Cubic critical, medium range

Percentage Accurate: 31.1% → 99.4%

Time: 7.6s

Alternatives: 12

Speedup: 3.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

3. Applied rewrites 31.9%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. +-inverses 99.4%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, \color{blue}{0}\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(-3 \cdot a\right) \cdot c + 0}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

   2. +-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. metadata-eval N/A

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

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

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 99.4%

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

7. Applied rewrites 99.4%

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

Alternative 2: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

3. Applied rewrites 31.9%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. +-inverses 99.4%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, \color{blue}{0}\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{3 \cdot a} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(-3 \cdot a\right) \cdot c + 0}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

   3. +-rgt-identity N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.2%

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

7. Applied rewrites 99.2%

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

Alternative 3: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

3. Applied rewrites 31.9%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. +-inverses 99.4%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, \color{blue}{0}\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{\color{blue}{3 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{3 \cdot a} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(-3 \cdot a\right) \cdot c + 0}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

   5. +-rgt-identity N/A

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

   6. associate-/l/ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. metadata-eval N/A

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

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

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

   19. *-commutative N/A

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

   20. metadata-eval N/A

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

7. Applied rewrites 99.1%

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

Alternative 4: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

3. Applied rewrites 31.9%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. +-inverses 99.4%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, \color{blue}{0}\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{\color{blue}{3 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{3 \cdot a} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(-3 \cdot a\right) \cdot c + 0}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

   5. +-rgt-identity N/A

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

   6. associate-/l/ N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 99.2%

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

7. Applied rewrites 99.2%

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

Alternative 5: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

3. Applied rewrites 31.9%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. +-inverses 99.4%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, \color{blue}{0}\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{\color{blue}{3 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{3 \cdot a} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(-3 \cdot a\right) \cdot c + 0}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

   5. +-rgt-identity N/A

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

   6. associate-/l/ N/A

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

   7. lift-*.f64 N/A

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

   8. metadata-eval N/A

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

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

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-neg-out N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

7. Applied rewrites 99.2%

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

Alternative 6: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

3. Applied rewrites 31.9%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. +-inverses 99.4%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, \color{blue}{0}\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{\color{blue}{3 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{3 \cdot a} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(-3 \cdot a\right) \cdot c + 0}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a} \]

   5. +-rgt-identity N/A

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

   6. associate-/l/ N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

7. Applied rewrites 99.2%

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

Alternative 7: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{-3}, 3, \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}\right) \cdot 1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5e-8)
   (/ (* (fma (/ b -3.0) 3.0 (sqrt (fma (* c -3.0) a (* b b)))) 1.0) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5e-8) {
		tmp = (fma((b / -3.0), 3.0, sqrt(fma((c * -3.0), a, (b * b)))) * 1.0) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5e-8)
		tmp = Float64(Float64(fma(Float64(b / -3.0), 3.0, sqrt(fma(Float64(c * -3.0), a, Float64(b * b)))) * 1.0) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5e-8], N[(N[(N[(N[(b / -3.0), $MachinePrecision] * 3.0 + N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.49999999999999987e-8

   1. Initial program 31.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. mult-flip N/A

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

      5. lift-+.f64 N/A

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

      6. div-add N/A

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

      7. add-to-fraction N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 31.9%

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

   if -1.49999999999999987e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.5%

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

   4. Applied rewrites 81.5%

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

Alternative 8: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5e-8)
   (/ 1.0 (/ 3.0 (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) a)))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5e-8) {
		tmp = 1.0 / (3.0 / ((sqrt(fma((c * -3.0), a, (b * b))) - b) / a));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5e-8)
		tmp = Float64(1.0 / Float64(3.0 / Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / a)));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5e-8], N[(1.0 / N[(3.0 / N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.49999999999999987e-8

   1. Initial program 31.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-/.f64 31.1%

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

   3. Applied rewrites 31.1%

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

   if -1.49999999999999987e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.5%

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

   4. Applied rewrites 81.5%

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

Alternative 9: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-b, 1, \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5e-8)
   (/ (/ (fma (- b) 1.0 (sqrt (fma (* c -3.0) a (* b b)))) 3.0) a)
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5e-8) {
		tmp = (fma(-b, 1.0, sqrt(fma((c * -3.0), a, (b * b)))) / 3.0) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5e-8)
		tmp = Float64(Float64(fma(Float64(-b), 1.0, sqrt(fma(Float64(c * -3.0), a, Float64(b * b)))) / 3.0) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5e-8], N[(N[(N[((-b) * 1.0 + N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.49999999999999987e-8

   1. Initial program 31.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. add-to-fraction N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 31.1%

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

   if -1.49999999999999987e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.5%

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

   4. Applied rewrites 81.5%

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

Alternative 10: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5e-8)
   (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5e-8) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5e-8)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5e-8], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.49999999999999987e-8

   1. Initial program 31.1%

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

   3. Applied rewrites 31.1%

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

   if -1.49999999999999987e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.5%

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

   4. Applied rewrites 81.5%

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

Alternative 11: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5e-8)
   (* (- (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5e-8) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5e-8)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5e-8], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.49999999999999987e-8

   1. Initial program 31.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

   3. Applied rewrites 31.1%

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

   if -1.49999999999999987e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 31.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.5%

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

   4. Applied rewrites 81.5%

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

Alternative 12: 81.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 81.5%

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

4. Applied rewrites 81.5%

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

Reproduce

herbie shell --seed 2025189 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))