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

Percentage Accurate: 70.2% → 77.5%

Time: 8.7s

Alternatives: 13

Speedup: 1.2×

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: 70.2% 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: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := -0.3333333333333333 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 2:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos t\_2 \cdot \cos y - \sin t\_2 \cdot \sin y\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot -2 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))) (t_2 (* -0.3333333333333333 (* t z))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 2.0)
     (-
      (* (* 2.0 (sqrt x)) (- (* (cos t_2) (cos y)) (* (sin t_2) (sin y))))
      t_1)
     (- (* (* (* x (cos y)) (/ 1.0 (sqrt x))) -2.0) t_1))))

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 = -0.3333333333333333 * (t * z);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 2.0) {
		tmp = ((2.0 * sqrt(x)) * ((cos(t_2) * cos(y)) - (sin(t_2) * sin(y)))) - t_1;
	} else {
		tmp = (((x * cos(y)) * (1.0 / sqrt(x))) * -2.0) - t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = a / (b * 3.0d0)
    t_2 = (-0.3333333333333333d0) * (t * z)
    if (cos((y - ((z * t) / 3.0d0))) <= 2.0d0) then
        tmp = ((2.0d0 * sqrt(x)) * ((cos(t_2) * cos(y)) - (sin(t_2) * sin(y)))) - t_1
    else
        tmp = (((x * cos(y)) * (1.0d0 / sqrt(x))) * (-2.0d0)) - t_1
    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 t_2 = -0.3333333333333333 * (t * z);
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 2.0) {
		tmp = ((2.0 * Math.sqrt(x)) * ((Math.cos(t_2) * Math.cos(y)) - (Math.sin(t_2) * Math.sin(y)))) - t_1;
	} else {
		tmp = (((x * Math.cos(y)) * (1.0 / Math.sqrt(x))) * -2.0) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	t_2 = -0.3333333333333333 * (t * z)
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 2.0:
		tmp = ((2.0 * math.sqrt(x)) * ((math.cos(t_2) * math.cos(y)) - (math.sin(t_2) * math.sin(y)))) - t_1
	else:
		tmp = (((x * math.cos(y)) * (1.0 / math.sqrt(x))) * -2.0) - t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(-0.3333333333333333 * Float64(t * z))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 2.0)
		tmp = Float64(Float64(Float64(2.0 * sqrt(x)) * Float64(Float64(cos(t_2) * cos(y)) - Float64(sin(t_2) * sin(y)))) - t_1);
	else
		tmp = Float64(Float64(Float64(Float64(x * cos(y)) * Float64(1.0 / sqrt(x))) * -2.0) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (b * 3.0);
	t_2 = -0.3333333333333333 * (t * z);
	tmp = 0.0;
	if (cos((y - ((z * t) / 3.0))) <= 2.0)
		tmp = ((2.0 * sqrt(x)) * ((cos(t_2) * cos(y)) - (sin(t_2) * sin(y)))) - t_1;
	else
		tmp = (((x * cos(y)) * (1.0 / sqrt(x))) * -2.0) - t_1;
	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]}, Block[{t$95$2 = N[(-0.3333333333333333 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[t$95$2], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 2

   1. Initial program 70.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. cos-sum N/A

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

      13. metadata-eval N/A

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

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

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

      15. cos-neg N/A

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

      16. cos-neg-rev N/A

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

      17. mul-1-neg N/A

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

      18. lower--.f64 N/A

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

   3. Applied rewrites 70.7%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot \cos y - \sin \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)} - \frac{a}{b \cdot 3} \]

   if 2 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sqrt.f64 51.1

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

   7. Applied rewrites 51.1%

      \[\leadsto \left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{-2} - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 2:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos y, \cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333\right)\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot -2 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 2.0)
     (-
      (*
       (* 2.0 (sqrt x))
       (fma
        (cos y)
        (cos (* -0.3333333333333333 (* t z)))
        (* (sin y) (sin (* (* t z) 0.3333333333333333)))))
      t_1)
     (- (* (* (* x (cos y)) (/ 1.0 (sqrt x))) -2.0) t_1))))

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 (cos((y - ((z * t) / 3.0))) <= 2.0) {
		tmp = ((2.0 * sqrt(x)) * fma(cos(y), cos((-0.3333333333333333 * (t * z))), (sin(y) * sin(((t * z) * 0.3333333333333333))))) - t_1;
	} else {
		tmp = (((x * cos(y)) * (1.0 / sqrt(x))) * -2.0) - t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 2.0)
		tmp = Float64(Float64(Float64(2.0 * sqrt(x)) * fma(cos(y), cos(Float64(-0.3333333333333333 * Float64(t * z))), Float64(sin(y) * sin(Float64(Float64(t * z) * 0.3333333333333333))))) - t_1);
	else
		tmp = Float64(Float64(Float64(Float64(x * cos(y)) * Float64(1.0 / sqrt(x))) * -2.0) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * N[Cos[N[(-0.3333333333333333 * N[(t * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[Sin[N[(N[(t * z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 2

   1. Initial program 70.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. cos-diff N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. cos-neg N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 70.7%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\cos y, \cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333\right)\right)} - \frac{a}{b \cdot 3} \]

   if 2 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sqrt.f64 51.1

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

   7. Applied rewrites 51.1%

      \[\leadsto \left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{-2} - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - t\_1\\ \mathbf{if}\;t\_2 \leq 10^{+152}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot -2 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0)))
        (t_2 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) t_1)))
   (if (<= t_2 1e+152)
     t_2
     (- (* (* (* x (cos y)) (/ 1.0 (sqrt x))) -2.0) t_1))))

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 = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - t_1;
	double tmp;
	if (t_2 <= 1e+152) {
		tmp = t_2;
	} else {
		tmp = (((x * cos(y)) * (1.0 / sqrt(x))) * -2.0) - t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = a / (b * 3.0d0)
    t_2 = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - t_1
    if (t_2 <= 1d+152) then
        tmp = t_2
    else
        tmp = (((x * cos(y)) * (1.0d0 / sqrt(x))) * (-2.0d0)) - t_1
    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 t_2 = ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - t_1;
	double tmp;
	if (t_2 <= 1e+152) {
		tmp = t_2;
	} else {
		tmp = (((x * Math.cos(y)) * (1.0 / Math.sqrt(x))) * -2.0) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	t_2 = ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - t_1
	tmp = 0
	if t_2 <= 1e+152:
		tmp = t_2
	else:
		tmp = (((x * math.cos(y)) * (1.0 / math.sqrt(x))) * -2.0) - t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - t_1)
	tmp = 0.0
	if (t_2 <= 1e+152)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(Float64(x * cos(y)) * Float64(1.0 / sqrt(x))) * -2.0) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (b * 3.0);
	t_2 = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - t_1;
	tmp = 0.0;
	if (t_2 <= 1e+152)
		tmp = t_2;
	else
		tmp = (((x * cos(y)) * (1.0 / sqrt(x))) * -2.0) - t_1;
	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]}, Block[{t$95$2 = 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] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 1e+152], t$95$2, N[(N[(N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64)))) < 1e152

   1. Initial program 70.2%

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

   if 1e152 < (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64))))

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sqrt.f64 51.1

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

   7. Applied rewrites 51.1%

      \[\leadsto \left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{-2} - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\\ \mathbf{if}\;t\_2 - t\_1 \leq 10^{+152}:\\ \;\;\;\;t\_2 - a \cdot \frac{0.3333333333333333}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot -2 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0)))
        (t_2 (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))))
   (if (<= (- t_2 t_1) 1e+152)
     (- t_2 (* a (/ 0.3333333333333333 b)))
     (- (* (* (* x (cos y)) (/ 1.0 (sqrt x))) -2.0) t_1))))

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 = (2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)));
	double tmp;
	if ((t_2 - t_1) <= 1e+152) {
		tmp = t_2 - (a * (0.3333333333333333 / b));
	} else {
		tmp = (((x * cos(y)) * (1.0 / sqrt(x))) * -2.0) - t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = a / (b * 3.0d0)
    t_2 = (2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))
    if ((t_2 - t_1) <= 1d+152) then
        tmp = t_2 - (a * (0.3333333333333333d0 / b))
    else
        tmp = (((x * cos(y)) * (1.0d0 / sqrt(x))) * (-2.0d0)) - t_1
    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 t_2 = (2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)));
	double tmp;
	if ((t_2 - t_1) <= 1e+152) {
		tmp = t_2 - (a * (0.3333333333333333 / b));
	} else {
		tmp = (((x * Math.cos(y)) * (1.0 / Math.sqrt(x))) * -2.0) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	t_2 = (2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))
	tmp = 0
	if (t_2 - t_1) <= 1e+152:
		tmp = t_2 - (a * (0.3333333333333333 / b))
	else:
		tmp = (((x * math.cos(y)) * (1.0 / math.sqrt(x))) * -2.0) - t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0))))
	tmp = 0.0
	if (Float64(t_2 - t_1) <= 1e+152)
		tmp = Float64(t_2 - Float64(a * Float64(0.3333333333333333 / b)));
	else
		tmp = Float64(Float64(Float64(Float64(x * cos(y)) * Float64(1.0 / sqrt(x))) * -2.0) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (b * 3.0);
	t_2 = (2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)));
	tmp = 0.0;
	if ((t_2 - t_1) <= 1e+152)
		tmp = t_2 - (a * (0.3333333333333333 / b));
	else
		tmp = (((x * cos(y)) * (1.0 / sqrt(x))) * -2.0) - t_1;
	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]}, Block[{t$95$2 = N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - t$95$1), $MachinePrecision], 1e+152], N[(t$95$2 - N[(a * N[(0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64)))) < 1e152

   1. Initial program 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 70.2

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - a \cdot \color{blue}{\frac{0.3333333333333333}{b}} \]

   3. Applied rewrites 70.2%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \color{blue}{a \cdot \frac{0.3333333333333333}{b}} \]

   if 1e152 < (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64))))

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sqrt.f64 51.1

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

   7. Applied rewrites 51.1%

      \[\leadsto \left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{-2} - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - t\_1 \leq 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, \cos \left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, z, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot -2 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))))
   (if (<= (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) t_1) 1e+152)
     (fma
      -0.3333333333333333
      (/ a b)
      (* (cos (fma (* -0.3333333333333333 t) z y)) (* (sqrt x) 2.0)))
     (- (* (* (* x (cos y)) (/ 1.0 (sqrt x))) -2.0) t_1))))

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 ((((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - t_1) <= 1e+152) {
		tmp = fma(-0.3333333333333333, (a / b), (cos(fma((-0.3333333333333333 * t), z, y)) * (sqrt(x) * 2.0)));
	} else {
		tmp = (((x * cos(y)) * (1.0 / sqrt(x))) * -2.0) - t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - t_1) <= 1e+152)
		tmp = fma(-0.3333333333333333, Float64(a / b), Float64(cos(fma(Float64(-0.3333333333333333 * t), z, y)) * Float64(sqrt(x) * 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(x * cos(y)) * Float64(1.0 / sqrt(x))) * -2.0) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] - t$95$1), $MachinePrecision], 1e+152], N[(-0.3333333333333333 * N[(a / b), $MachinePrecision] + N[(N[Cos[N[(N[(-0.3333333333333333 * t), $MachinePrecision] * z + y), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64)))) < 1e152

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. mult-flip N/A

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

   4. Applied rewrites 70.2%

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

   if 1e152 < (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64))))

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sqrt.f64 51.1

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

   7. Applied rewrites 51.1%

      \[\leadsto \left(\left(x \cdot \cos y\right) \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{-2} - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

2. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lift-sqrt.f64 76.5

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

4. Applied rewrites 76.5%

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

Alternative 7: 76.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

2. Taylor expanded in z around 0

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

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

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 76.4

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

4. Applied rewrites 76.4%

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

Alternative 8: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-36}:\\ \;\;\;\;t\_2 - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, z, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))) (t_2 (* 2.0 (sqrt x))))
   (if (<= t_1 -1e-36)
     (- t_2 (/ (/ a b) 3.0))
     (if (<= t_1 2e-95)
       (* (cos (fma (* -0.3333333333333333 t) z y)) (* (sqrt x) 2.0))
       (- t_2 t_1)))))

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 = 2.0 * sqrt(x);
	double tmp;
	if (t_1 <= -1e-36) {
		tmp = t_2 - ((a / b) / 3.0);
	} else if (t_1 <= 2e-95) {
		tmp = cos(fma((-0.3333333333333333 * t), z, y)) * (sqrt(x) * 2.0);
	} else {
		tmp = t_2 - t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (t_1 <= -1e-36)
		tmp = Float64(t_2 - Float64(Float64(a / b) / 3.0));
	elseif (t_1 <= 2e-95)
		tmp = Float64(cos(fma(Float64(-0.3333333333333333 * t), z, y)) * Float64(sqrt(x) * 2.0));
	else
		tmp = Float64(t_2 - t_1);
	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[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-36], N[(t$95$2 - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-95], N[(N[Cos[N[(N[(-0.3333333333333333 * t), $MachinePrecision] * z + y), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -9.9999999999999994e-37

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 76.5

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x}} - \frac{\frac{a}{b}}{3} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.3%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x}} - \frac{\frac{a}{b}}{3} \]

   if -9.9999999999999994e-37 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 1.99999999999999998e-95

   1. Initial program 70.2%

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

   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 49.7

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

   4. Applied rewrites 49.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 49.7

         \[\leadsto -0.3333333333333333 \cdot \left(a \cdot \frac{1}{\color{blue}{b}}\right) \]

   6. Applied rewrites 49.7%

      \[\leadsto -0.3333333333333333 \cdot \left(a \cdot \color{blue}{\frac{1}{b}}\right) \]

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 28.8%

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

   if 1.99999999999999998e-95 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x}} - \frac{a}{b \cdot 3} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.3%

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

Alternative 9: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-36}:\\ \;\;\;\;t\_2 - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\left(2 \cdot \cos y\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))) (t_2 (* 2.0 (sqrt x))))
   (if (<= t_1 -1e-36)
     (- t_2 (/ (/ a b) 3.0))
     (if (<= t_1 2e-95) (* (* 2.0 (cos y)) (sqrt x)) (- t_2 t_1)))))

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 = 2.0 * sqrt(x);
	double tmp;
	if (t_1 <= -1e-36) {
		tmp = t_2 - ((a / b) / 3.0);
	} else if (t_1 <= 2e-95) {
		tmp = (2.0 * cos(y)) * sqrt(x);
	} else {
		tmp = t_2 - t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = a / (b * 3.0d0)
    t_2 = 2.0d0 * sqrt(x)
    if (t_1 <= (-1d-36)) then
        tmp = t_2 - ((a / b) / 3.0d0)
    else if (t_1 <= 2d-95) then
        tmp = (2.0d0 * cos(y)) * sqrt(x)
    else
        tmp = t_2 - t_1
    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 t_2 = 2.0 * Math.sqrt(x);
	double tmp;
	if (t_1 <= -1e-36) {
		tmp = t_2 - ((a / b) / 3.0);
	} else if (t_1 <= 2e-95) {
		tmp = (2.0 * Math.cos(y)) * Math.sqrt(x);
	} else {
		tmp = t_2 - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	t_2 = 2.0 * math.sqrt(x)
	tmp = 0
	if t_1 <= -1e-36:
		tmp = t_2 - ((a / b) / 3.0)
	elif t_1 <= 2e-95:
		tmp = (2.0 * math.cos(y)) * math.sqrt(x)
	else:
		tmp = t_2 - t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (t_1 <= -1e-36)
		tmp = Float64(t_2 - Float64(Float64(a / b) / 3.0));
	elseif (t_1 <= 2e-95)
		tmp = Float64(Float64(2.0 * cos(y)) * sqrt(x));
	else
		tmp = Float64(t_2 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (b * 3.0);
	t_2 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (t_1 <= -1e-36)
		tmp = t_2 - ((a / b) / 3.0);
	elseif (t_1 <= 2e-95)
		tmp = (2.0 * cos(y)) * sqrt(x);
	else
		tmp = t_2 - t_1;
	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]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-36], N[(t$95$2 - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-95], N[(N[(2.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(t$95$2 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -9.9999999999999994e-37

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 76.5

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x}} - \frac{\frac{a}{b}}{3} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.3%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x}} - \frac{\frac{a}{b}}{3} \]

   if -9.9999999999999994e-37 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 1.99999999999999998e-95

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 76.5

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in z around 0

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

      1. associate-/l/ N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2 N/A

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 76.4

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

   9. Applied rewrites 76.4%

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

   10. Taylor expanded in a around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-sqrt.f64 29.2

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

   12. Applied rewrites 29.2%

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

   if 1.99999999999999998e-95 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lift-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x}} - \frac{a}{b \cdot 3} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.3%

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

Alternative 10: 65.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} - \frac{a}{b \cdot 3} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

2. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lift-sqrt.f64 76.5

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in y around 0

   \[\leadsto 2 \cdot \sqrt{\color{blue}{x}} - \frac{a}{b \cdot 3} \]
6. Step-by-step derivation

7. Applied rewrites 65.3%

   \[\leadsto 2 \cdot \sqrt{\color{blue}{x}} - \frac{a}{b \cdot 3} \]
8. Add Preprocessing

Alternative 11: 65.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

2. Taylor expanded in z around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

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

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

4. Applied rewrites 66.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\sin y \cdot z\right) \cdot \sqrt{x}\right) \cdot t, 0.6666666666666666, \mathsf{fma}\left(2 \cdot \cos y, \sqrt{x}, -0.3333333333333333 \cdot \frac{a}{b}\right)\right)} \]

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{a}{b} \cdot \frac{-1}{3} + 2 \cdot \sqrt{\color{blue}{x}} \]

   2. lower-fma.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 65.3

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

7. Applied rewrites 65.3%

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

Alternative 12: 49.7% accurate, 8.0× 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 70.2%

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

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 49.7

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

4. Applied rewrites 49.7%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
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 49.7

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

6. Applied rewrites 49.7%

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

Alternative 13: 49.7% accurate, 8.0× 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 70.2%

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

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 49.7

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

4. Applied rewrites 49.7%

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

Reproduce

herbie shell --seed 2025136 
(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))))