Cubic critical, narrow range

Percentage Accurate: 55.1% → 92.1%

Time: 5.9s

Alternatives: 8

Speedup: 3.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Initial Program: 55.1% 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: 92.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.215:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, -0.5625 \cdot a\right)}{b \cdot b}, c, -0.375\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.215)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (fma
    a
    (/
     (*
      (fma
       (/ (fma -1.0546875 (/ (* (* a a) c) (* b b)) (* -0.5625 a)) (* b b))
       c
       -0.375)
      (* c c))
     (* (* b b) b))
    (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.215) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma(a, ((fma((fma(-1.0546875, (((a * a) * c) / (b * b)), (-0.5625 * a)) / (b * b)), c, -0.375) * (c * c)) / ((b * b) * b)), ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.215)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = fma(a, Float64(Float64(fma(Float64(fma(-1.0546875, Float64(Float64(Float64(a * a) * c) / Float64(b * b)), Float64(-0.5625 * a)) / Float64(b * b)), c, -0.375) * Float64(c * c)) / Float64(Float64(b * b) * b)), Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.215], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(N[(N[(-1.0546875 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c + -0.375), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.214999999999999997

   1. Initial program 83.0%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.2

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

   3. Applied rewrites 83.2%

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

   if 0.214999999999999997 < b

   1. Initial program 51.2%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 6.328125\right) \cdot a}{b}, -0.16666666666666666, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -0.5625\right), \frac{-0.375 \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right), \frac{c}{b} \cdot -0.5\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{4}} + \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right)}{\color{blue}{{b}^{3}}}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{{c}^{2} \cdot \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-9}{16} \cdot \frac{a}{{b}^{2}}\right) - \frac{3}{8}\right)}{\left(b \cdot b\right) \cdot b}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-9}{16} \cdot \frac{a}{{b}^{2}}\right) - \frac{3}{8}\right) \cdot {c}^{2}}{\left(b \cdot b\right) \cdot b}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-9}{16} \cdot \frac{a}{{b}^{2}}\right) - \frac{3}{8}\right) \cdot {c}^{2}}{\left(b \cdot b\right) \cdot b}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

   10. Applied rewrites 93.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

           \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot a\right)}{b \cdot b}, c, \frac{-3}{8}\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

       11. lift-*.f64 93.3

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

   13. Applied rewrites 93.3%

       \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, -0.5625 \cdot a\right)}{b \cdot b}, c, -0.375\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}, \frac{c}{b} \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 84:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 84.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/
    (fma
     (/ (* (* a a) (* (* c c) c)) (* (* b b) (* b b)))
     -0.5625
     (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 84.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -0.5625, fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 84.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), -0.5625, fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 84.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 84

   1. Initial program 76.1%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.3

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

   3. Applied rewrites 76.3%

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

   if 84 < b

   1. Initial program 46.0%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 84:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -0.5625, -0.375\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 84.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (fma
    a
    (/ (* (fma (* a (/ c (* b b))) -0.5625 -0.375) (* c c)) (* (* b b) b))
    (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 84.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma(a, ((fma((a * (c / (b * b))), -0.5625, -0.375) * (c * c)) / ((b * b) * b)), ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 84.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = fma(a, Float64(Float64(fma(Float64(a * Float64(c / Float64(b * b))), -0.5625, -0.375) * Float64(c * c)) / Float64(Float64(b * b) * b)), Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 84.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + -0.375), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 84

   1. Initial program 76.1%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.3

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

   3. Applied rewrites 76.3%

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

   if 84 < b

   1. Initial program 46.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 6.328125\right) \cdot a}{b}, -0.16666666666666666, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -0.5625\right), \frac{-0.375 \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right), \frac{c}{b} \cdot -0.5\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{4}} + \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right)}{\color{blue}{{b}^{3}}}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

   7. Applied rewrites 95.2%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 93.2

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

   10. Applied rewrites 93.2%

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

Alternative 4: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 84:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 84.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 84.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 84.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 84.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 84

   1. Initial program 76.1%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.3

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

   3. Applied rewrites 76.3%

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

   if 84 < b

   1. Initial program 46.0%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 84:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -0.375, -0.5\right)}{b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 84.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (* (/ (fma (* a (/ c (* b b))) -0.375 -0.5) b) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 84.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = (fma((a * (c / (b * b))), -0.375, -0.5) / b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 84.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(a * Float64(c / Float64(b * b))), -0.375, -0.5) / b) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 84.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -0.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 84

   1. Initial program 76.1%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.3

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

   3. Applied rewrites 76.3%

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

   if 84 < b

   1. Initial program 46.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow3 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 88.4%

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -0.375, -0.5\right)}{b} \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

2. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. unpow3 N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 81.8

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

4. Applied rewrites 81.8%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

   11. lift-/.f64 N/A

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

   12. div-sub N/A

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

   13. lower-/.f64 N/A

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

6. Applied rewrites 81.7%

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

Alternative 7: 64.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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 / b) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.7

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

4. Applied rewrites 64.7%

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

Alternative 8: 64.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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) / b) * c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

2. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. unpow3 N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 81.8

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

4. Applied rewrites 81.8%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 64.6

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

7. Applied rewrites 64.6%

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

Reproduce

herbie shell --seed 2025110 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))