Cubic critical, medium range

Percentage Accurate: 30.8% → 93.2%

Time: 6.0s

Alternatives: 14

Speedup: 2.9×

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: 30.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\left(\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, -3 \cdot \left(a \cdot c\right)\right) + b \cdot b\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{5}}, -0.375 \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0013)
   (/
    (fma (* (- b) b) b (pow (fma (* c a) -3.0 (* b b)) 1.5))
    (*
     (+
      (fma b (+ b (sqrt (fma -3.0 (* a c) (* b b)))) (* -3.0 (* a c)))
      (* b b))
     (* a 3.0)))
   (fma
    -0.5
    (/ c b)
    (*
     a
     (fma
      -0.5625
      (/ (* a (pow c 3.0)) (pow b 5.0))
      (* -0.375 (/ (* c c) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0013) {
		tmp = fma((-b * b), b, pow(fma((c * a), -3.0, (b * b)), 1.5)) / ((fma(b, (b + sqrt(fma(-3.0, (a * c), (b * b)))), (-3.0 * (a * c))) + (b * b)) * (a * 3.0));
	} else {
		tmp = fma(-0.5, (c / b), (a * fma(-0.5625, ((a * pow(c, 3.0)) / pow(b, 5.0)), (-0.375 * ((c * c) / pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(fma(Float64(Float64(-b) * b), b, (fma(Float64(c * a), -3.0, Float64(b * b)) ^ 1.5)) / Float64(Float64(fma(b, Float64(b + sqrt(fma(-3.0, Float64(a * c), Float64(b * b)))), Float64(-3.0 * Float64(a * c))) + Float64(b * b)) * Float64(a * 3.0)));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(a * fma(-0.5625, Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0)), Float64(-0.375 * Float64(Float64(c * c) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0013], N[(N[(N[((-b) * b), $MachinePrecision] * b + N[Power[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

   3. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*r* N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-*.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(-b\right) \cdot b}, b, {\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\color{blue}{\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. Applied rewrites 70.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. Applied rewrites 69.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\color{blue}{\left(\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, -3 \cdot \left(a \cdot c\right)\right) + b \cdot b\right)} \cdot \left(a \cdot 3\right)} \]

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      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

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 95.3

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

   6. Applied rewrites 95.3%

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

Alternative 2: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{6}}\\ \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{c \cdot c}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, -0.16666666666666666 \cdot \frac{a \cdot \mathsf{fma}\left(1.265625, t\_0, 5.0625 \cdot t\_0\right)}{b}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (pow c 4.0) (pow b 6.0))))
   (fma
    -0.5
    (/ c b)
    (*
     a
     (fma
      -0.375
      (/ (* c c) (pow b 3.0))
      (*
       a
       (fma
        -0.5625
        (/ (pow c 3.0) (pow b 5.0))
        (*
         -0.16666666666666666
         (/ (* a (fma 1.265625 t_0 (* 5.0625 t_0))) b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(c, 4.0) / pow(b, 6.0);
	return fma(-0.5, (c / b), (a * fma(-0.375, ((c * c) / pow(b, 3.0)), (a * fma(-0.5625, (pow(c, 3.0) / pow(b, 5.0)), (-0.16666666666666666 * ((a * fma(1.265625, t_0, (5.0625 * t_0))) / b)))))));
}

Julia

function code(a, b, c)
	t_0 = Float64((c ^ 4.0) / (b ^ 6.0))
	return fma(-0.5, Float64(c / b), Float64(a * fma(-0.375, Float64(Float64(c * c) / (b ^ 3.0)), Float64(a * fma(-0.5625, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(-0.16666666666666666 * Float64(Float64(a * fma(1.265625, t_0, Float64(5.0625 * t_0))) / b)))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(a * N[(1.265625 * t$95$0 + N[(5.0625 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 30.8

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

3. Applied rewrites 30.8%

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

4. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

6. Applied rewrites 95.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{c \cdot c}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, -0.16666666666666666 \cdot \frac{a \cdot \mathsf{fma}\left(1.265625, \frac{{c}^{4}}{{b}^{6}}, 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)} \]
7. Add Preprocessing

Alternative 3: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot c\right)}^{4}\\ \frac{\mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, t\_0, 5.0625 \cdot t\_0\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 4.0)))
   (/
    (fma
     -0.5625
     (/ (* (* a a) (pow c 3.0)) (pow b 4.0))
     (fma
      -0.5
      c
      (fma
       -0.375
       (/ (* a (* c c)) (* b b))
       (*
        -0.16666666666666666
        (/ (fma 1.265625 t_0 (* 5.0625 t_0)) (* a (pow b 6.0)))))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 4.0);
	return fma(-0.5625, (((a * a) * pow(c, 3.0)) / pow(b, 4.0)), fma(-0.5, c, fma(-0.375, ((a * (c * c)) / (b * b)), (-0.16666666666666666 * (fma(1.265625, t_0, (5.0625 * t_0)) / (a * pow(b, 6.0))))))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 4.0
	return Float64(fma(-0.5625, Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 4.0)), fma(-0.5, c, fma(-0.375, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(-0.16666666666666666 * Float64(fma(1.265625, t_0, Float64(5.0625 * t_0)) / Float64(a * (b ^ 6.0))))))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, N[(N[(-0.5625 * N[(N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(1.265625 * t$95$0 + N[(5.0625 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 30.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 30.8

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

3. Applied rewrites 30.8%

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

4. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

6. Applied rewrites 95.5%

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

Alternative 4: 93.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\left(\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, -3 \cdot \left(a \cdot c\right)\right) + b \cdot b\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0013)
   (/
    (fma (* (- b) b) b (pow (fma (* c a) -3.0 (* b b)) 1.5))
    (*
     (+
      (fma b (+ b (sqrt (fma -3.0 (* a c) (* b b)))) (* -3.0 (* a c)))
      (* b b))
     (* a 3.0)))
   (/
    (fma
     -0.5625
     (/ (* (* a a) (pow c 3.0)) (pow b 4.0))
     (fma -0.5 c (* -0.375 (/ (* a (* c c)) (* b b)))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0013) {
		tmp = fma((-b * b), b, pow(fma((c * a), -3.0, (b * b)), 1.5)) / ((fma(b, (b + sqrt(fma(-3.0, (a * c), (b * b)))), (-3.0 * (a * c))) + (b * b)) * (a * 3.0));
	} else {
		tmp = fma(-0.5625, (((a * a) * pow(c, 3.0)) / pow(b, 4.0)), fma(-0.5, c, (-0.375 * ((a * (c * c)) / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(fma(Float64(Float64(-b) * b), b, (fma(Float64(c * a), -3.0, Float64(b * b)) ^ 1.5)) / Float64(Float64(fma(b, Float64(b + sqrt(fma(-3.0, Float64(a * c), Float64(b * b)))), Float64(-3.0 * Float64(a * c))) + Float64(b * b)) * Float64(a * 3.0)));
	else
		tmp = Float64(fma(-0.5625, Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 4.0)), fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0013], N[(N[(N[((-b) * b), $MachinePrecision] * b + N[Power[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

   3. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*r* N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-*.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(-b\right) \cdot b}, b, {\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\color{blue}{\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. Applied rewrites 70.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. Applied rewrites 69.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\color{blue}{\left(\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, -3 \cdot \left(a \cdot c\right)\right) + b \cdot b\right)} \cdot \left(a \cdot 3\right)} \]

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 95.3%

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

Alternative 5: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\left(\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, -3 \cdot \left(a \cdot c\right)\right) + b \cdot b\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0013)
   (/
    (fma (* (- b) b) b (pow (fma (* c a) -3.0 (* b b)) 1.5))
    (*
     (+
      (fma b (+ b (sqrt (fma -3.0 (* a c) (* b b)))) (* -3.0 (* a c)))
      (* b b))
     (* a 3.0)))
   (fma -0.5 (/ c b) (* -0.375 (/ (* a (* c c)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0013) {
		tmp = fma((-b * b), b, pow(fma((c * a), -3.0, (b * b)), 1.5)) / ((fma(b, (b + sqrt(fma(-3.0, (a * c), (b * b)))), (-3.0 * (a * c))) + (b * b)) * (a * 3.0));
	} else {
		tmp = fma(-0.5, (c / b), (-0.375 * ((a * (c * c)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(fma(Float64(Float64(-b) * b), b, (fma(Float64(c * a), -3.0, Float64(b * b)) ^ 1.5)) / Float64(Float64(fma(b, Float64(b + sqrt(fma(-3.0, Float64(a * c), Float64(b * b)))), Float64(-3.0 * Float64(a * c))) + Float64(b * b)) * Float64(a * 3.0)));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0013], N[(N[(N[((-b) * b), $MachinePrecision] * b + N[Power[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

   3. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*r* N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-*.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(-b\right) \cdot b}, b, {\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\color{blue}{\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. Applied rewrites 70.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. Applied rewrites 69.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\color{blue}{\left(\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, -3 \cdot \left(a \cdot c\right)\right) + b \cdot b\right)} \cdot \left(a \cdot 3\right)} \]

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      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. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 6: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(a \cdot c, -3, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, b \cdot b\right)\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0013)
   (/
    (fma (* (- b) b) b (pow (fma (* c a) -3.0 (* b b)) 1.5))
    (*
     (fma (* a c) -3.0 (fma b (+ b (sqrt (fma -3.0 (* a c) (* b b)))) (* b b)))
     (* a 3.0)))
   (fma -0.5 (/ c b) (* -0.375 (/ (* a (* c c)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0013) {
		tmp = fma((-b * b), b, pow(fma((c * a), -3.0, (b * b)), 1.5)) / (fma((a * c), -3.0, fma(b, (b + sqrt(fma(-3.0, (a * c), (b * b)))), (b * b))) * (a * 3.0));
	} else {
		tmp = fma(-0.5, (c / b), (-0.375 * ((a * (c * c)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(fma(Float64(Float64(-b) * b), b, (fma(Float64(c * a), -3.0, Float64(b * b)) ^ 1.5)) / Float64(fma(Float64(a * c), -3.0, fma(b, Float64(b + sqrt(fma(-3.0, Float64(a * c), Float64(b * b)))), Float64(b * b))) * Float64(a * 3.0)));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0013], N[(N[(N[((-b) * b), $MachinePrecision] * b + N[Power[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

   3. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*r* N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-*.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(-b\right) \cdot b}, b, {\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\color{blue}{\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. Applied rewrites 70.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. Applied rewrites 69.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\color{blue}{\mathsf{fma}\left(a \cdot c, -3, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, b \cdot b\right)\right)} \cdot \left(a \cdot 3\right)} \]

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      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. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 7: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, -3 \cdot \left(a \cdot c\right)\right)\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0013)
   (/
    (fma (* (- b) b) b (pow (fma (* c a) -3.0 (* b b)) 1.5))
    (*
     (fma b b (fma b (+ b (sqrt (fma -3.0 (* a c) (* b b)))) (* -3.0 (* a c))))
     (* a 3.0)))
   (fma -0.5 (/ c b) (* -0.375 (/ (* a (* c c)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0013) {
		tmp = fma((-b * b), b, pow(fma((c * a), -3.0, (b * b)), 1.5)) / (fma(b, b, fma(b, (b + sqrt(fma(-3.0, (a * c), (b * b)))), (-3.0 * (a * c)))) * (a * 3.0));
	} else {
		tmp = fma(-0.5, (c / b), (-0.375 * ((a * (c * c)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(fma(Float64(Float64(-b) * b), b, (fma(Float64(c * a), -3.0, Float64(b * b)) ^ 1.5)) / Float64(fma(b, b, fma(b, Float64(b + sqrt(fma(-3.0, Float64(a * c), Float64(b * b)))), Float64(-3.0 * Float64(a * c)))) * Float64(a * 3.0)));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0013], N[(N[(N[((-b) * b), $MachinePrecision] * b + N[Power[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(b * N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

   3. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*r* N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-*.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(-b\right) \cdot b}, b, {\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\color{blue}{\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. Applied rewrites 70.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. Applied rewrites 70.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, -3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \left(a \cdot 3\right)} \]

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      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. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 8: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\left(\mathsf{fma}\left(-3 \cdot c, a, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, b \cdot b\right)\right) \cdot 3\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0013)
   (/
    (fma (* (- b) b) b (pow (fma (* c a) -3.0 (* b b)) 1.5))
    (*
     (*
      (fma
       (* -3.0 c)
       a
       (fma b (+ b (sqrt (fma -3.0 (* a c) (* b b)))) (* b b)))
      3.0)
     a))
   (fma -0.5 (/ c b) (* -0.375 (/ (* a (* c c)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0013) {
		tmp = fma((-b * b), b, pow(fma((c * a), -3.0, (b * b)), 1.5)) / ((fma((-3.0 * c), a, fma(b, (b + sqrt(fma(-3.0, (a * c), (b * b)))), (b * b))) * 3.0) * a);
	} else {
		tmp = fma(-0.5, (c / b), (-0.375 * ((a * (c * c)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(fma(Float64(Float64(-b) * b), b, (fma(Float64(c * a), -3.0, Float64(b * b)) ^ 1.5)) / Float64(Float64(fma(Float64(-3.0 * c), a, fma(b, Float64(b + sqrt(fma(-3.0, Float64(a * c), Float64(b * b)))), Float64(b * b))) * 3.0) * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0013], N[(N[(N[((-b) * b), $MachinePrecision] * b + N[Power[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

   3. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*r* N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-*.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(-b\right) \cdot b}, b, {\left(\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 70.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\color{blue}{\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}}^{1.5}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. Applied rewrites 70.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. Applied rewrites 70.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{1.5}\right)}{\color{blue}{\left(\mathsf{fma}\left(-3 \cdot c, a, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}, b \cdot b\right)\right) \cdot 3\right) \cdot a}} \]

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      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. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 9: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} - \left(-b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* c a) (* b b))))
   (if (<= b 0.0013)
     (/ (/ (- t_0 (* b b)) (- (sqrt t_0) (- b))) (* 3.0 a))
     (fma -0.5 (/ c b) (* -0.375 (/ (* a (* c c)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (c * a), (b * b));
	double tmp;
	if (b <= 0.0013) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) - -b)) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (-0.375 * ((a * (c * c)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(c * a), Float64(b * b))
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) - Float64(-b))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0013], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] - (-b)), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 69.9%

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

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      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. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 10: 90.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} - \left(-b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* c a) (* b b))))
   (if (<= b 0.0013)
     (/ (/ (- t_0 (* b b)) (- (sqrt t_0) (- b))) (* 3.0 a))
     (/ (fma -0.5 c (* -0.375 (/ (* a (* c c)) (* b b)))) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (c * a), (b * b));
	double tmp;
	if (b <= 0.0013) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) - -b)) / (3.0 * a);
	} else {
		tmp = fma(-0.5, c, (-0.375 * ((a * (c * c)) / (b * b)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(c * a), Float64(b * b))
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) - Float64(-b))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0013], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] - (-b)), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 69.9%

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

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   5. 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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 11: 90.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* c a) (* b b))))
   (if (<= b 0.0013)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* a 3.0)))
     (/ (fma -0.5 c (* -0.375 (/ (* a (* c c)) (* b b)))) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (c * a), (b * b));
	double tmp;
	if (b <= 0.0013) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (a * 3.0));
	} else {
		tmp = fma(-0.5, c, (-0.375 * ((a * (c * c)) / (b * b)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(c * a), Float64(b * b))
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(a * 3.0)));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0013], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 69.9%

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

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   5. 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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 12: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0013:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0013)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (fma -0.5 c (* -0.375 (/ (* a (* c c)) (* b b)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0013) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, c, (-0.375 * ((a * (c * c)) / (b * b)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0013)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0013], 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[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0012999999999999999

   1. Initial program 68.3%

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

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.5

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

   3. Applied rewrites 68.5%

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

   if 0.0012999999999999999 < b

   1. Initial program 27.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.4

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   5. 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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 13: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.315:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.315)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.315) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.315)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.315], 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[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.315000000000000002

   1. Initial program 63.0%

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

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 63.1

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

   3. Applied rewrites 63.1%

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

   if 0.315000000000000002 < b

   1. Initial program 25.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 25.7

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

   3. Applied rewrites 25.7%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 85.5

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

   6. Applied rewrites 85.5%

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

Alternative 14: 81.7% accurate, 2.9× 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 30.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 30.8

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

3. Applied rewrites 30.8%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 81.7

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

6. Applied rewrites 81.7%

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

Reproduce

herbie shell --seed 2025106 
(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)))