Cubic critical, wide range

Percentage Accurate: 17.7% → 99.3%

Time: 7.2s

Alternatives: 7

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: 17.7% 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: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((a * c) * -3.0) / (3.0 * a)) / (sqrt(fma((a * -3.0), c, (b * b))) + b);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 17.3

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

4. Applied rewrites 17.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 17.8%

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

7. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/l/ N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 99.3

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

    7. lift--.f64 N/A

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

    8. sub-neg N/A

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

    9. lift-neg.f64 N/A

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

    10. remove-double-neg N/A

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

11. Applied rewrites 99.3%

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

Alternative 2: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1e-14)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1e-14) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1e-14)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1e-14], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

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)) < -9.99999999999999999e-15

   1. Initial program 66.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 67.1

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

   4. Applied rewrites 67.1%

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

   if -9.99999999999999999e-15 < (/.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 5.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1e-14)
   (* (- (sqrt (fma (* -3.0 c) a (* b b))) b) (/ 0.3333333333333333 a))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1e-14) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1e-14)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1e-14], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

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)) < -9.99999999999999999e-15

   1. Initial program 66.7%

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

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 66.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 66.7

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

   4. Applied rewrites 66.7%

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

   if -9.99999999999999999e-15 < (/.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 5.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((a * c) * -3.0) / ((sqrt(fma((a * -3.0), c, (b * b))) + b) * (3.0 * a));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 17.3

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

4. Applied rewrites 17.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 17.8%

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

7. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/l/ N/A

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

    4. lower-/.f64 N/A

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

    5. lower-*.f64 99.3

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

    6. lift--.f64 N/A

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

    7. sub-neg N/A

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

    8. lift-neg.f64 N/A

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

    9. remove-double-neg N/A

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

    10. lower-+.f64 99.3

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

11. Applied rewrites 99.3%

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

12. Final simplification 99.3%

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

Alternative 5: 95.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 94.9

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

5. Applied rewrites 94.9%

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

7. Applied rewrites 94.9%

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

8. Final simplification 94.9%

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

Alternative 6: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 94.9

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

5. Applied rewrites 94.9%

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

7. Applied rewrites 94.9%

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

8. Final simplification 94.9%

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

Alternative 7: 90.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 90.4

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

5. Applied rewrites 90.4%

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

6. Final simplification 90.4%

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

Reproduce

herbie shell --seed 2024304 
(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)))