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

Percentage Accurate: 70.0% → 76.4%

Time: 11.8s

Alternatives: 8

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: 70.0% 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: 76.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

2. Taylor expanded in y around inf

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

4. Applied rewrites 76.4%

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

Alternative 2: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{x} \cdot 2\\ t_2 := \frac{a}{b \cdot 3}\\ t_3 := \mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, t\_1\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-79}:\\ \;\;\;\;\cos y \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (sqrt x) 2.0))
        (t_2 (/ a (* b 3.0)))
        (t_3 (fma (/ a b) -0.3333333333333333 t_1)))
   (if (<= t_2 -2e-8) t_3 (if (<= t_2 5e-79) (* (cos y) t_1) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = sqrt(x) * 2.0;
	double t_2 = a / (b * 3.0);
	double t_3 = fma((a / b), -0.3333333333333333, t_1);
	double tmp;
	if (t_2 <= -2e-8) {
		tmp = t_3;
	} else if (t_2 <= 5e-79) {
		tmp = cos(y) * t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(sqrt(x) * 2.0)
	t_2 = Float64(a / Float64(b * 3.0))
	t_3 = fma(Float64(a / b), -0.3333333333333333, t_1)
	tmp = 0.0
	if (t_2 <= -2e-8)
		tmp = t_3;
	elseif (t_2 <= 5e-79)
		tmp = Float64(cos(y) * t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a / b), $MachinePrecision] * -0.3333333333333333 + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-8], t$95$3, If[LessEqual[t$95$2, 5e-79], N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2e-8 or 4.99999999999999999e-79 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 81.1%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{-1}{3} \cdot \frac{a}{b} + \color{blue}{2 \cdot \sqrt{x}} \]
   9. 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. lift-sqrt.f64 N/A

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

      6. lift-*.f64 87.7

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

   10. Applied rewrites 87.7%

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

   if -2e-8 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4.99999999999999999e-79

   1. Initial program 55.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 57.1%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-*.f64 50.4

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

   10. Applied rewrites 50.4%

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

Alternative 3: 76.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

2. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. metadata-eval N/A

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

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

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 69.9

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

4. Applied rewrites 69.9%

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

5. Taylor expanded in y around inf

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

7. Applied rewrites 76.3%

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

Alternative 4: 59.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot a}{b}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))))
   (if (<= t_1 -5e-64)
     (/ (* -0.3333333333333333 a) b)
     (if (<= t_1 5e-38) (* (sqrt x) 2.0) (* (/ -0.3333333333333333 b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double tmp;
	if (t_1 <= -5e-64) {
		tmp = (-0.3333333333333333 * a) / b;
	} else if (t_1 <= 5e-38) {
		tmp = sqrt(x) * 2.0;
	} else {
		tmp = (-0.3333333333333333 / b) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (b * 3.0d0)
    if (t_1 <= (-5d-64)) then
        tmp = ((-0.3333333333333333d0) * a) / b
    else if (t_1 <= 5d-38) then
        tmp = sqrt(x) * 2.0d0
    else
        tmp = ((-0.3333333333333333d0) / b) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double tmp;
	if (t_1 <= -5e-64) {
		tmp = (-0.3333333333333333 * a) / b;
	} else if (t_1 <= 5e-38) {
		tmp = Math.sqrt(x) * 2.0;
	} else {
		tmp = (-0.3333333333333333 / b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	tmp = 0
	if t_1 <= -5e-64:
		tmp = (-0.3333333333333333 * a) / b
	elif t_1 <= 5e-38:
		tmp = math.sqrt(x) * 2.0
	else:
		tmp = (-0.3333333333333333 / b) * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	tmp = 0.0
	if (t_1 <= -5e-64)
		tmp = Float64(Float64(-0.3333333333333333 * a) / b);
	elseif (t_1 <= 5e-38)
		tmp = Float64(sqrt(x) * 2.0);
	else
		tmp = Float64(Float64(-0.3333333333333333 / b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (b * 3.0);
	tmp = 0.0;
	if (t_1 <= -5e-64)
		tmp = (-0.3333333333333333 * a) / b;
	elseif (t_1 <= 5e-38)
		tmp = sqrt(x) * 2.0;
	else
		tmp = (-0.3333333333333333 / b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-64], N[(N[(-0.3333333333333333 * a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 5e-38], N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(-0.3333333333333333 / b), $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -5.00000000000000033e-64

   1. Initial program 80.4%

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

   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 80.3

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

   4. Applied rewrites 80.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
   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 80.4

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

   6. Applied rewrites 80.4%

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

   if -5.00000000000000033e-64 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 5.00000000000000033e-38

   1. Initial program 55.3%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 56.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{-1}{3} \cdot \frac{a}{b} + \color{blue}{2 \cdot \sqrt{x}} \]
   9. 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. lift-sqrt.f64 N/A

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

      6. lift-*.f64 36.1

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in x around inf

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot 2 \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{x} \cdot 2 \]

       3. lift-*.f64 30.8

          \[\leadsto \sqrt{x} \cdot 2 \]

   13. Applied rewrites 30.8%

       \[\leadsto \sqrt{x} \cdot 2 \]

   if 5.00000000000000033e-38 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 81.8%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 81.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-/.f64 83.2

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

   10. Applied rewrites 83.2%

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

Alternative 5: 59.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \frac{-0.3333333333333333 \cdot a}{b}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))) (t_2 (/ (* -0.3333333333333333 a) b)))
   (if (<= t_1 -5e-64) t_2 (if (<= t_1 5e-38) (* (sqrt x) 2.0) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double t_2 = (-0.3333333333333333 * a) / b;
	double tmp;
	if (t_1 <= -5e-64) {
		tmp = t_2;
	} else if (t_1 <= 5e-38) {
		tmp = sqrt(x) * 2.0;
	} else {
		tmp = t_2;
	}
	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) * a) / b
    if (t_1 <= (-5d-64)) then
        tmp = t_2
    else if (t_1 <= 5d-38) then
        tmp = sqrt(x) * 2.0d0
    else
        tmp = t_2
    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 * a) / b;
	double tmp;
	if (t_1 <= -5e-64) {
		tmp = t_2;
	} else if (t_1 <= 5e-38) {
		tmp = Math.sqrt(x) * 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	t_2 = (-0.3333333333333333 * a) / b
	tmp = 0
	if t_1 <= -5e-64:
		tmp = t_2
	elif t_1 <= 5e-38:
		tmp = math.sqrt(x) * 2.0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(Float64(-0.3333333333333333 * a) / b)
	tmp = 0.0
	if (t_1 <= -5e-64)
		tmp = t_2;
	elseif (t_1 <= 5e-38)
		tmp = Float64(sqrt(x) * 2.0);
	else
		tmp = t_2;
	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 * a) / b;
	tmp = 0.0;
	if (t_1 <= -5e-64)
		tmp = t_2;
	elseif (t_1 <= 5e-38)
		tmp = sqrt(x) * 2.0;
	else
		tmp = t_2;
	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[(-0.3333333333333333 * a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-64], t$95$2, If[LessEqual[t$95$1, 5e-38], N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -5.00000000000000033e-64 or 5.00000000000000033e-38 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 81.1%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.6

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

   4. Applied rewrites 81.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
   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 81.7

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

   6. Applied rewrites 81.7%

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

   if -5.00000000000000033e-64 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 5.00000000000000033e-38

   1. Initial program 55.3%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 56.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{-1}{3} \cdot \frac{a}{b} + \color{blue}{2 \cdot \sqrt{x}} \]
   9. 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. lift-sqrt.f64 N/A

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

      6. lift-*.f64 36.1

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in x around inf

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot 2 \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{x} \cdot 2 \]

       3. lift-*.f64 30.8

          \[\leadsto \sqrt{x} \cdot 2 \]

   13. Applied rewrites 30.8%

       \[\leadsto \sqrt{x} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 59.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := -0.3333333333333333 \cdot \frac{a}{b}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))) (t_2 (* -0.3333333333333333 (/ a b))))
   (if (<= t_1 -5e-64) t_2 (if (<= t_1 5e-38) (* (sqrt x) 2.0) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double t_2 = -0.3333333333333333 * (a / b);
	double tmp;
	if (t_1 <= -5e-64) {
		tmp = t_2;
	} else if (t_1 <= 5e-38) {
		tmp = sqrt(x) * 2.0;
	} else {
		tmp = t_2;
	}
	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) * (a / b)
    if (t_1 <= (-5d-64)) then
        tmp = t_2
    else if (t_1 <= 5d-38) then
        tmp = sqrt(x) * 2.0d0
    else
        tmp = t_2
    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 * (a / b);
	double tmp;
	if (t_1 <= -5e-64) {
		tmp = t_2;
	} else if (t_1 <= 5e-38) {
		tmp = Math.sqrt(x) * 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	t_2 = -0.3333333333333333 * (a / b)
	tmp = 0
	if t_1 <= -5e-64:
		tmp = t_2
	elif t_1 <= 5e-38:
		tmp = math.sqrt(x) * 2.0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(-0.3333333333333333 * Float64(a / b))
	tmp = 0.0
	if (t_1 <= -5e-64)
		tmp = t_2;
	elseif (t_1 <= 5e-38)
		tmp = Float64(sqrt(x) * 2.0);
	else
		tmp = t_2;
	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 * (a / b);
	tmp = 0.0;
	if (t_1 <= -5e-64)
		tmp = t_2;
	elseif (t_1 <= 5e-38)
		tmp = sqrt(x) * 2.0;
	else
		tmp = t_2;
	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[(a / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-64], t$95$2, If[LessEqual[t$95$1, 5e-38], N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -5.00000000000000033e-64 or 5.00000000000000033e-38 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 81.1%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.6

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

   4. Applied rewrites 81.6%

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

   if -5.00000000000000033e-64 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 5.00000000000000033e-38

   1. Initial program 55.3%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 56.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{-1}{3} \cdot \frac{a}{b} + \color{blue}{2 \cdot \sqrt{x}} \]
   9. 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. lift-sqrt.f64 N/A

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

      6. lift-*.f64 36.1

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in x around inf

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot 2 \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{x} \cdot 2 \]

       3. lift-*.f64 30.8

          \[\leadsto \sqrt{x} \cdot 2 \]

   13. Applied rewrites 30.8%

       \[\leadsto \sqrt{x} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.2% accurate, 4.8× 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 70.0%

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

2. Taylor expanded in y around inf

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 58.7%

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

8. Taylor expanded in z around 0

   \[\leadsto \frac{-1}{3} \cdot \frac{a}{b} + \color{blue}{2 \cdot \sqrt{x}} \]
9. 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. lift-sqrt.f64 N/A

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

   6. lift-*.f64 65.2

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

10. Applied rewrites 65.2%

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

Alternative 8: 18.0% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot 2 \end{array} \]

FPCore

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

C

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

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 = sqrt(x) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

2. Taylor expanded in y around inf

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 58.7%

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

8. Taylor expanded in z around 0

   \[\leadsto \frac{-1}{3} \cdot \frac{a}{b} + \color{blue}{2 \cdot \sqrt{x}} \]
9. 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. lift-sqrt.f64 N/A

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

   6. lift-*.f64 65.2

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

10. Applied rewrites 65.2%

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

11. Taylor expanded in x around inf

    \[\leadsto 2 \cdot \sqrt{x} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \sqrt{x} \cdot 2 \]

    2. lift-sqrt.f64 N/A

       \[\leadsto \sqrt{x} \cdot 2 \]

    3. lift-*.f64 18.0

       \[\leadsto \sqrt{x} \cdot 2 \]

13. Applied rewrites 18.0%

    \[\leadsto \sqrt{x} \cdot 2 \]
14. Add Preprocessing

Developer Target 1: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.3333333333333333}{z}}{t}\\ t_2 := \frac{\frac{a}{3}}{b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;t\_3 \cdot \cos \left(\frac{1}{y} - t\_1\right) - t\_2\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - t\_1\right) \cdot t\_3 - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ 0.3333333333333333 z) t))
        (t_2 (/ (/ a 3.0) b))
        (t_3 (* 2.0 (sqrt x))))
   (if (< z -1.3793337487235141e+129)
     (- (* t_3 (cos (- (/ 1.0 y) t_1))) t_2)
     (if (< z 3.516290613555987e+106)
       (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) t_2)
       (- (* (cos (- y t_1)) t_3) (/ (/ a b) 3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = (0.3333333333333333d0 / z) / t
    t_2 = (a / 3.0d0) / b
    t_3 = 2.0d0 * sqrt(x)
    if (z < (-1.3793337487235141d+129)) then
        tmp = (t_3 * cos(((1.0d0 / y) - t_1))) - t_2
    else if (z < 3.516290613555987d+106) then
        tmp = ((sqrt(x) * 2.0d0) * cos((y - ((t / 3.0d0) * z)))) - t_2
    else
        tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0d0)
    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 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * Math.sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * Math.cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((Math.sqrt(x) * 2.0) * Math.cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (Math.cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (0.3333333333333333 / z) / t
	t_2 = (a / 3.0) / b
	t_3 = 2.0 * math.sqrt(x)
	tmp = 0
	if z < -1.3793337487235141e+129:
		tmp = (t_3 * math.cos(((1.0 / y) - t_1))) - t_2
	elif z < 3.516290613555987e+106:
		tmp = ((math.sqrt(x) * 2.0) * math.cos((y - ((t / 3.0) * z)))) - t_2
	else:
		tmp = (math.cos((y - t_1)) * t_3) - ((a / b) / 3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(0.3333333333333333 / z) / t)
	t_2 = Float64(Float64(a / 3.0) / b)
	t_3 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (z < -1.3793337487235141e+129)
		tmp = Float64(Float64(t_3 * cos(Float64(Float64(1.0 / y) - t_1))) - t_2);
	elseif (z < 3.516290613555987e+106)
		tmp = Float64(Float64(Float64(sqrt(x) * 2.0) * cos(Float64(y - Float64(Float64(t / 3.0) * z)))) - t_2);
	else
		tmp = Float64(Float64(cos(Float64(y - t_1)) * t_3) - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (0.3333333333333333 / z) / t;
	t_2 = (a / 3.0) / b;
	t_3 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (z < -1.3793337487235141e+129)
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	elseif (z < 3.516290613555987e+106)
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	else
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.3793337487235141e+129], N[(N[(t$95$3 * N[Cos[N[(N[(1.0 / y), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[z, 3.516290613555987e+106], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(y - N[(N[(t / 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[Cos[N[(y - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2025096 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -1379333748723514100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 3333333333333333/10000000000000000 z) t)))) (/ (/ a 3) b)) (if (< z 35162906135559870000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 3333333333333333/10000000000000000 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3)))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))