Cubic critical, medium range

Percentage Accurate: 30.9% → 95.5%

Time: 9.8s

Alternatives: 19

Speedup: 3.3×

Specification

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

Math

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

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: 30.9% 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: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \frac{\mathsf{fma}\left(\frac{-1.0546875}{\left(\left(t\_0 \cdot a\right) \cdot b\right) \cdot \left(b \cdot b\right)}, {\left(c \cdot a\right)}^{4}, \left(\left(\frac{a}{t\_0 \cdot b} \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)\right) \cdot -0.5625\right)}{b} + \frac{\mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, -0.5 \cdot c\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (+
    (/
     (fma
      (/ -1.0546875 (* (* (* t_0 a) b) (* b b)))
      (pow (* c a) 4.0)
      (* (* (* (/ a (* t_0 b)) a) (* (* c c) c)) -0.5625))
     b)
    (/ (fma (* (* (/ c (* b b)) c) -0.375) a (* -0.5 c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return (fma((-1.0546875 / (((t_0 * a) * b) * (b * b))), pow((c * a), 4.0), ((((a / (t_0 * b)) * a) * ((c * c) * c)) * -0.5625)) / b) + (fma((((c / (b * b)) * c) * -0.375), a, (-0.5 * c)) / b);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return Float64(Float64(fma(Float64(-1.0546875 / Float64(Float64(Float64(t_0 * a) * b) * Float64(b * b))), (Float64(c * a) ^ 4.0), Float64(Float64(Float64(Float64(a / Float64(t_0 * b)) * a) * Float64(Float64(c * c) * c)) * -0.5625)) / b) + Float64(fma(Float64(Float64(Float64(c / Float64(b * b)) * c) * -0.375), a, Float64(-0.5 * c)) / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(N[(-1.0546875 / N[(N[(N[(t$95$0 * a), $MachinePrecision] * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(N[(N[(a / N[(t$95$0 * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * -0.375), $MachinePrecision] * a + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

6. Applied rewrites 95.5%

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

8. Applied rewrites 95.5%

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

Alternative 2: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \frac{\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, -0.5 \cdot c\right) + \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{\left(a \cdot t\_0\right) \cdot t\_0}, -0.16666666666666666, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \frac{a \cdot a}{t\_0 \cdot b}\right) \cdot -0.5625\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (/
    (+
     (fma (* -0.375 a) (* c (/ c (* b b))) (* -0.5 c))
     (fma
      (* (pow (* c a) 4.0) (/ 6.328125 (* (* a t_0) t_0)))
      -0.16666666666666666
      (* (* (* (* c c) c) (/ (* a a) (* t_0 b))) -0.5625)))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return (fma((-0.375 * a), (c * (c / (b * b))), (-0.5 * c)) + fma((pow((c * a), 4.0) * (6.328125 / ((a * t_0) * t_0))), -0.16666666666666666, ((((c * c) * c) * ((a * a) / (t_0 * b))) * -0.5625))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return Float64(Float64(fma(Float64(-0.375 * a), Float64(c * Float64(c / Float64(b * b))), Float64(-0.5 * c)) + fma(Float64((Float64(c * a) ^ 4.0) * Float64(6.328125 / Float64(Float64(a * t_0) * t_0))), -0.16666666666666666, Float64(Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) / Float64(t_0 * b))) * -0.5625))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(N[(-0.375 * a), $MachinePrecision] * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / N[(N[(a * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[(t$95$0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

6. Applied rewrites 95.5%

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

Alternative 3: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \frac{\mathsf{fma}\left(\frac{c}{b \cdot b}, \left(-0.375 \cdot a\right) \cdot c, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{-1.0546875}{\left(\left(t\_0 \cdot a\right) \cdot b\right) \cdot \left(b \cdot b\right)}, {\left(c \cdot a\right)}^{4}, \left(\left(\frac{a}{t\_0 \cdot b} \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)\right) \cdot -0.5625\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (/
    (fma
     (/ c (* b b))
     (* (* -0.375 a) c)
     (fma
      -0.5
      c
      (fma
       (/ -1.0546875 (* (* (* t_0 a) b) (* b b)))
       (pow (* c a) 4.0)
       (* (* (* (/ a (* t_0 b)) a) (* (* c c) c)) -0.5625))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return fma((c / (b * b)), ((-0.375 * a) * c), fma(-0.5, c, fma((-1.0546875 / (((t_0 * a) * b) * (b * b))), pow((c * a), 4.0), ((((a / (t_0 * b)) * a) * ((c * c) * c)) * -0.5625)))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return Float64(fma(Float64(c / Float64(b * b)), Float64(Float64(-0.375 * a) * c), fma(-0.5, c, fma(Float64(-1.0546875 / Float64(Float64(Float64(t_0 * a) * b) * Float64(b * b))), (Float64(c * a) ^ 4.0), Float64(Float64(Float64(Float64(a / Float64(t_0 * b)) * a) * Float64(Float64(c * c) * c)) * -0.5625)))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.375 * a), $MachinePrecision] * c), $MachinePrecision] + N[(-0.5 * c + N[(N[(-1.0546875 / N[(N[(N[(t$95$0 * a), $MachinePrecision] * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(N[(N[(a / N[(t$95$0 * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

6. Applied rewrites 95.5%

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

8. Applied rewrites 95.5%

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

Alternative 4: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \frac{\mathsf{fma}\left(\left(\left(\frac{a \cdot a}{t\_0 \cdot b} \cdot c\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-1.0546875, \frac{{\left(c \cdot a\right)}^{4}}{\left(\left(\left(t\_0 \cdot a\right) \cdot b\right) \cdot b\right) \cdot b}, \mathsf{fma}\left(-0.375 \cdot \left(\frac{c}{b \cdot b} \cdot c\right), a, -0.5 \cdot c\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (/
    (fma
     (* (* (* (/ (* a a) (* t_0 b)) c) c) c)
     -0.5625
     (fma
      -1.0546875
      (/ (pow (* c a) 4.0) (* (* (* (* t_0 a) b) b) b))
      (fma (* -0.375 (* (/ c (* b b)) c)) a (* -0.5 c))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return fma((((((a * a) / (t_0 * b)) * c) * c) * c), -0.5625, fma(-1.0546875, (pow((c * a), 4.0) / ((((t_0 * a) * b) * b) * b)), fma((-0.375 * ((c / (b * b)) * c)), a, (-0.5 * c)))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return Float64(fma(Float64(Float64(Float64(Float64(Float64(a * a) / Float64(t_0 * b)) * c) * c) * c), -0.5625, fma(-1.0546875, Float64((Float64(c * a) ^ 4.0) / Float64(Float64(Float64(Float64(t_0 * a) * b) * b) * b)), fma(Float64(-0.375 * Float64(Float64(c / Float64(b * b)) * c)), a, Float64(-0.5 * c)))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] / N[(t$95$0 * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * -0.5625 + N[(-1.0546875 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[(N[(N[(N[(t$95$0 * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

6. Applied rewrites 95.5%

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

8. Applied rewrites 95.5%

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

10. Applied rewrites 95.5%

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

Alternative 5: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \mathsf{fma}\left(\frac{-1.0546875}{\left(\left(t\_0 \cdot a\right) \cdot b\right) \cdot \left(b \cdot b\right)}, {\left(c \cdot a\right)}^{4}, \left(\left(\frac{a}{t\_0 \cdot b} \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)\right) \cdot -0.5625\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (/
    (fma
     c
     -0.5
     (fma
      (* (* (/ c (* b b)) c) -0.375)
      a
      (fma
       (/ -1.0546875 (* (* (* t_0 a) b) (* b b)))
       (pow (* c a) 4.0)
       (* (* (* (/ a (* t_0 b)) a) (* (* c c) c)) -0.5625))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return fma(c, -0.5, fma((((c / (b * b)) * c) * -0.375), a, fma((-1.0546875 / (((t_0 * a) * b) * (b * b))), pow((c * a), 4.0), ((((a / (t_0 * b)) * a) * ((c * c) * c)) * -0.5625)))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return Float64(fma(c, -0.5, fma(Float64(Float64(Float64(c / Float64(b * b)) * c) * -0.375), a, fma(Float64(-1.0546875 / Float64(Float64(Float64(t_0 * a) * b) * Float64(b * b))), (Float64(c * a) ^ 4.0), Float64(Float64(Float64(Float64(a / Float64(t_0 * b)) * a) * Float64(Float64(c * c) * c)) * -0.5625)))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(c * -0.5 + N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * -0.375), $MachinePrecision] * a + N[(N[(-1.0546875 / N[(N[(N[(t$95$0 * a), $MachinePrecision] * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(N[(N[(a / N[(t$95$0 * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

6. Applied rewrites 95.5%

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

8. Applied rewrites 95.5%

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

Alternative 6: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{5}}, -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (*
   a
   (fma
    -0.5625
    (/ (* a (pow c 3.0)) (pow b 5.0))
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (a * fma(-0.5625, ((a * pow(c, 3.0)) / pow(b, 5.0)), (-0.375 * (pow(c, 2.0) / pow(b, 3.0))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(a * fma(-0.5625, Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0)), Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

5. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

7. Applied rewrites 94.0%

   \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{c}{b}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{5}}, -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]
8. Add Preprocessing

Alternative 7: 94.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (fma (* -0.375 a) (* c (/ c (* b b))) (* -0.5 c))
   (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0))))
  b))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

6. Applied rewrites 95.5%

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

7. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 94.0

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

9. Applied rewrites 94.0%

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

Alternative 8: 93.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (*
     c
     (fma
      -0.5625
      (/ (* (pow a 2.0) c) (pow b 4.0))
      (* -0.375 (/ a (pow b 2.0)))))
    0.5))
  b))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

5. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

7. Applied rewrites 93.9%

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

Alternative 9: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma -0.5 (/ c b) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

5. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 91.0

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

7. Applied rewrites 91.0%

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

Alternative 10: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (- (* -0.375 (/ (* a c) (pow b 2.0))) 0.5)) b))

C

double code(double a, double b, double c) {
	return (c * ((-0.375 * ((a * c) / pow(b, 2.0))) - 0.5)) / 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 = (c * (((-0.375d0) * ((a * c) / (b ** 2.0d0))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((-0.375 * ((a * c) / Math.pow(b, 2.0))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((-0.375 * ((a * c) / math.pow(b, 2.0))) - 0.5)) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((-0.375 * ((a * c) / (b ^ 2.0))) - 0.5)) / b;
end

Wolfram

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

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.5%

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

5. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 90.9

      \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]

7. Applied rewrites 90.9%

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

Alternative 11: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \left(a \cdot 3\right) \cdot \frac{c}{b \cdot b}}, \left|b\right|, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1e-9)
   (/
    (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)) <= -1e-9) {
		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)) <= -1e-9)
		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], -1e-9], 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)) < -1.00000000000000006e-9

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 31.4%

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

   if -1.00000000000000006e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 30.9%

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

   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 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 12: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{-b} + 1\right) \cdot \left(\left(-b\right) \cdot \frac{0.3333333333333333}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1e-9)
   (*
    (+ (/ (sqrt (fma -3.0 (* c a) (* b b))) (- b)) 1.0)
    (* (- 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)) <= -1e-9) {
		tmp = ((sqrt(fma(-3.0, (c * a), (b * b))) / -b) + 1.0) * (-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)) <= -1e-9)
		tmp = Float64(Float64(Float64(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))) / Float64(-b)) + 1.0) * Float64(Float64(-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], -1e-9], N[(N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-b)), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-b) * N[(0.3333333333333333 / a), $MachinePrecision]), $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)) < -1.00000000000000006e-9

   1. Initial program 30.9%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{-b} + 1\right) \cdot \left(\left(-b\right) \cdot \frac{0.3333333333333333}{a}\right)} \]

   if -1.00000000000000006e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 30.9%

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

   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 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 13: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{-\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}{b} + 1\right) \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1e-9)
   (*
    (+ (/ (- (sqrt (fma (* c a) -3.0 (* b b)))) b) 1.0)
    (* -0.3333333333333333 (/ b a)))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1e-9) {
		tmp = ((-sqrt(fma((c * a), -3.0, (b * b))) / b) + 1.0) * (-0.3333333333333333 * (b / a));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1e-9)
		tmp = Float64(Float64(Float64(Float64(-sqrt(fma(Float64(c * a), -3.0, Float64(b * b)))) / b) + 1.0) * Float64(-0.3333333333333333 * Float64(b / a)));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1e-9], N[(N[(N[((-N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / b), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.3333333333333333 * N[(b / a), $MachinePrecision]), $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)) < -1.00000000000000006e-9

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 32.5%

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 30.8

         \[\leadsto \left(\frac{-\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}{b} + 1\right) \cdot \left(-0.3333333333333333 \cdot \frac{b}{\color{blue}{a}}\right) \]

   8. Applied rewrites 30.8%

      \[\leadsto \left(\frac{-\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}{b} + 1\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{b}{a}\right)} \]

   if -1.00000000000000006e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 30.9%

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

   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 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 14: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-0.3333333333333333 \cdot b\right) \cdot \left(1 - \frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{b}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1e-9)
   (/
    (*
     (* -0.3333333333333333 b)
     (- 1.0 (/ (sqrt (fma (* -3.0 a) c (* b b))) b)))
    a)
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1e-9) {
		tmp = ((-0.3333333333333333 * b) * (1.0 - (sqrt(fma((-3.0 * a), c, (b * b))) / b))) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1e-9)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * b) * Float64(1.0 - Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) / b))) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1e-9], N[(N[(N[(-0.3333333333333333 * b), $MachinePrecision] * N[(1.0 - N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $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)) < -1.00000000000000006e-9

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 32.5%

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

   5. Applied rewrites 30.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. associate-/r* N/A

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

      11. lift-neg.f64 N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{\left(-0.3333333333333333 \cdot b\right) \cdot \left(1 - \frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{b}\right)}{a}} \]

   if -1.00000000000000006e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 30.9%

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

   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 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 15: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{a}, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \frac{0.3333333333333333}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1e-9)
   (fma
    (/ -0.3333333333333333 a)
    b
    (* (sqrt (fma (* -3.0 a) c (* 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)) <= -1e-9) {
		tmp = fma((-0.3333333333333333 / a), b, (sqrt(fma((-3.0 * a), c, (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)) <= -1e-9)
		tmp = fma(Float64(-0.3333333333333333 / a), b, Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(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], -1e-9], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * b + N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $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)) < -1.00000000000000006e-9

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 32.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-/r* N/A

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

      5. mult-flip N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 30.7

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 30.7

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

   5. Applied rewrites 30.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-lft-neg-out N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 30.7

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. lower-*.f64 32.5

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

   7. Applied rewrites 32.5%

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

   if -1.00000000000000006e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 30.9%

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

   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 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 16: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1e-9)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1e-9) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1e-9)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1e-9], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

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)) < -1.00000000000000006e-9

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 31.0

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

   3. Applied rewrites 31.0%

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

   if -1.00000000000000006e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 30.9%

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

   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 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 17: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1e-9)
   (* (- (sqrt (fma (* -3.0 a) c (* 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)) <= -1e-9) {
		tmp = (sqrt(fma((-3.0 * a), c, (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)) <= -1e-9)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, 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], -1e-9], N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + 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)) < -1.00000000000000006e-9

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 32.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-/r* N/A

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

      5. mult-flip N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 30.7

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 30.7

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

   5. Applied rewrites 30.7%

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

   7. Applied rewrites 30.9%

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

   if -1.00000000000000006e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 30.9%

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

   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 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 18: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1e-9)
   (* (- (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)) <= -1e-9) {
		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)) <= -1e-9)
		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], -1e-9], 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)) < -1.00000000000000006e-9

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 30.9%

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

   if -1.00000000000000006e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 30.9%

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

   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 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 19: 81.7% 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 30.9%

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

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 81.7

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

4. Applied rewrites 81.7%

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

Reproduce

herbie shell --seed 2025149 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))