (< b 9007199254740992.0)))
     (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (* -0.5 (/ c b)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(a, b, c):
	a in [11102230246251565404236316680908203125e-53, 9007199254740992],
	b in [11102230246251565404236316680908203125e-53, 9007199254740992],
	c in [11102230246251565404236316680908203125e-53, 9007199254740992]
code: THEORY
BEGIN
f(a, b, c: real): real =
	(-5e-1) * (c / b)
END code

Derivation

1. Initial program 31.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 81.6%

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

Reproduce

herbie shell --seed 2026047 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))