Quadratic roots, medium range

Percentage Accurate: 32.0% → 95.4%

Time: 5.9s

Alternatives: 9

Speedup: 4.6×

Specification

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

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Initial Program: 32.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Alternative 1: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \frac{\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{c}{t\_0}, -2, \left(a \cdot \frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{t\_0 \cdot \left(b \cdot b\right)}\right) \cdot -5\right) \cdot a - \frac{c \cdot c}{b \cdot b}\right) \cdot a - c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* (* b b) b) b)))
   (/
    (-
     (*
      (-
       (*
        (fma
         (* (* c c) (/ c t_0))
         -2.0
         (* (* a (/ (* (* c c) (* c c)) (* t_0 (* b b)))) -5.0))
        a)
       (/ (* c c) (* b b)))
      a)
     c)
    b)))

C

double code(double a, double b, double c) {
	double t_0 = ((b * b) * b) * b;
	return ((((fma(((c * c) * (c / t_0)), -2.0, ((a * (((c * c) * (c * c)) / (t_0 * (b * b)))) * -5.0)) * a) - ((c * c) / (b * b))) * a) - c) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(b * b) * b) * b)
	return Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(c * c) * Float64(c / t_0)), -2.0, Float64(Float64(a * Float64(Float64(Float64(c * c) * Float64(c * c)) / Float64(t_0 * Float64(b * b)))) * -5.0)) * a) - Float64(Float64(c * c) / Float64(b * b))) * a) - c) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 32.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   2. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   8. pow3 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   11. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   15. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   16. lift-*.f64 95.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

8. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   6. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   8. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   10. lower-pow.f64 95.3

       \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

10. Applied rewrites 95.3%

    \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

11. Taylor expanded in a around 0

    \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]

13. Applied rewrites 95.4%

    \[\leadsto \frac{\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b}, -2, \left(a \cdot \frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)}\right) \cdot -5\right) \cdot a - \frac{c \cdot c}{b \cdot b}\right) \cdot a - c}{b} \]
14. Add Preprocessing

Alternative 2: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \frac{\left(\left(\mathsf{fma}\left(\frac{a \cdot a}{t\_0}, -2, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{t\_0 \cdot \left(b \cdot b\right)} \cdot -5\right) \cdot c - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* (* b b) b) b)))
   (/
    (*
     (-
      (*
       (-
        (*
         (fma
          (/ (* a a) t_0)
          -2.0
          (* (/ (* (* (* a a) a) c) (* t_0 (* b b))) -5.0))
         c)
        (/ a (* b b)))
       c)
      1.0)
     c)
    b)))

C

double code(double a, double b, double c) {
	double t_0 = ((b * b) * b) * b;
	return (((((fma(((a * a) / t_0), -2.0, (((((a * a) * a) * c) / (t_0 * (b * b))) * -5.0)) * c) - (a / (b * b))) * c) - 1.0) * c) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(b * b) * b) * b)
	return Float64(Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(a * a) / t_0), -2.0, Float64(Float64(Float64(Float64(Float64(a * a) * a) * c) / Float64(t_0 * Float64(b * b))) * -5.0)) * c) - Float64(a / Float64(b * b))) * c) - 1.0) * c) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] / t$95$0), $MachinePrecision] * -2.0 + N[(N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * c), $MachinePrecision] / N[(t$95$0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 32.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   2. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   8. pow3 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   11. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   15. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   16. lift-*.f64 95.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

8. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   6. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   8. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   10. lower-pow.f64 95.3

       \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

10. Applied rewrites 95.3%

    \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

11. Taylor expanded in c around 0

    \[\leadsto \frac{c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]

13. Applied rewrites 95.3%

    \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{a \cdot a}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b}, -2, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)} \cdot -5\right) \cdot c - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b} \]
14. Add Preprocessing

Alternative 3: 93.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   2. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   8. pow3 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   11. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   15. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   16. lift-*.f64 95.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

8. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   6. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   8. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   10. lower-pow.f64 95.3

       \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

10. Applied rewrites 95.3%

    \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

11. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

13. Applied rewrites 93.8%

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

Alternative 4: 93.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b, double c) {
	return ((((((((c * c) * c) * a) / (((b * b) * b) * b)) * -2.0) - ((c * c) / (b * b))) * a) - c) / b;
}

Python

def code(a, b, c):
	return ((((((((c * c) * c) * a) / (((b * b) * b) * b)) * -2.0) - ((c * c) / (b * b))) * a) - c) / b

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   2. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   8. pow3 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   11. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   15. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   16. lift-*.f64 95.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

8. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   6. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   8. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   10. lower-pow.f64 95.3

       \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

10. Applied rewrites 95.3%

    \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

11. Taylor expanded in a around 0

    \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]
12. Step-by-step derivation

    1. lower--.f64 N/A

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

13. Applied rewrites 93.8%

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

Alternative 5: 93.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   (-
    (* (- (/ (* (* (* a a) c) -2.0) (* (* (* b b) b) b)) (/ a (* b b))) c)
    1.0)
   c)
  b))

C

double code(double a, double b, double c) {
	return ((((((((a * a) * c) * -2.0) / (((b * b) * b) * b)) - (a / (b * b))) * c) - 1.0) * 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 = ((((((((a * a) * c) * (-2.0d0)) / (((b * b) * b) * b)) - (a / (b * b))) * c) - 1.0d0) * c) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((((((((a * a) * c) * -2.0) / (((b * b) * b) * b)) - (a / (b * b))) * c) - 1.0) * c) / b;
}

Python

def code(a, b, c):
	return ((((((((a * a) * c) * -2.0) / (((b * b) * b) * b)) - (a / (b * b))) * c) - 1.0) * c) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * -2.0) / Float64(Float64(Float64(b * b) * b) * b)) - Float64(a / Float64(b * b))) * c) - 1.0) * c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((((((a * a) * c) * -2.0) / (((b * b) * b) * b)) - (a / (b * b))) * c) - 1.0) * c) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 32.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   2. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   8. pow3 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   11. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   15. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   16. lift-*.f64 95.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

8. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   6. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   8. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   10. lower-pow.f64 95.3

       \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

10. Applied rewrites 95.3%

    \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

11. Taylor expanded in c around 0

    \[\leadsto \frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]
12. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

13. Applied rewrites 93.7%

    \[\leadsto \frac{\left(\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b} \]
14. Add Preprocessing

Alternative 6: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, \frac{c}{b}, -1 \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma -1.0 (/ c b) (* -1.0 (/ (* a (* c c)) (* (* b b) b)))))

C

double code(double a, double b, double c) {
	return fma(-1.0, (c / b), (-1.0 * ((a * (c * c)) / ((b * b) * b))));
}

Julia

function code(a, b, c)
	return fma(-1.0, Float64(c / b), Float64(-1.0 * Float64(Float64(a * Float64(c * c)) / Float64(Float64(b * b) * b))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 * N[(c / b), $MachinePrecision] + N[(-1.0 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

5. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow3 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 90.6

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

7. Applied rewrites 90.6%

   \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{c}{b}}, -1 \cdot \frac{a \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right) \]
8. Add Preprocessing

Alternative 7: 90.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + {c}^{2} \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   2. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   8. pow3 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot {c}^{2}}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   11. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{{b}^{2}}\right)}{b} \]

   15. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   16. lift-*.f64 95.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

8. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   6. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{2}}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   8. pow2 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

   10. lower-pow.f64 95.3

       \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

10. Applied rewrites 95.3%

    \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}} - 1\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]

11. Taylor expanded in b around inf

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

    1. distribute-lft-out N/A

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

    2. associate-*r/ N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. associate-/l* N/A

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

    9. lower-fma.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 N/A

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

    12. pow2 N/A

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

    13. lift-/.f64 N/A

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

    14. lift-*.f64 90.6

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

13. Applied rewrites 90.6%

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

Alternative 8: 80.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -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 = -c / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\color{blue}{b}} \]

   4. lower-neg.f64 80.9

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

4. Applied rewrites 80.9%

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

Alternative 9: 3.2% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 32.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right)} \cdot c}}{2 \cdot a} \]

   6. pow1/2 N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{\left(-b\right) + {\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}}{2 \cdot a} \]

   8. pow-neg N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. pow2 N/A

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

   12. associate-*l* N/A

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

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

       \[\leadsto \frac{\left(-b\right) + \frac{1}{{\color{blue}{\left({b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)\right)}}^{\frac{-1}{2}}}}{2 \cdot a} \]

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. associate-*r* N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 32.0

       \[\leadsto \frac{\left(-b\right) + \frac{1}{{\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{-0.5}}}{2 \cdot a} \]

3. Applied rewrites 32.0%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{1}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{-0.5}}}}{2 \cdot a} \]

4. Taylor expanded in b around inf

   \[\leadsto \color{blue}{0} \]
5. Step-by-step derivation

6. Applied rewrites 3.2%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))