Cubic critical, narrow range

Percentage Accurate: 54.8% → 99.3%

Time: 7.0s

Alternatives: 9

Speedup: 3.3×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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]

Initial Program: 54.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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]

Alternative 1: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. 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. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 56.3%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.1

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

6. Applied rewrites 99.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

8. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

10. Applied rewrites 99.3%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\left(3 \cdot a\right) \cdot c}{b + \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}} \cdot \frac{-0.3333333333333333}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ (* (* 3.0 a) c) (+ b (sqrt (fma (* c -3.0) a (* b b)))))
  (/ -0.3333333333333333 a)))

C

double code(double a, double b, double c) {
	return (((3.0 * a) * c) / (b + sqrt(fma((c * -3.0), a, (b * b))))) * (-0.3333333333333333 / a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(3.0 * a) * c) / Float64(b + sqrt(fma(Float64(c * -3.0), a, Float64(b * b))))) * Float64(-0.3333333333333333 / a))
end

Wolfram

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

Derivation

1. Initial program 54.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. 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. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 56.3%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.1

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

6. Applied rewrites 99.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

8. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. mult-flip N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 N/A

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

10. Applied rewrites 99.0%

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

Alternative 3: 76.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \left(a \cdot 3\right) \cdot \frac{c}{b \cdot b}}, \left|b\right|, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -4.8e-7)
   (/
    (fma (sqrt (- 1.0 (* (* a 3.0) (/ c (* b b))))) (fabs b) (- b))
    (* 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)) <= -4.8e-7) {
		tmp = fma(sqrt((1.0 - ((a * 3.0) * (c / (b * b))))), fabs(b), -b) / (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)) <= -4.8e-7)
		tmp = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(a * 3.0) * Float64(c / Float64(b * b))))), abs(b), Float64(-b)) / 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], -4.8e-7], N[(N[(N[Sqrt[N[(1.0 - N[(N[(a * 3.0), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $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)) < -4.79999999999999957e-7

   1. Initial program 54.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-to-mult N/A

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

      6. sqrt-prod N/A

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

      7. lift-*.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 55.5%

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

   if -4.79999999999999957e-7 < (/.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 54.8%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.9

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

   4. Applied rewrites 64.9%

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

Alternative 4: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -4.8e-7)
   (/ (+ (- 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)) <= -4.8e-7) {
		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)) <= -4.8e-7)
		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], -4.8e-7], 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]]

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)) < -4.79999999999999957e-7

   1. Initial program 54.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. 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-flip 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-out 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 54.9

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

   3. Applied rewrites 54.9%

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

   if -4.79999999999999957e-7 < (/.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 54.8%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.9

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

   4. Applied rewrites 64.9%

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

Alternative 5: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -4.8e-7)
   (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) (* 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)) <= -4.8e-7) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) / (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)) <= -4.8e-7)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / 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], -4.8e-7], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $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)) < -4.79999999999999957e-7

   1. Initial program 54.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. 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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 55.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 54.7%

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

   if -4.79999999999999957e-7 < (/.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 54.8%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.9

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

   4. Applied rewrites 64.9%

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

Alternative 6: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -4.8e-7)
   (/ (- (sqrt (fma -3.0 (* c a) (* b b))) b) (* a 3.0))
   (* -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)) <= -4.8e-7) {
		tmp = (sqrt(fma(-3.0, (c * a), (b * b))) - b) / (a * 3.0);
	} 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)) <= -4.8e-7)
		tmp = Float64(Float64(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))) - b) / Float64(a * 3.0));
	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], -4.8e-7], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $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)) < -4.79999999999999957e-7

   1. Initial program 54.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 54.8

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 54.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 54.7

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

   3. Applied rewrites 54.7%

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

   if -4.79999999999999957e-7 < (/.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 54.8%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.9

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

   4. Applied rewrites 64.9%

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

Alternative 7: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -4.8e-7)
   (* (- (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 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)) <= -4.8e-7) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) * (0.3333333333333333 / 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)) <= -4.8e-7)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / 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], -4.8e-7], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $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)) < -4.79999999999999957e-7

   1. Initial program 54.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. 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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 55.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-out N/A

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

   7. Applied rewrites 54.7%

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

   if -4.79999999999999957e-7 < (/.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 54.8%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.9

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

   4. Applied rewrites 64.9%

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

Alternative 8: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -4.8e-7)
   (* (- (sqrt (fma -3.0 (* c a) (* b b))) b) (/ 0.3333333333333333 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)) <= -4.8e-7) {
		tmp = (sqrt(fma(-3.0, (c * a), (b * b))) - b) * (0.3333333333333333 / 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)) <= -4.8e-7)
		tmp = Float64(Float64(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))) - b) * Float64(0.3333333333333333 / 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], -4.8e-7], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $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)) < -4.79999999999999957e-7

   1. Initial program 54.8%

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

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

      3. lower-*.f64 N/A

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

   3. Applied rewrites 54.7%

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

   if -4.79999999999999957e-7 < (/.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 54.8%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.9

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

   4. Applied rewrites 64.9%

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

Alternative 9: 64.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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]

Derivation

1. Initial program 54.8%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.9

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

4. Applied rewrites 64.9%

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

Reproduce

herbie shell --seed 2025151 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))