VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 4.2s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

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

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

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, 0.7× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	return (((1.0 / sin(B)) * tan(B)) - x) / tan(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 / sin(b)) * tan(b)) - x) / tan(b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. sub-flip-reverse N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip-rev N/A

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

   8. sub-to-fraction N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\frac{1}{\sin B} - \frac{x}{\tan B} \]

FPCore

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

C

double code(double B, double x) {
	return (1.0 / sin(B)) - (x / tan(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 / sin(b)) - (x / tan(b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. sub-flip-reverse N/A

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

   5. lower--.f64 99.7%

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|B\right|\right)\\ t_1 := \cos \left(\left|B\right|\right)\\ \mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 0.000325:\\ \;\;\;\;\frac{1 - x}{t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - t\_1}{t\_0} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
  :precision binary64
  (let* ((t_0 (sin (fabs B))) (t_1 (cos (fabs B))))
  (*
   (copysign 1.0 B)
   (if (<= (fabs B) 0.000325)
     (* (/ (- 1.0 x) t_0) t_1)
     (* (/ (- (/ 1.0 x) t_1) t_0) x)))))

C

double code(double B, double x) {
	double t_0 = sin(fabs(B));
	double t_1 = cos(fabs(B));
	double tmp;
	if (fabs(B) <= 0.000325) {
		tmp = ((1.0 - x) / t_0) * t_1;
	} else {
		tmp = (((1.0 / x) - t_1) / t_0) * x;
	}
	return copysign(1.0, B) * tmp;
}

Java

public static double code(double B, double x) {
	double t_0 = Math.sin(Math.abs(B));
	double t_1 = Math.cos(Math.abs(B));
	double tmp;
	if (Math.abs(B) <= 0.000325) {
		tmp = ((1.0 - x) / t_0) * t_1;
	} else {
		tmp = (((1.0 / x) - t_1) / t_0) * x;
	}
	return Math.copySign(1.0, B) * tmp;
}

Python

def code(B, x):
	t_0 = math.sin(math.fabs(B))
	t_1 = math.cos(math.fabs(B))
	tmp = 0
	if math.fabs(B) <= 0.000325:
		tmp = ((1.0 - x) / t_0) * t_1
	else:
		tmp = (((1.0 / x) - t_1) / t_0) * x
	return math.copysign(1.0, B) * tmp

Julia

function code(B, x)
	t_0 = sin(abs(B))
	t_1 = cos(abs(B))
	tmp = 0.0
	if (abs(B) <= 0.000325)
		tmp = Float64(Float64(Float64(1.0 - x) / t_0) * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / x) - t_1) / t_0) * x);
	end
	return Float64(copysign(1.0, B) * tmp)
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = sin(abs(B));
	t_1 = cos(abs(B));
	tmp = 0.0;
	if (abs(B) <= 0.000325)
		tmp = ((1.0 - x) / t_0) * t_1;
	else
		tmp = (((1.0 / x) - t_1) / t_0) * x;
	end
	tmp_2 = (sign(B) * abs(1.0)) * tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[Sin[N[Abs[B], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Abs[B], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 0.000325], N[(N[(N[(1.0 - x), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(1.0 / x), $MachinePrecision] - t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.2499999999999999e-4

   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

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. sub-to-fraction N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 77.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-tan.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lower-/.f64 76.9%

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

   8. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. tan-quot N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 77.0%

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

   10. Applied rewrites 77.0%

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

   if 3.2499999999999999e-4 < B

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 87.7%

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

   4. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.7%

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 87.7%

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

   6. Applied rewrites 87.7%

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

Alternative 4: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(\left|B\right|\right)\\ t_1 := \frac{1}{\sin \left(\left|B\right|\right)}\\ t_2 := \left(-x \cdot \frac{1}{t\_0}\right) + t\_1\\ \mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 - x}{t\_0}\\ \mathbf{elif}\;t\_2 \leq 40:\\ \;\;\;\;t\_1 - \frac{x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|B\right|} - \frac{x}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
  :precision binary64
  (let* ((t_0 (tan (fabs B)))
       (t_1 (/ 1.0 (sin (fabs B))))
       (t_2 (+ (- (* x (/ 1.0 t_0))) t_1)))
  (*
   (copysign 1.0 B)
   (if (<= t_2 -5e+22)
     (/ (- 1.0 x) t_0)
     (if (<= t_2 40.0)
       (- t_1 (/ x (fabs B)))
       (- (/ 1.0 (fabs B)) (/ x t_0)))))))

C

double code(double B, double x) {
	double t_0 = tan(fabs(B));
	double t_1 = 1.0 / sin(fabs(B));
	double t_2 = -(x * (1.0 / t_0)) + t_1;
	double tmp;
	if (t_2 <= -5e+22) {
		tmp = (1.0 - x) / t_0;
	} else if (t_2 <= 40.0) {
		tmp = t_1 - (x / fabs(B));
	} else {
		tmp = (1.0 / fabs(B)) - (x / t_0);
	}
	return copysign(1.0, B) * tmp;
}

Java

public static double code(double B, double x) {
	double t_0 = Math.tan(Math.abs(B));
	double t_1 = 1.0 / Math.sin(Math.abs(B));
	double t_2 = -(x * (1.0 / t_0)) + t_1;
	double tmp;
	if (t_2 <= -5e+22) {
		tmp = (1.0 - x) / t_0;
	} else if (t_2 <= 40.0) {
		tmp = t_1 - (x / Math.abs(B));
	} else {
		tmp = (1.0 / Math.abs(B)) - (x / t_0);
	}
	return Math.copySign(1.0, B) * tmp;
}

Python

def code(B, x):
	t_0 = math.tan(math.fabs(B))
	t_1 = 1.0 / math.sin(math.fabs(B))
	t_2 = -(x * (1.0 / t_0)) + t_1
	tmp = 0
	if t_2 <= -5e+22:
		tmp = (1.0 - x) / t_0
	elif t_2 <= 40.0:
		tmp = t_1 - (x / math.fabs(B))
	else:
		tmp = (1.0 / math.fabs(B)) - (x / t_0)
	return math.copysign(1.0, B) * tmp

Julia

function code(B, x)
	t_0 = tan(abs(B))
	t_1 = Float64(1.0 / sin(abs(B)))
	t_2 = Float64(Float64(-Float64(x * Float64(1.0 / t_0))) + t_1)
	tmp = 0.0
	if (t_2 <= -5e+22)
		tmp = Float64(Float64(1.0 - x) / t_0);
	elseif (t_2 <= 40.0)
		tmp = Float64(t_1 - Float64(x / abs(B)));
	else
		tmp = Float64(Float64(1.0 / abs(B)) - Float64(x / t_0));
	end
	return Float64(copysign(1.0, B) * tmp)
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = tan(abs(B));
	t_1 = 1.0 / sin(abs(B));
	t_2 = -(x * (1.0 / t_0)) + t_1;
	tmp = 0.0;
	if (t_2 <= -5e+22)
		tmp = (1.0 - x) / t_0;
	elseif (t_2 <= 40.0)
		tmp = t_1 - (x / abs(B));
	else
		tmp = (1.0 / abs(B)) - (x / t_0);
	end
	tmp_2 = (sign(B) * abs(1.0)) * tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]) + t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -5e+22], N[(N[(1.0 - x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 40.0], N[(t$95$1 - N[(x / N[Abs[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $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))) < -4.9999999999999996e22

   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

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. sub-to-fraction N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 77.0%

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

   if -4.9999999999999996e22 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 40

   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

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 99.7%

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 75.2%

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

   6. Applied rewrites 75.2%

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

   if 40 < (+.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

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 99.7%

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 75.2%

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

   6. Applied rewrites 75.2%

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

Alternative 5: 96.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{1 - x}{\tan B}\\ \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (B x)
  :precision binary64
  (let* ((t_0 (/ (- 1.0 x) (tan B))))
  (if (<= x -2.9) t_0 (if (<= x 3.2e+20) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 - x) / tan(B);
	double tmp;
	if (x <= -2.9) {
		tmp = t_0;
	} else if (x <= 3.2e+20) {
		tmp = 1.0 / sin(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 = (1.0d0 - x) / tan(b)
    if (x <= (-2.9d0)) then
        tmp = t_0
    else if (x <= 3.2d+20) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 - x) / Math.tan(B);
	double tmp;
	if (x <= -2.9) {
		tmp = t_0;
	} else if (x <= 3.2e+20) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 - x) / math.tan(B)
	tmp = 0
	if x <= -2.9:
		tmp = t_0
	elif x <= 3.2e+20:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 - x) / tan(B))
	tmp = 0.0
	if (x <= -2.9)
		tmp = t_0;
	elseif (x <= 3.2e+20)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 - x) / tan(B);
	tmp = 0.0;
	if (x <= -2.9)
		tmp = t_0;
	elseif (x <= 3.2e+20)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9], t$95$0, If[LessEqual[x, 3.2e+20], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999999 or 3.2e20 < x

   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

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. sub-to-fraction N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 77.0%

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

   if -2.8999999999999999 < x < 3.2e20

   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

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 99.7%

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 99.8%

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

   3. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. sub-flip N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 50.2%

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

   8. Applied rewrites 50.2%

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

Alternative 6: 74.7% accurate, 1.7× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 820000:\\ \;\;\;\;\frac{\left(1 + {\left(\left|B\right|\right)}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (B x)
  :precision binary64
  (*
 (copysign 1.0 B)
 (if (<= (fabs B) 820000.0)
   (/
    (-
     (+
      1.0
      (*
       (pow (fabs B) 2.0)
       (+ 0.16666666666666666 (* 0.3333333333333333 x))))
     x)
    (fabs B))
   (/ 1.0 (sin (fabs B))))))

C

double code(double B, double x) {
	double tmp;
	if (fabs(B) <= 820000.0) {
		tmp = ((1.0 + (pow(fabs(B), 2.0) * (0.16666666666666666 + (0.3333333333333333 * x)))) - x) / fabs(B);
	} else {
		tmp = 1.0 / sin(fabs(B));
	}
	return copysign(1.0, B) * tmp;
}

Java

public static double code(double B, double x) {
	double tmp;
	if (Math.abs(B) <= 820000.0) {
		tmp = ((1.0 + (Math.pow(Math.abs(B), 2.0) * (0.16666666666666666 + (0.3333333333333333 * x)))) - x) / Math.abs(B);
	} else {
		tmp = 1.0 / Math.sin(Math.abs(B));
	}
	return Math.copySign(1.0, B) * tmp;
}

Python

def code(B, x):
	tmp = 0
	if math.fabs(B) <= 820000.0:
		tmp = ((1.0 + (math.pow(math.fabs(B), 2.0) * (0.16666666666666666 + (0.3333333333333333 * x)))) - x) / math.fabs(B)
	else:
		tmp = 1.0 / math.sin(math.fabs(B))
	return math.copysign(1.0, B) * tmp

Julia

function code(B, x)
	tmp = 0.0
	if (abs(B) <= 820000.0)
		tmp = Float64(Float64(Float64(1.0 + Float64((abs(B) ^ 2.0) * Float64(0.16666666666666666 + Float64(0.3333333333333333 * x)))) - x) / abs(B));
	else
		tmp = Float64(1.0 / sin(abs(B)));
	end
	return Float64(copysign(1.0, B) * tmp)
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (abs(B) <= 820000.0)
		tmp = ((1.0 + ((abs(B) ^ 2.0) * (0.16666666666666666 + (0.3333333333333333 * x)))) - x) / abs(B);
	else
		tmp = 1.0 / sin(abs(B));
	end
	tmp_2 = (sign(B) * abs(1.0)) * tmp;
end

Wolfram

code[B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 820000.0], N[(N[(N[(1.0 + N[(N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.16666666666666666 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 8.2e5

   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 51.4%

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   6. 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. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 51.5%

         \[\leadsto \frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B} \]

   7. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   if 8.2e5 < 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

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 99.7%

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 99.8%

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

   3. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. sub-flip N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 50.2%

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

   8. Applied rewrites 50.2%

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

Alternative 7: 51.5% accurate, 2.3× speedup

Math

\[\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{1}{B} \]

FPCore

(FPCore (B x)
  :precision binary64
  (+ (/ (- (* 0.3333333333333333 (* (pow B 2.0) x)) x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	return (((0.3333333333333333 * (pow(B, 2.0) * x)) - x) / B) + (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 = (((0.3333333333333333d0 * ((b ** 2.0d0) * x)) - x) / b) + (1.0d0 / b)
end function

Java

public static double code(double B, double x) {
	return (((0.3333333333333333 * (Math.pow(B, 2.0) * x)) - x) / B) + (1.0 / B);
}

Python

def code(B, x):
	return (((0.3333333333333333 * (math.pow(B, 2.0) * x)) - x) / B) + (1.0 / B)

Julia

function code(B, x)
	return Float64(Float64(Float64(Float64(0.3333333333333333 * Float64((B ^ 2.0) * x)) - x) / B) + Float64(1.0 / B))
end

MATLAB

function tmp = code(B, x)
	tmp = (((0.3333333333333333 * ((B ^ 2.0) * x)) - x) / B) + (1.0 / B);
end

Wolfram

code[B_, x_] := N[(N[(N[(N[(0.3333333333333333 * N[(N[Power[B, 2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(1.0 / 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 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 60.3%

      \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{1}{\sin B} \]

4. Applied rewrites 60.3%

   \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{1}{\sin B} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \color{blue}{\frac{1}{B}} \]
6. Step-by-step derivation

   1. lower-/.f64 51.4%

      \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]

7. Applied rewrites 51.4%

   \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \color{blue}{\frac{1}{B}} \]
8. Add Preprocessing

Alternative 8: 51.5% accurate, 2.4× speedup

Math

\[\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B} \]

FPCore

(FPCore (B x)
  :precision binary64
  (/
 (-
  (+
   1.0
   (* (pow B 2.0) (+ 0.16666666666666666 (* 0.3333333333333333 x))))
  x)
 B))

C

double code(double B, double x) {
	return ((1.0 + (pow(B, 2.0) * (0.16666666666666666 + (0.3333333333333333 * x)))) - 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 + ((b ** 2.0d0) * (0.16666666666666666d0 + (0.3333333333333333d0 * x)))) - x) / b
end function

Java

public static double code(double B, double x) {
	return ((1.0 + (Math.pow(B, 2.0) * (0.16666666666666666 + (0.3333333333333333 * x)))) - x) / B;
}

Python

def code(B, x):
	return ((1.0 + (math.pow(B, 2.0) * (0.16666666666666666 + (0.3333333333333333 * x)))) - x) / B

Julia

function code(B, x)
	return Float64(Float64(Float64(1.0 + Float64((B ^ 2.0) * Float64(0.16666666666666666 + Float64(0.3333333333333333 * x)))) - x) / B)
end

MATLAB

function tmp = code(B, x)
	tmp = ((1.0 + ((B ^ 2.0) * (0.16666666666666666 + (0.3333333333333333 * x)))) - x) / B;
end

Wolfram

code[B_, x_] := N[(N[(N[(1.0 + N[(N[Power[B, 2.0], $MachinePrecision] * N[(0.16666666666666666 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 51.4%

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

4. Applied rewrites 51.4%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
6. 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. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 51.5%

      \[\leadsto \frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B} \]

7. Applied rewrites 51.5%

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]
8. Add Preprocessing

Alternative 9: 51.4% accurate, 2.6× speedup

Math

\[\frac{1 + -0.3333333333333333 \cdot {B}^{2}}{B} \cdot \left(1 - x\right) \]

FPCore

(FPCore (B x)
  :precision binary64
  (* (/ (+ 1.0 (* -0.3333333333333333 (pow B 2.0))) B) (- 1.0 x)))

C

double code(double B, double x) {
	return ((1.0 + (-0.3333333333333333 * pow(B, 2.0))) / B) * (1.0 - x);
}

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 + ((-0.3333333333333333d0) * (b ** 2.0d0))) / b) * (1.0d0 - x)
end function

Java

public static double code(double B, double x) {
	return ((1.0 + (-0.3333333333333333 * Math.pow(B, 2.0))) / B) * (1.0 - x);
}

Python

def code(B, x):
	return ((1.0 + (-0.3333333333333333 * math.pow(B, 2.0))) / B) * (1.0 - x)

Julia

function code(B, x)
	return Float64(Float64(Float64(1.0 + Float64(-0.3333333333333333 * (B ^ 2.0))) / B) * Float64(1.0 - x))
end

MATLAB

function tmp = code(B, x)
	tmp = ((1.0 + (-0.3333333333333333 * (B ^ 2.0))) / B) * (1.0 - x);
end

Wolfram

code[B_, x_] := N[(N[(N[(1.0 + N[(-0.3333333333333333 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] * N[(1.0 - x), $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

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. sub-flip-reverse N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip-rev N/A

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

   8. sub-to-fraction N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in B around 0

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

6. Applied rewrites 77.0%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-tan.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lower-/.f64 76.9%

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

8. Applied rewrites 76.9%

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

9. Taylor expanded in B around 0

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

    1. lower-/.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-pow.f64 51.3%

       \[\leadsto \frac{1 + -0.3333333333333333 \cdot {B}^{2}}{B} \cdot \left(1 - x\right) \]

11. Applied rewrites 51.3%

    \[\leadsto \color{blue}{\frac{1 + -0.3333333333333333 \cdot {B}^{2}}{B}} \cdot \left(1 - x\right) \]
12. Add Preprocessing

Alternative 10: 51.4% accurate, 6.1× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	return ((B * (1.0 - 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 * (1.0d0 - x)) / b) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(N[(B * N[(1.0 - 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 51.4%

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

4. Applied rewrites 51.4%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   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. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 36.8%

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 36.8%

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

6. Applied rewrites 36.8%

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

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 51.5%

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-out-- N/A

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

   11. lift--.f64 N/A

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

   12. lower-*.f64 51.5%

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

8. Applied rewrites 51.5%

   \[\leadsto \frac{\frac{B \cdot \left(1 - x\right)}{B}}{\color{blue}{B}} \]
9. Add Preprocessing

Alternative 11: 51.3% accurate, 11.6× speedup

Math

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

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 51.4%

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

4. Applied rewrites 51.4%

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

Alternative 12: 27.0% accurate, 14.8× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	return -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 = -x / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[((-x) / 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 51.4%

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

4. Applied rewrites 51.4%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 51.3%

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

6. Applied rewrites 51.3%

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

7. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 27.0%

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

9. Applied rewrites 27.0%

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

    1. lift-*.f64 N/A

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

    2. mul-1-neg N/A

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

    3. lift-/.f64 N/A

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

    4. distribute-neg-frac N/A

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

    5. lower-/.f64 N/A

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

    6. lower-neg.f64 27.0%

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

11. Applied rewrites 27.0%

    \[\leadsto \frac{-x}{B} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))