Cubic critical, wide range

Percentage Accurate: 18.3% → 97.6%

Time: 9.5s

Alternatives: 15

Speedup: 3.3×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

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

C

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

Fortran

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: 18.3% 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: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-b\right) \cdot a}{-3}, \frac{3}{-a}, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (/
    (fma (/ (* (- b) a) -3.0) (/ 3.0 (- a)) (sqrt (fma (* -3.0 c) a (* b b))))
    (* 3.0 a))
   (/
    (fma
     (* (* (* (* (pow b -4.0) c) (* a a)) c) c)
     -0.5625
     (fma -0.5 c (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = fma(((-b * a) / -3.0), (3.0 / -a), sqrt(fma((-3.0 * c), a, (b * b)))) / (3.0 * a);
	} else {
		tmp = fma(((((pow(b, -4.0) * c) * (a * a)) * c) * c), -0.5625, fma(-0.5, c, (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(fma(Float64(Float64(Float64(-b) * a) / -3.0), Float64(3.0 / Float64(-a)), sqrt(fma(Float64(-3.0 * c), a, Float64(b * b)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64((b ^ -4.0) * c) * Float64(a * a)) * c) * c), -0.5625, fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[(N[((-b) * a), $MachinePrecision] / -3.0), $MachinePrecision] * N[(3.0 / (-a)), $MachinePrecision] + N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

   3. Applied rewrites 19.5%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(c \cdot \left(a \cdot a\right)\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

      12. lower-*.f64 97.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \left(\frac{-3}{8} \cdot a\right) \cdot \left(c \cdot \frac{c}{b \cdot b}\right) + \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)}{b} \]

   8. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \frac{-1.0546875}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right)\right)\right)}{b} \]

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-pow.f64 96.8

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

   11. Applied rewrites 96.8%

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

Alternative 2: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-b\right) \cdot a}{-3}, \frac{3}{-a}, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot c}{{b}^{4}}, -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (/
    (fma (/ (* (- b) a) -3.0) (/ 3.0 (- a)) (sqrt (fma (* -3.0 c) a (* b b))))
    (* 3.0 a))
   (/
    (*
     c
     (-
      (*
       c
       (fma
        -0.5625
        (/ (* (pow a 2.0) c) (pow b 4.0))
        (* -0.375 (/ a (pow b 2.0)))))
      0.5))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = fma(((-b * a) / -3.0), (3.0 / -a), sqrt(fma((-3.0 * c), a, (b * b)))) / (3.0 * a);
	} else {
		tmp = (c * ((c * fma(-0.5625, ((pow(a, 2.0) * c) / pow(b, 4.0)), (-0.375 * (a / pow(b, 2.0))))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(fma(Float64(Float64(Float64(-b) * a) / -3.0), Float64(3.0 / Float64(-a)), sqrt(fma(Float64(-3.0 * c), a, Float64(b * b)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * fma(-0.5625, Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0)), Float64(-0.375 * Float64(a / (b ^ 2.0))))) - 0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[(N[((-b) * a), $MachinePrecision] / -3.0), $MachinePrecision] * N[(3.0 / (-a)), $MachinePrecision] + N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

   3. Applied rewrites 19.5%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 96.7%

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

Alternative 3: 94.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {a}^{4} \cdot {c}^{4}\\ \frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, t\_0, 5.0625 \cdot t\_0\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (pow a 4.0) (pow c 4.0))))
   (/
    (fma
     -0.5625
     (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0))
     (fma
      -0.5
      c
      (fma
       -0.375
       (/ (* a (pow c 2.0)) (pow b 2.0))
       (*
        -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, 4.0) * pow(c, 4.0);
	return fma(-0.5625, ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0)), fma(-0.5, c, fma(-0.375, ((a * pow(c, 2.0)) / pow(b, 2.0)), (-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 ^ 4.0) * (c ^ 4.0))
	return Float64(fma(-0.5625, Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0)), fma(-0.5, c, fma(-0.375, Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)), 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[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $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 18.3%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.6%

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

Alternative 4: 94.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot a\right) \cdot a\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left({\left(c \cdot a\right)}^{4}, \frac{-1.0546875}{{b}^{6} \cdot a}, \left(-0.375 \cdot \left(\frac{c}{b \cdot b} \cdot c\right)\right) \cdot a\right)\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (* (* (* (* (* (pow b -4.0) c) a) a) c) c)
   -0.5625
   (fma
    -0.5
    c
    (fma
     (pow (* c a) 4.0)
     (/ -1.0546875 (* (pow b 6.0) a))
     (* (* -0.375 (* (/ c (* b b)) c)) a))))
  b))

C

double code(double a, double b, double c) {
	return fma((((((pow(b, -4.0) * c) * a) * a) * c) * c), -0.5625, fma(-0.5, c, fma(pow((c * a), 4.0), (-1.0546875 / (pow(b, 6.0) * a)), ((-0.375 * ((c / (b * b)) * c)) * a)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(Float64(Float64((b ^ -4.0) * c) * a) * a) * c) * c), -0.5625, fma(-0.5, c, fma((Float64(c * a) ^ 4.0), Float64(-1.0546875 / Float64((b ^ 6.0) * a)), Float64(Float64(-0.375 * Float64(Float64(c / Float64(b * b)) * c)) * a)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(-1.0546875 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.3%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.6%

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

6. Applied rewrites 97.6%

   \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(c \cdot \left(a \cdot a\right)\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   12. lower-*.f64 97.6

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   13. lift-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \left(\frac{-3}{8} \cdot a\right) \cdot \left(c \cdot \frac{c}{b \cdot b}\right) + \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)}{b} \]

8. Applied rewrites 97.6%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \frac{-1.0546875}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right)\right)\right)}{b} \]
9. Step-by-step derivation

10. Applied rewrites 97.6%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot a\right) \cdot a\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left({\left(c \cdot a\right)}^{4}, \frac{-1.0546875}{{b}^{6} \cdot a}, \left(-0.375 \cdot \left(\frac{c}{b \cdot b} \cdot c\right)\right) \cdot a\right)\right)\right)}{\color{blue}{b}} \]
11. Add Preprocessing

Alternative 5: 94.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-b\right) \cdot a}{-3}, \frac{3}{-a}, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (/
    (fma (/ (* (- b) a) -3.0) (/ 3.0 (- a)) (sqrt (fma (* -3.0 c) a (* b b))))
    (* 3.0 a))
   (/ (fma -0.5 c (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = fma(((-b * a) / -3.0), (3.0 / -a), sqrt(fma((-3.0 * c), a, (b * b)))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, c, (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(fma(Float64(Float64(Float64(-b) * a) / -3.0), Float64(3.0 / Float64(-a)), sqrt(fma(Float64(-3.0 * c), a, Float64(b * b)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[(N[((-b) * a), $MachinePrecision] / -3.0), $MachinePrecision] * N[(3.0 / (-a)), $MachinePrecision] + N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

   3. Applied rewrites 19.5%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. 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. lower-pow.f64 N/A

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

      7. lower-pow.f64 95.0

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

   4. Applied rewrites 95.0%

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

Alternative 6: 93.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-b\right) \cdot a}{-3}, \frac{3}{-a}, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot a\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b}, c, \frac{-0.5 \cdot c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (/
    (fma (/ (* (- b) a) -3.0) (/ 3.0 (- a)) (sqrt (fma (* -3.0 c) a (* b b))))
    (* 3.0 a))
   (fma (* (* -0.375 a) (/ c (* (* b b) b))) c (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = fma(((-b * a) / -3.0), (3.0 / -a), sqrt(fma((-3.0 * c), a, (b * b)))) / (3.0 * a);
	} else {
		tmp = fma(((-0.375 * a) * (c / ((b * b) * b))), c, ((-0.5 * c) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(fma(Float64(Float64(Float64(-b) * a) / -3.0), Float64(3.0 / Float64(-a)), sqrt(fma(Float64(-3.0 * c), a, Float64(b * b)))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(Float64(-0.375 * a) * Float64(c / Float64(Float64(b * b) * b))), c, Float64(Float64(-0.5 * c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[(N[((-b) * a), $MachinePrecision] / -3.0), $MachinePrecision] * N[(3.0 / (-a)), $MachinePrecision] + N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * a), $MachinePrecision] * N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

   3. Applied rewrites 19.5%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.7

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied rewrites 94.7%

      \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-flip N/A

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

       4. distribute-rgt-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

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

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

       8. metadata-eval N/A

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

       9. associate-*r* N/A

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

       10. lift-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-/r/ N/A

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

       13. lift-/.f64 N/A

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

       14. lift-/.f64 N/A

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

       15. lower-fma.f64 94.7

           \[\leadsto \mathsf{fma}\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}}, c, \frac{1}{\frac{b}{-0.5 \cdot c}}\right) \]

   11. Applied rewrites 95.0%

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

Alternative 7: 93.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}{a}, a, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot a\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b}, c, \frac{-0.5 \cdot c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (/ (fma (/ (sqrt (fma (* c a) -3.0 (* b b))) a) a (- b)) (* a 3.0))
   (fma (* (* -0.375 a) (/ c (* (* b b) b))) c (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = fma((sqrt(fma((c * a), -3.0, (b * b))) / a), a, -b) / (a * 3.0);
	} else {
		tmp = fma(((-0.375 * a) * (c / ((b * b) * b))), c, ((-0.5 * c) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(fma(Float64(sqrt(fma(Float64(c * a), -3.0, Float64(b * b))) / a), a, Float64(-b)) / Float64(a * 3.0));
	else
		tmp = fma(Float64(Float64(-0.375 * a) * Float64(c / Float64(Float64(b * b) * b))), c, Float64(Float64(-0.5 * c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] * a + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * a), $MachinePrecision] * N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

   3. Applied rewrites 18.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. mult-flip N/A

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

      16. frac-2neg N/A

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

      17. lift-neg.f64 N/A

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

      18. metadata-eval N/A

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

      19. associate-/r* N/A

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

   5. Applied rewrites 19.8%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.7

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied rewrites 94.7%

      \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-flip N/A

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

       4. distribute-rgt-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

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

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

       8. metadata-eval N/A

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

       9. associate-*r* N/A

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

       10. lift-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-/r/ N/A

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

       13. lift-/.f64 N/A

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

       14. lift-/.f64 N/A

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

       15. lower-fma.f64 94.7

           \[\leadsto \mathsf{fma}\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}}, c, \frac{1}{\frac{b}{-0.5 \cdot c}}\right) \]

   11. Applied rewrites 95.0%

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

Alternative 8: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{3 \cdot \frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a}}{9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot a\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b}, c, \frac{-0.5 \cdot c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (/ (* 3.0 (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a)) 9.0)
   (fma (* (* -0.375 a) (/ c (* (* b b) b))) c (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = (3.0 * ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a)) / 9.0;
	} else {
		tmp = fma(((-0.375 * a) * (c / ((b * b) * b))), c, ((-0.5 * c) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(3.0 * Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a)) / 9.0);
	else
		tmp = fma(Float64(Float64(-0.375 * a) * Float64(c / Float64(Float64(b * b) * b))), c, Float64(Float64(-0.5 * c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(3.0 * N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / 9.0), $MachinePrecision], N[(N[(N[(-0.375 * a), $MachinePrecision] * N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

   3. Applied rewrites 18.3%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.7

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied rewrites 94.7%

      \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-flip N/A

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

       4. distribute-rgt-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

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

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

       8. metadata-eval N/A

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

       9. associate-*r* N/A

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

       10. lift-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-/r/ N/A

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

       13. lift-/.f64 N/A

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

       14. lift-/.f64 N/A

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

       15. lower-fma.f64 94.7

           \[\leadsto \mathsf{fma}\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}}, c, \frac{1}{\frac{b}{-0.5 \cdot c}}\right) \]

   11. Applied rewrites 95.0%

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

Alternative 9: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{3 \cdot \frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a}}{9}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(\frac{c}{\left(b \cdot b\right) \cdot b} \cdot a, -0.375, \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (/ (* 3.0 (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a)) 9.0)
   (* c (fma (* (/ c (* (* b b) b)) a) -0.375 (/ -0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = (3.0 * ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a)) / 9.0;
	} else {
		tmp = c * fma(((c / ((b * b) * b)) * a), -0.375, (-0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(3.0 * Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a)) / 9.0);
	else
		tmp = Float64(c * fma(Float64(Float64(c / Float64(Float64(b * b) * b)) * a), -0.375, Float64(-0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(3.0 * N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / 9.0), $MachinePrecision], N[(c * N[(N[(N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -0.375 + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

   3. Applied rewrites 18.3%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.7

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied rewrites 94.7%

      \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. sub-flip N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lift-pow.f64 N/A

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

       13. unpow3 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. lift-/.f64 N/A

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

       18. mult-flip-rev N/A

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

       19. distribute-neg-frac N/A

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

       20. metadata-eval N/A

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

       21. lower-/.f64 94.7

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

   11. Applied rewrites 94.7%

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

Alternative 10: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(\frac{c}{\left(b \cdot b\right) \cdot b} \cdot a, -0.375, \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) (* a 3.0))
   (* c (fma (* (/ c (* (* b b) b)) a) -0.375 (/ -0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = c * fma(((c / ((b * b) * b)) * a), -0.375, (-0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * fma(Float64(Float64(c / Float64(Float64(b * b) * b)) * a), -0.375, Float64(-0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -0.375 + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

   3. Applied rewrites 18.3%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.7

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied rewrites 94.7%

      \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. sub-flip N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lift-pow.f64 N/A

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

       13. unpow3 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. lift-/.f64 N/A

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

       18. mult-flip-rev N/A

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

       19. distribute-neg-frac N/A

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

       20. metadata-eval N/A

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

       21. lower-/.f64 94.7

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

   11. Applied rewrites 94.7%

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

Alternative 11: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c \cdot a}{b}, \frac{-0.375}{b}, -0.5\right)}{b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) (* a 3.0))
   (* (/ (fma (/ (* c a) b) (/ -0.375 b) -0.5) b) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = (fma(((c * a) / b), (-0.375 / b), -0.5) / b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(c * a) / b), Float64(-0.375 / b), -0.5) / b) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * N[(-0.375 / b), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

   3. Applied rewrites 18.3%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.7

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied rewrites 94.7%

      \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 94.7

          \[\leadsto \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \cdot c \]

   11. Applied rewrites 94.7%

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

Alternative 12: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c \cdot a}{b}, \frac{-0.375}{b}, -0.5\right)}{b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
   (* (/ (fma (/ (* c a) b) (/ -0.375 b) -0.5) b) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (fma(((c * a) / b), (-0.375 / b), -0.5) / b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(Float64(c * a) / b), Float64(-0.375 / b), -0.5) / b) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * N[(-0.375 / b), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

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

      3. associate-/r* N/A

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

      4. mult-flip N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 18.3%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.7

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied rewrites 94.7%

      \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 94.7

          \[\leadsto \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \cdot c \]

   11. Applied rewrites 94.7%

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

Alternative 13: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c \cdot a}{b}, \frac{-0.375}{b}, -0.5\right)}{b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.08)
   (* (- (sqrt (fma (* -3.0 c) a (* b b))) b) (/ 0.3333333333333333 a))
   (* (/ (fma (/ (* c a) b) (/ -0.375 b) -0.5) b) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.08) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (fma(((c * a) / b), (-0.375 / b), -0.5) / b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(c * a) / b), Float64(-0.375 / b), -0.5) / b) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * N[(-0.375 / b), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0800000000000000017

   1. Initial program 18.3%

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

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

      3. lower-*.f64 N/A

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

   3. Applied rewrites 18.3%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.7

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied rewrites 94.7%

      \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 94.7

          \[\leadsto \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \cdot c \]

   11. Applied rewrites 94.7%

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

Alternative 14: 93.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.6%

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

6. Applied rewrites 97.6%

   \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

7. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 94.7

      \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

9. Applied rewrites 94.7%

   \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 94.7

       \[\leadsto \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \cdot c \]

11. Applied rewrites 94.7%

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

Alternative 15: 90.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 90.0

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

4. Applied rewrites 90.0%

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

Reproduce

herbie shell --seed 2025152 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))