VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 2.7s

Alternatives: 12

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
end function

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

function code(B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
end

Wolfram

code[B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
end function

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

function code(B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
end

Wolfram

code[B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{\tan B} + \frac{1}{\sin B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (+ (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	return (-x / tan(B)) + (1.0 / sin(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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-x / tan(b)) + (1.0d0 / sin(b))
end function

Java

public static double code(double B, double x) {
	return (-x / Math.tan(B)) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return (-x / math.tan(B)) + (1.0 / math.sin(B))

Julia

function code(B, x)
	return Float64(Float64(Float64(-x) / tan(B)) + Float64(1.0 / sin(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = (-x / tan(B)) + (1.0 / sin(B));
end

Wolfram

code[B_, x_] := N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

   3. lift-tan.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{1}{\sin B} \]

   4. associate-*r/ N/A

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B} \]

   5. lower-/.f64 N/A

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{1}{\sin B} \]

   7. lift-tan.f64 99.8

      \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{1}{\sin B} \]

3. Applied rewrites 99.8%

   \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B} \]
4. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot 1}{\tan B}\right)\right)} + \frac{1}{\sin B} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot 1}}{\tan B}\right)\right) + \frac{1}{\sin B} \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot 1}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   4. lift-tan.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x \cdot 1}{\color{blue}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   5. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x \cdot 1\right)}{\tan B}} + \frac{1}{\sin B} \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{x}\right)}{\tan B} + \frac{1}{\sin B} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{1}{\sin B} \]

   8. lower-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{-x}}{\tan B} + \frac{1}{\sin B} \]

   9. lift-tan.f64 99.8

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} + \frac{1}{\sin B} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{-x}{\tan B} + \frac{1}{\sin B}} \]
6. Add Preprocessing

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos B \cdot x}{\sin B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 (* (cos B) x)) (sin B)))

C

double code(double B, double x) {
	return (1.0 - (cos(B) * x)) / sin(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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - (cos(b) * x)) / sin(b)
end function

Java

public static double code(double B, double x) {
	return (1.0 - (Math.cos(B) * x)) / Math.sin(B);
}

Python

def code(B, x):
	return (1.0 - (math.cos(B) * x)) / math.sin(B)

Julia

function code(B, x)
	return Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - (cos(B) * x)) / sin(B);
end

Wolfram

code[B_, x_] := N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

2. Taylor expanded in B around inf

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
3. Step-by-step derivation

   1. sub-div N/A

      \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{1 - x \cdot \cos B}{\sin \color{blue}{B}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   7. lift-sin.f64 99.7

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B} + \frac{1}{B}\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 20000000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))
        (t_1 (+ (/ (- x) (tan B)) (/ 1.0 B))))
   (if (<= t_0 -100.0)
     t_1
     (if (<= t_0 20000000.0) (/ (- 1.0 x) (sin B)) t_1))))

C

double code(double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
	double t_1 = (-x / tan(B)) + (1.0 / B);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_1;
	} else if (t_0 <= 20000000.0) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = 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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
    t_1 = (-x / tan(b)) + (1.0d0 / b)
    if (t_0 <= (-100.0d0)) then
        tmp = t_1
    else if (t_0 <= 20000000.0d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	double t_1 = (-x / Math.tan(B)) + (1.0 / B);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_1;
	} else if (t_0 <= 20000000.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))
	t_1 = (-x / math.tan(B)) + (1.0 / B)
	tmp = 0
	if t_0 <= -100.0:
		tmp = t_1
	elif t_0 <= 20000000.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = t_1
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
	t_1 = Float64(Float64(Float64(-x) / tan(B)) + Float64(1.0 / B))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = t_1;
	elseif (t_0 <= 20000000.0)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
	t_1 = (-x / tan(B)) + (1.0 / B);
	tmp = 0.0;
	if (t_0 <= -100.0)
		tmp = t_1;
	elseif (t_0 <= 20000000.0)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], t$95$1, If[LessEqual[t$95$0, 20000000.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -100 or 2e7 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{1}{\sin B} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{1}{\sin B} \]

      7. lift-tan.f64 99.9

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{1}{\sin B} \]

   3. Applied rewrites 99.9%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B} \]
   4. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot 1}{\tan B}\right)\right)} + \frac{1}{\sin B} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot 1}}{\tan B}\right)\right) + \frac{1}{\sin B} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot 1}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

      4. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x \cdot 1}{\color{blue}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

      5. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x \cdot 1\right)}{\tan B}} + \frac{1}{\sin B} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{x}\right)}{\tan B} + \frac{1}{\sin B} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{1}{\sin B} \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{-x}}{\tan B} + \frac{1}{\sin B} \]

      9. lift-tan.f64 99.9

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} + \frac{1}{\sin B} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B} + \frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{-x}{\tan B} + \color{blue}{\frac{1}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 99.2

         \[\leadsto \frac{-x}{\tan B} + \frac{1}{\color{blue}{B}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{-x}{\tan B} + \color{blue}{\frac{1}{B}} \]

   if -100 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 2e7

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in B around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. sub-div N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 99.6

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\sin B} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.4%

      \[\leadsto \frac{1 - x}{\sin B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{\sin B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 x) (sin B)))

C

double code(double B, double x) {
	return (1.0 - x) / sin(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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - x) / sin(b)
end function

Java

public static double code(double B, double x) {
	return (1.0 - x) / Math.sin(B);
}

Python

def code(B, x):
	return (1.0 - x) / math.sin(B)

Julia

function code(B, x)
	return Float64(Float64(1.0 - x) / sin(B))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - x) / sin(B);
end

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

2. Taylor expanded in B around inf

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
3. Step-by-step derivation

   1. sub-div N/A

      \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{1 - x \cdot \cos B}{\sin \color{blue}{B}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   7. lift-sin.f64 99.7

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{1 - x}{\sin B} \]
6. Step-by-step derivation

7. Applied rewrites 76.9%

   \[\leadsto \frac{1 - x}{\sin B} \]
8. Add Preprocessing

Alternative 5: 64.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \left(-x \cdot \frac{1}{\tan B}\right) + t\_0\\ \mathbf{if}\;t\_1 \leq -4000:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right), B \cdot B, 0.16666666666666666\right), B \cdot B, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (+ (- (* x (/ 1.0 (tan B)))) t_0)))
   (if (<= t_1 -4000.0)
     (- (/ x (sin B)))
     (if (<= t_1 2e+27)
       t_0
       (+
        (- (* x (/ 1.0 B)))
        (/
         (fma
          (fma
           (fma 0.00205026455026455 (* B B) 0.019444444444444445)
           (* B B)
           0.16666666666666666)
          (* B B)
          1.0)
         B))))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -(x * (1.0 / tan(B))) + t_0;
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = -(x / sin(B));
	} else if (t_1 <= 2e+27) {
		tmp = t_0;
	} else {
		tmp = -(x * (1.0 / B)) + (fma(fma(fma(0.00205026455026455, (B * B), 0.019444444444444445), (B * B), 0.16666666666666666), (B * B), 1.0) / B);
	}
	return tmp;
}

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + t_0)
	tmp = 0.0
	if (t_1 <= -4000.0)
		tmp = Float64(-Float64(x / sin(B)));
	elseif (t_1 <= 2e+27)
		tmp = t_0;
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64(fma(fma(fma(0.00205026455026455, Float64(B * B), 0.019444444444444445), Float64(B * B), 0.16666666666666666), Float64(B * B), 1.0) / B));
	end
	return tmp
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4000.0], (-N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 2e+27], t$95$0, N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(N[(N[(N[(0.00205026455026455 * N[(B * B), $MachinePrecision] + 0.019444444444444445), $MachinePrecision] * N[(B * B), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -4e3

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 66.4

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -\frac{x}{\sin B} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto -\frac{x}{\sin B} \]

   if -4e3 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 2e27

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\sin B} \]

      2. lift-/.f64 92.6

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

   4. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 2e27 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{1}{\sin B} \]
   3. Step-by-step derivation

   4. Applied rewrites 68.8%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{1}{\sin B} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{1}{6} + {B}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right)\right)}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{1 + {B}^{2} \cdot \left(\frac{1}{6} + {B}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right)\right)}{\color{blue}{B}} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{{B}^{2} \cdot \left(\frac{1}{6} + {B}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right)\right) + 1}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\left(\frac{1}{6} + {B}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right)\right) \cdot {B}^{2} + 1}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\frac{1}{6} + {B}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right), {B}^{2}, 1\right)}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left({B}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right) + \frac{1}{6}, {B}^{2}, 1\right)}{B} \]

      6. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right) \cdot {B}^{2} + \frac{1}{6}, {B}^{2}, 1\right)}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}, {B}^{2}, \frac{1}{6}\right), {B}^{2}, 1\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120} \cdot {B}^{2} + \frac{7}{360}, {B}^{2}, \frac{1}{6}\right), {B}^{2}, 1\right)}{B} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, {B}^{2}, \frac{7}{360}\right), {B}^{2}, \frac{1}{6}\right), {B}^{2}, 1\right)}{B} \]

      10. pow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, B \cdot B, \frac{7}{360}\right), {B}^{2}, \frac{1}{6}\right), {B}^{2}, 1\right)}{B} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, B \cdot B, \frac{7}{360}\right), {B}^{2}, \frac{1}{6}\right), {B}^{2}, 1\right)}{B} \]

      12. pow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, B \cdot B, \frac{7}{360}\right), B \cdot B, \frac{1}{6}\right), {B}^{2}, 1\right)}{B} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, B \cdot B, \frac{7}{360}\right), B \cdot B, \frac{1}{6}\right), {B}^{2}, 1\right)}{B} \]

      14. pow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, B \cdot B, \frac{7}{360}\right), B \cdot B, \frac{1}{6}\right), B \cdot B, 1\right)}{B} \]

      15. lift-*.f64 68.8

          \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right), B \cdot B, 0.16666666666666666\right), B \cdot B, 1\right)}{B} \]

   7. Applied rewrites 68.8%

      \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right), B \cdot B, 0.16666666666666666\right), B \cdot B, 1\right)}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \left(-x \cdot \frac{1}{\tan B}\right) + t\_0\\ \mathbf{if}\;t\_1 \leq -4000:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{elif}\;t\_1 \leq 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right) + 0.019444444444444445 \cdot \left(B \cdot B\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (+ (- (* x (/ 1.0 (tan B)))) t_0)))
   (if (<= t_1 -4000.0)
     (- (/ x (sin B)))
     (if (<= t_1 1e+115)
       t_0
       (/
        (-
         (fma
          (+
           (fma 0.3333333333333333 x 0.16666666666666666)
           (* 0.019444444444444445 (* B B)))
          (* B B)
          1.0)
         x)
        B)))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -(x * (1.0 / tan(B))) + t_0;
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = -(x / sin(B));
	} else if (t_1 <= 1e+115) {
		tmp = t_0;
	} else {
		tmp = (fma((fma(0.3333333333333333, x, 0.16666666666666666) + (0.019444444444444445 * (B * B))), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + t_0)
	tmp = 0.0
	if (t_1 <= -4000.0)
		tmp = Float64(-Float64(x / sin(B)));
	elseif (t_1 <= 1e+115)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(Float64(fma(0.3333333333333333, x, 0.16666666666666666) + Float64(0.019444444444444445 * Float64(B * B))), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4000.0], (-N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 1e+115], t$95$0, N[(N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] + N[(0.019444444444444445 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -4e3

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 66.4

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -\frac{x}{\sin B} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto -\frac{x}{\sin B} \]

   if -4e3 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 1e115

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\sin B} \]

      2. lift-/.f64 82.3

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

   4. Applied rewrites 82.3%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 1e115 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{\color{blue}{B}} \]

   4. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right) + \mathsf{fma}\left(x, 0.022222222222222223, 0.019444444444444445\right) \cdot \left(B \cdot B\right), B \cdot B, 1\right) - x}{B}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right) + \frac{7}{360} \cdot \left(B \cdot B\right), B \cdot B, 1\right) - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.5%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right) + 0.019444444444444445 \cdot \left(B \cdot B\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 14.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right) + \mathsf{fma}\left(x, 0.022222222222222223, 0.019444444444444445\right) \cdot \left(B \cdot B\right), B \cdot B, 1\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 14.2)
   (/
    (-
     (fma
      (+
       (fma 0.3333333333333333 x 0.16666666666666666)
       (* (fma x 0.022222222222222223 0.019444444444444445) (* B B)))
      (* B B)
      1.0)
     x)
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 14.2) {
		tmp = (fma((fma(0.3333333333333333, x, 0.16666666666666666) + (fma(x, 0.022222222222222223, 0.019444444444444445) * (B * B))), (B * B), 1.0) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 14.2)
		tmp = Float64(Float64(fma(Float64(fma(0.3333333333333333, x, 0.16666666666666666) + Float64(fma(x, 0.022222222222222223, 0.019444444444444445) * Float64(B * B))), Float64(B * B), 1.0) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 14.2], N[(N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] + N[(N[(x * 0.022222222222222223 + 0.019444444444444445), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 14.199999999999999

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{\color{blue}{B}} \]

   4. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right) + \mathsf{fma}\left(x, 0.022222222222222223, 0.019444444444444445\right) \cdot \left(B \cdot B\right), B \cdot B, 1\right) - x}{B}} \]

   if 14.199999999999999 < B

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\sin B} \]

      2. lift-/.f64 50.0

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (/ (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x) B))

C

double code(double B, double x) {
	return (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
}

Julia

function code(B, x)
	return Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B)
end

Wolfram

code[B_, x_] := N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{\color{blue}{B}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right) - x}{B} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\left(\frac{1}{6} + \frac{1}{3} \cdot x\right) \cdot {B}^{2} + 1\right) - x}{B} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{3} \cdot x, {B}^{2}, 1\right) - x}{B} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3} \cdot x + \frac{1}{6}, {B}^{2}, 1\right) - x}{B} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right), {B}^{2}, 1\right) - x}{B} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right), B \cdot B, 1\right) - x}{B} \]

   9. lower-*.f64 51.3

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]

4. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}} \]
5. Add Preprocessing

Alternative 9: 51.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{B - B \cdot x}{B}}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (/ (- B (* B x)) B) B))

C

double code(double B, double x) {
	return ((B - (B * x)) / B) / 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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = ((b - (b * x)) / b) / b
end function

Java

public static double code(double B, double x) {
	return ((B - (B * x)) / B) / B;
}

Python

def code(B, x):
	return ((B - (B * x)) / B) / B

Julia

function code(B, x)
	return Float64(Float64(Float64(B - Float64(B * x)) / B) / B)
end

MATLAB

function tmp = code(B, x)
	tmp = ((B - (B * x)) / B) / B;
end

Wolfram

code[B_, x_] := N[(N[(N[(B - N[(B * x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]

   2. lower--.f64 51.2

      \[\leadsto \frac{1 - x}{B} \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{1 - x}{B} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]

   3. div-sub N/A

      \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

   4. frac-sub N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{\color{blue}{B \cdot B}} \]

   5. pow2 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{{B}^{\color{blue}{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{\color{blue}{{B}^{2}}} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{{\color{blue}{B}}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{{B}^{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{{B}^{2}} \]

   10. pow2 N/A

       \[\leadsto \frac{1 \cdot B - B \cdot x}{B \cdot \color{blue}{B}} \]

   11. lift-*.f64 36.4

       \[\leadsto \frac{1 \cdot B - B \cdot x}{B \cdot \color{blue}{B}} \]

6. Applied rewrites 36.4%

   \[\leadsto \frac{1 \cdot B - B \cdot x}{\color{blue}{B \cdot B}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{\color{blue}{B \cdot B}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{B \cdot B} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{B \cdot B} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{\color{blue}{B} \cdot B} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot B - B \cdot x}{B \cdot \color{blue}{B}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{1 \cdot B - B \cdot x}{B}}{\color{blue}{B}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1 \cdot B - B \cdot x}{B}}{\color{blue}{B}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1 \cdot B - B \cdot x}{B}}{B} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{\frac{1 \cdot B - B \cdot x}{B}}{B} \]

   10. *-lft-identity N/A

       \[\leadsto \frac{\frac{B - B \cdot x}{B}}{B} \]

   11. lift-*.f64 51.3

       \[\leadsto \frac{\frac{B - B \cdot x}{B}}{B} \]

8. Applied rewrites 51.3%

   \[\leadsto \frac{\frac{B - B \cdot x}{B}}{\color{blue}{B}} \]
9. Add Preprocessing

Alternative 10: 51.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 x) B))

C

double code(double B, double x) {
	return (1.0 - x) / 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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - x) / b
end function

Java

public static double code(double B, double x) {
	return (1.0 - x) / B;
}

Python

def code(B, x):
	return (1.0 - x) / B

Julia

function code(B, x)
	return Float64(Float64(1.0 - x) / B)
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - x) / B;
end

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]

   2. lower--.f64 51.2

      \[\leadsto \frac{1 - x}{B} \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
5. Add Preprocessing

Alternative 11: 49.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -3.25 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -3.25e+16) t_0 (if (<= x 1.0) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -3.25e+16) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / b
    if (x <= (-3.25d+16)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -3.25e+16) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -3.25e+16:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (x <= -3.25e+16)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (x <= -3.25e+16)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[x, -3.25e+16], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.25e16 or 1 < x

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - x}{\color{blue}{B}} \]

      2. lower--.f64 51.3

         \[\leadsto \frac{1 - x}{B} \]

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 50.8

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 50.8%

      \[\leadsto \frac{-x}{B} \]

   if -3.25e16 < x < 1

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - x}{\color{blue}{B}} \]

      2. lower--.f64 51.1

         \[\leadsto \frac{1 - x}{B} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.6%

      \[\leadsto \frac{1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 26.5% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ 1.0 B))

C

double code(double B, double x) {
	return 1.0 / 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(b, x)
use fmin_fmax_functions
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double B, double x) {
	return 1.0 / B;
}

Python

def code(B, x):
	return 1.0 / B

Julia

function code(B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(B, x)
	tmp = 1.0 / B;
end

Wolfram

code[B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]

   2. lower--.f64 51.2

      \[\leadsto \frac{1 - x}{B} \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 26.5%

   \[\leadsto \frac{1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))