Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, K

Percentage Accurate: 70.8% → 77.0%

Time: 9.0s

Alternatives: 8

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 77.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\frac{a}{b}}{3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos y)) (/ (/ a b) 3.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos(y)) - ((a / b) / 3.0);
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos(y)) - ((a / b) / 3.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos(y)) - ((a / b) / 3.0);
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos(y)) - ((a / b) / 3.0)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(y)) - Float64(Float64(a / b) / 3.0))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos(y)) - ((a / b) / 3.0);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

2. Taylor expanded in y around inf

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
3. Step-by-step derivation

4. Applied rewrites 77.0%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{\color{blue}{b \cdot 3}} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{a}{b \cdot 3}} \]

   3. associate-/r* N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

   5. lift-/.f64 77.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\color{blue}{\frac{a}{b}}}{3} \]

6. Applied rewrites 77.0%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]
7. Add Preprocessing

Alternative 2: 77.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos y)) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos(y)) - (a / (b * 3.0));
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos(y)) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos(y)) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos(y)) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(y)) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos(y)) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

2. Taylor expanded in y around inf

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
3. Step-by-step derivation

4. Applied rewrites 77.0%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{b} \cdot -0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (* (sqrt x) 2.0) (cos y) (* (/ a b) -0.3333333333333333)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((sqrt(x) * 2.0), cos(y), ((a / b) * -0.3333333333333333));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(sqrt(x) * 2.0), cos(y), Float64(Float64(a / b) * -0.3333333333333333))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

2. Taylor expanded in y around inf

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
3. Step-by-step derivation

4. Applied rewrites 77.0%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

   3. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

   4. cos-diff-rev N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

   6. lift-/.f64 N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

   8. lift-/.f64 N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

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

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

7. Applied rewrites 76.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \left(\cos y \cdot \sqrt{x}\right) \cdot 2 + \color{blue}{\frac{a}{b} \cdot \frac{-1}{3}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\cos y \cdot \sqrt{x}\right) \cdot 2 + \frac{\color{blue}{a}}{b} \cdot \frac{-1}{3} \]

   3. lift-cos.f64 N/A

      \[\leadsto \left(\cos y \cdot \sqrt{x}\right) \cdot 2 + \frac{a}{b} \cdot \frac{-1}{3} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\cos y \cdot \sqrt{x}\right) \cdot 2 + \frac{a}{b} \cdot \frac{-1}{3} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\cos y \cdot \sqrt{x}\right) \cdot 2 + \frac{a}{b} \cdot \color{blue}{\frac{-1}{3}} \]

   6. lift-/.f64 N/A

      \[\leadsto \left(\cos y \cdot \sqrt{x}\right) \cdot 2 + \frac{a}{b} \cdot \frac{-1}{3} \]

   7. *-commutative N/A

      \[\leadsto \left(\cos y \cdot \sqrt{x}\right) \cdot 2 + \frac{-1}{3} \cdot \color{blue}{\frac{a}{b}} \]

   8. *-commutative N/A

      \[\leadsto 2 \cdot \left(\cos y \cdot \sqrt{x}\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   9. *-commutative N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \frac{-1}{3} \cdot \frac{a}{b} \]

   10. associate-*r* N/A

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \color{blue}{\cos y}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos \color{blue}{y}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   13. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos \color{blue}{y}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   15. lift-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{b} \cdot \frac{-1}{3}\right) \]

   17. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{b} \cdot \frac{-1}{3}\right) \]

   18. lift-*.f64 76.9

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{b} \cdot -0.3333333333333333\right) \]

9. Applied rewrites 76.9%

   \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \color{blue}{\cos y}, \frac{a}{b} \cdot -0.3333333333333333\right) \]
10. Add Preprocessing

Alternative 4: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \frac{a}{b}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (* (cos y) (sqrt x)) 2.0 (* -0.3333333333333333 (/ a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((cos(y) * sqrt(x)), 2.0, (-0.3333333333333333 * (a / b)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(cos(y) * sqrt(x)), 2.0, Float64(-0.3333333333333333 * Float64(a / b)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Cos[y], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
3. Step-by-step derivation

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

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

   2. *-commutative N/A

      \[\leadsto \left(\sqrt{x} \cdot \cos y\right) \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot \frac{a}{b} \]

   3. metadata-eval N/A

      \[\leadsto \left(\sqrt{x} \cdot \cos y\right) \cdot 2 + \frac{-1}{3} \cdot \frac{\color{blue}{a}}{b} \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot \cos y, \color{blue}{2}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   7. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   10. lower-/.f64 76.9

       \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \frac{a}{b}\right) \]

4. Applied rewrites 76.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \frac{a}{b}\right)} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-51}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0)))
        (t_2 (fma (sqrt x) 2.0 (* (/ a b) -0.3333333333333333))))
   (if (<= t_1 -5e-105)
     t_2
     (if (<= t_1 4e-51) (* (* (sqrt x) 2.0) (cos y)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double t_2 = fma(sqrt(x), 2.0, ((a / b) * -0.3333333333333333));
	double tmp;
	if (t_1 <= -5e-105) {
		tmp = t_2;
	} else if (t_1 <= 4e-51) {
		tmp = (sqrt(x) * 2.0) * cos(y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = fma(sqrt(x), 2.0, Float64(Float64(a / b) * -0.3333333333333333))
	tmp = 0.0
	if (t_1 <= -5e-105)
		tmp = t_2;
	elseif (t_1 <= 4e-51)
		tmp = Float64(Float64(sqrt(x) * 2.0) * cos(y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * 2.0 + N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-105], t$95$2, If[LessEqual[t$95$1, 4e-51], N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -4.99999999999999963e-105 or 4e-51 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 79.5%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

   2. Taylor expanded in y around inf

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
   3. Step-by-step derivation

   4. Applied rewrites 89.0%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

      3. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

      4. cos-diff-rev N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

      6. lift-/.f64 N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

      8. lift-/.f64 N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b} \]

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

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

   7. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)} \]

   8. Taylor expanded in y around 0

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

      1. lift-sqrt.f64 83.7

         \[\leadsto \mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right) \]

   10. Applied rewrites 83.7%

       \[\leadsto \mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right) \]

   if -4.99999999999999963e-105 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4e-51

   1. Initial program 56.0%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

   2. Taylor expanded in y around inf

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
   3. Step-by-step derivation

   4. Applied rewrites 56.6%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{\color{blue}{b \cdot 3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{a}{b \cdot 3}} \]

      3. associate-/r* N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

      5. lift-/.f64 56.6

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\color{blue}{\frac{a}{b}}}{3} \]

   6. Applied rewrites 56.6%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) \]

      2. cos-diff-rev N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \color{blue}{\left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \color{blue}{\left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\color{blue}{y} - \frac{1}{3} \cdot \left(t \cdot z\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \left(t \cdot z\right)\right) \]

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

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y + \frac{-1}{3} \cdot \left(t \cdot z\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\frac{-1}{3} \cdot \left(t \cdot z\right) + y\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\left(t \cdot z\right) \cdot \frac{-1}{3} + y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(t \cdot z, \frac{-1}{3}, y\right)\right) \]

      15. lower-*.f64 53.0

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(t \cdot z, -0.3333333333333333, y\right)\right) \]

   9. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(t \cdot z, -0.3333333333333333, y\right)\right)} \]

   10. Taylor expanded in y around inf

       \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos y \]
   11. Step-by-step derivation

   12. Applied rewrites 53.4%

       \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot a}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))))
   (if (<= t_1 -2e-44)
     (/ (* -0.3333333333333333 a) b)
     (if (<= t_1 2e-47)
       (* (* (sqrt x) 2.0) (cos y))
       (* -0.3333333333333333 (/ a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double tmp;
	if (t_1 <= -2e-44) {
		tmp = (-0.3333333333333333 * a) / b;
	} else if (t_1 <= 2e-47) {
		tmp = (sqrt(x) * 2.0) * cos(y);
	} else {
		tmp = -0.3333333333333333 * (a / b);
	}
	return tmp;
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (b * 3.0d0)
    if (t_1 <= (-2d-44)) then
        tmp = ((-0.3333333333333333d0) * a) / b
    else if (t_1 <= 2d-47) then
        tmp = (sqrt(x) * 2.0d0) * cos(y)
    else
        tmp = (-0.3333333333333333d0) * (a / b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double tmp;
	if (t_1 <= -2e-44) {
		tmp = (-0.3333333333333333 * a) / b;
	} else if (t_1 <= 2e-47) {
		tmp = (Math.sqrt(x) * 2.0) * Math.cos(y);
	} else {
		tmp = -0.3333333333333333 * (a / b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	tmp = 0
	if t_1 <= -2e-44:
		tmp = (-0.3333333333333333 * a) / b
	elif t_1 <= 2e-47:
		tmp = (math.sqrt(x) * 2.0) * math.cos(y)
	else:
		tmp = -0.3333333333333333 * (a / b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	tmp = 0.0
	if (t_1 <= -2e-44)
		tmp = Float64(Float64(-0.3333333333333333 * a) / b);
	elseif (t_1 <= 2e-47)
		tmp = Float64(Float64(sqrt(x) * 2.0) * cos(y));
	else
		tmp = Float64(-0.3333333333333333 * Float64(a / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (b * 3.0);
	tmp = 0.0;
	if (t_1 <= -2e-44)
		tmp = (-0.3333333333333333 * a) / b;
	elseif (t_1 <= 2e-47)
		tmp = (sqrt(x) * 2.0) * cos(y);
	else
		tmp = -0.3333333333333333 * (a / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-44], N[(N[(-0.3333333333333333 * a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e-47], N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.99999999999999991e-44

   1. Initial program 81.3%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{a}{b}} \]

      2. lower-/.f64 83.0

         \[\leadsto -0.3333333333333333 \cdot \frac{a}{\color{blue}{b}} \]

   4. Applied rewrites 83.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{a}{b}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{a}{\color{blue}{b}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot a}{\color{blue}{b}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot a}{\color{blue}{b}} \]

      5. lower-*.f64 83.0

         \[\leadsto \frac{-0.3333333333333333 \cdot a}{b} \]

   6. Applied rewrites 83.0%

      \[\leadsto \frac{-0.3333333333333333 \cdot a}{\color{blue}{b}} \]

   if -1.99999999999999991e-44 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 1.9999999999999999e-47

   1. Initial program 56.8%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

   2. Taylor expanded in y around inf

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
   3. Step-by-step derivation

   4. Applied rewrites 57.5%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{y} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{\color{blue}{b \cdot 3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{a}{b \cdot 3}} \]

      3. associate-/r* N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

      5. lift-/.f64 57.6

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\color{blue}{\frac{a}{b}}}{3} \]

   6. Applied rewrites 57.6%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) \]

      2. cos-diff-rev N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \color{blue}{\left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \color{blue}{\left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\color{blue}{y} - \frac{1}{3} \cdot \left(t \cdot z\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \left(t \cdot z\right)\right) \]

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

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y + \frac{-1}{3} \cdot \left(t \cdot z\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\frac{-1}{3} \cdot \left(t \cdot z\right) + y\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\left(t \cdot z\right) \cdot \frac{-1}{3} + y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(t \cdot z, \frac{-1}{3}, y\right)\right) \]

      15. lower-*.f64 51.7

          \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(t \cdot z, -0.3333333333333333, y\right)\right) \]

   9. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(t \cdot z, -0.3333333333333333, y\right)\right)} \]

   10. Taylor expanded in y around inf

       \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos y \]
   11. Step-by-step derivation

   12. Applied rewrites 52.1%

       \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \cos y \]

   if 1.9999999999999999e-47 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 80.7%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{a}{b}} \]

      2. lower-/.f64 81.0

         \[\leadsto -0.3333333333333333 \cdot \frac{a}{\color{blue}{b}} \]

   4. Applied rewrites 81.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 50.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333 \cdot a}{b} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ (* -0.3333333333333333 a) b))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (-0.3333333333333333 * a) / 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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((-0.3333333333333333d0) * a) / b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (-0.3333333333333333 * a) / b;
}

Python

def code(x, y, z, t, a, b):
	return (-0.3333333333333333 * a) / b

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(-0.3333333333333333 * a) / b)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (-0.3333333333333333 * a) / b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(-0.3333333333333333 * a), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{a}{b}} \]

   2. lower-/.f64 50.9

      \[\leadsto -0.3333333333333333 \cdot \frac{a}{\color{blue}{b}} \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{a}{b}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{a}{\color{blue}{b}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot a}{\color{blue}{b}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot a}{\color{blue}{b}} \]

   5. lower-*.f64 50.9

      \[\leadsto \frac{-0.3333333333333333 \cdot a}{b} \]

6. Applied rewrites 50.9%

   \[\leadsto \frac{-0.3333333333333333 \cdot a}{\color{blue}{b}} \]
7. Add Preprocessing

Alternative 8: 50.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{a}{b} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* -0.3333333333333333 (/ a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return -0.3333333333333333 * (a / 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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-0.3333333333333333d0) * (a / b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return -0.3333333333333333 * (a / b);
}

Python

def code(x, y, z, t, a, b):
	return -0.3333333333333333 * (a / b)

Julia

function code(x, y, z, t, a, b)
	return Float64(-0.3333333333333333 * Float64(a / b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = -0.3333333333333333 * (a / b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{a}{b}} \]

   2. lower-/.f64 50.9

      \[\leadsto -0.3333333333333333 \cdot \frac{a}{\color{blue}{b}} \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64
  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))