VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.5s

Alternatives: 10

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

real(8) function code(b, x)
    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

real(8) function code(b, x)
    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

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

FPCore

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

C

double code(double B, double x) {
	return (-x / tan(B)) + pow(sin(B), -1.0);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-x / tan(b)) + (sin(b) ** (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. 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. un-div-inv N/A

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

   4. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin B}^{-1}\\ t_1 := x \cdot \frac{-1}{\tan B} + t\_0\\ \mathbf{if}\;t\_1 \leq -10000000 \lor \neg \left(t\_1 \leq 1000\right):\\ \;\;\;\;{B}^{-1} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (pow (sin B) -1.0)) (t_1 (+ (* x (/ -1.0 (tan B))) t_0)))
   (if (or (<= t_1 -10000000.0) (not (<= t_1 1000.0)))
     (- (pow B -1.0) (/ x (tan B)))
     (+ (- (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = pow(sin(B), -1.0);
	double t_1 = (x * (-1.0 / tan(B))) + t_0;
	double tmp;
	if ((t_1 <= -10000000.0) || !(t_1 <= 1000.0)) {
		tmp = pow(B, -1.0) - (x / tan(B));
	} else {
		tmp = -(x / B) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(b) ** (-1.0d0)
    t_1 = (x * ((-1.0d0) / tan(b))) + t_0
    if ((t_1 <= (-10000000.0d0)) .or. (.not. (t_1 <= 1000.0d0))) then
        tmp = (b ** (-1.0d0)) - (x / tan(b))
    else
        tmp = -(x / b) + t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = Math.pow(Math.sin(B), -1.0);
	double t_1 = (x * (-1.0 / Math.tan(B))) + t_0;
	double tmp;
	if ((t_1 <= -10000000.0) || !(t_1 <= 1000.0)) {
		tmp = Math.pow(B, -1.0) - (x / Math.tan(B));
	} else {
		tmp = -(x / B) + t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = math.pow(math.sin(B), -1.0)
	t_1 = (x * (-1.0 / math.tan(B))) + t_0
	tmp = 0
	if (t_1 <= -10000000.0) or not (t_1 <= 1000.0):
		tmp = math.pow(B, -1.0) - (x / math.tan(B))
	else:
		tmp = -(x / B) + t_0
	return tmp

Julia

function code(B, x)
	t_0 = sin(B) ^ -1.0
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + t_0)
	tmp = 0.0
	if ((t_1 <= -10000000.0) || !(t_1 <= 1000.0))
		tmp = Float64((B ^ -1.0) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(-Float64(x / B)) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = sin(B) ^ -1.0;
	t_1 = (x * (-1.0 / tan(B))) + t_0;
	tmp = 0.0;
	if ((t_1 <= -10000000.0) || ~((t_1 <= 1000.0)))
		tmp = (B ^ -1.0) - (x / tan(B));
	else
		tmp = -(x / B) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -10000000.0], N[Not[LessEqual[t$95$1, 1000.0]], $MachinePrecision]], N[(N[Power[B, -1.0], $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + t$95$0), $MachinePrecision]]]]

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))) < -1e7 or 1e3 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 67.9

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left({\sin B}^{2}\right)}^{\color{blue}{-0.5}} \]

   4. Applied rewrites 67.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{2}\right)}^{-0.5}} \]

   5. Taylor expanded in B around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.7

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\color{blue}{\left(B \cdot B\right)}}^{-0.5} \]

   7. Applied rewrites 67.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\color{blue}{\left(B \cdot B\right)}}^{-0.5} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 67.7

         \[\leadsto \color{blue}{{\left(B \cdot B\right)}^{-0.5} - x \cdot \frac{1}{\tan B}} \]

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 67.7

         \[\leadsto {\left(B \cdot B\right)}^{-0.5} - \color{blue}{\frac{x}{\tan B}} \]

   9. Applied rewrites 67.7%

      \[\leadsto \color{blue}{{\left(B \cdot B\right)}^{-0.5} - \frac{x}{\tan B}} \]

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 99.3

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

   12. Applied rewrites 99.3%

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

   if -1e7 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 1e3

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 96.0

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

   5. Applied rewrites 96.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{1}{\sin B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq -10000000 \lor \neg \left(x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq 1000\right):\\ \;\;\;\;{B}^{-1} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;{B}^{-1} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sin B}{1 - x}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 1.2)))
   (- (pow B -1.0) (/ x (tan B)))
   (pow (/ (sin B) (- 1.0 x)) -1.0)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.2)) {
		tmp = pow(B, -1.0) - (x / tan(B));
	} else {
		tmp = pow((sin(B) / (1.0 - x)), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 1.2d0))) then
        tmp = (b ** (-1.0d0)) - (x / tan(b))
    else
        tmp = (sin(b) / (1.0d0 - x)) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.2)) {
		tmp = Math.pow(B, -1.0) - (x / Math.tan(B));
	} else {
		tmp = Math.pow((Math.sin(B) / (1.0 - x)), -1.0);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.5) or not (x <= 1.2):
		tmp = math.pow(B, -1.0) - (x / math.tan(B))
	else:
		tmp = math.pow((math.sin(B) / (1.0 - x)), -1.0)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 1.2))
		tmp = Float64((B ^ -1.0) - Float64(x / tan(B)));
	else
		tmp = Float64(sin(B) / Float64(1.0 - x)) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 1.2)))
		tmp = (B ^ -1.0) - (x / tan(B));
	else
		tmp = (sin(B) / (1.0 - x)) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 1.2]], $MachinePrecision]], N[(N[Power[B, -1.0], $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sin[B], $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 1.19999999999999996 < x

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 86.9

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left({\sin B}^{2}\right)}^{\color{blue}{-0.5}} \]

   4. Applied rewrites 86.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{2}\right)}^{-0.5}} \]

   5. Taylor expanded in B around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\color{blue}{\left(B \cdot B\right)}}^{-0.5} \]

   7. Applied rewrites 86.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\color{blue}{\left(B \cdot B\right)}}^{-0.5} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 86.6

         \[\leadsto \color{blue}{{\left(B \cdot B\right)}^{-0.5} - x \cdot \frac{1}{\tan B}} \]

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 86.6

         \[\leadsto {\left(B \cdot B\right)}^{-0.5} - \color{blue}{\frac{x}{\tan B}} \]

   9. Applied rewrites 86.6%

      \[\leadsto \color{blue}{{\left(B \cdot B\right)}^{-0.5} - \frac{x}{\tan B}} \]

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 99.0

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

   12. Applied rewrites 99.0%

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

   if -1.5 < x < 1.19999999999999996

   1. Initial program 99.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing
   3. 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. un-div-inv N/A

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

      4. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/l* N/A

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

      18. sub-div N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in B around 0

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

       1. lower--.f64 97.7

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

   11. Applied rewrites 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;{B}^{-1} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sin B}{1 - x}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;{B}^{-1} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 1.2)))
   (- (pow B -1.0) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.2)) {
		tmp = pow(B, -1.0) - (x / tan(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 1.2d0))) then
        tmp = (b ** (-1.0d0)) - (x / tan(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.2)) {
		tmp = Math.pow(B, -1.0) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.5) or not (x <= 1.2):
		tmp = math.pow(B, -1.0) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 1.2))
		tmp = Float64((B ^ -1.0) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 1.2)))
		tmp = (B ^ -1.0) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 1.2]], $MachinePrecision]], N[(N[Power[B, -1.0], $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 1.19999999999999996 < x

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 86.9

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left({\sin B}^{2}\right)}^{\color{blue}{-0.5}} \]

   4. Applied rewrites 86.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{2}\right)}^{-0.5}} \]

   5. Taylor expanded in B around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\color{blue}{\left(B \cdot B\right)}}^{-0.5} \]

   7. Applied rewrites 86.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\color{blue}{\left(B \cdot B\right)}}^{-0.5} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 86.6

         \[\leadsto \color{blue}{{\left(B \cdot B\right)}^{-0.5} - x \cdot \frac{1}{\tan B}} \]

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 86.6

         \[\leadsto {\left(B \cdot B\right)}^{-0.5} - \color{blue}{\frac{x}{\tan B}} \]

   9. Applied rewrites 86.6%

      \[\leadsto \color{blue}{{\left(B \cdot B\right)}^{-0.5} - \frac{x}{\tan B}} \]

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 99.0

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

   12. Applied rewrites 99.0%

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

   if -1.5 < x < 1.19999999999999996

   1. Initial program 99.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing
   3. 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. un-div-inv N/A

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

      4. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/l* N/A

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

      18. sub-div N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 97.7

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

   9. Applied rewrites 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;{B}^{-1} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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

real(8) function code(b, x)
    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. Add Preprocessing
3. 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. un-div-inv N/A

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

   4. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. clear-num N/A

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

   17. associate-/l* N/A

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

   18. sub-div N/A

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

   19. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

Alternative 6: 76.7% accurate, 2.0× 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

real(8) function code(b, x)
    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. Add Preprocessing
3. 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. un-div-inv N/A

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

   4. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. clear-num N/A

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

   17. associate-/l* N/A

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

   18. sub-div N/A

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

   19. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in B around 0

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

   1. lower--.f64 76.4

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

9. Applied rewrites 76.4%

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

10. Final simplification 76.4%

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

Alternative 7: 50.0% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(N[(N[(0.3333333333333333 * x), $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. Add Preprocessing
3. 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. un-div-inv N/A

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

   4. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. clear-num N/A

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

   17. associate-/l* N/A

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

   18. sub-div N/A

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

   19. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 48.2

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, 1 - x, 0.5 \cdot x\right), \color{blue}{B \cdot B}, 1\right) - x}{B} \]

9. Applied rewrites 48.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, 1 - x, 0.5 \cdot x\right), B \cdot B, 1\right) - x}{B}} \]

10. Taylor expanded in x around inf

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

12. Applied rewrites 48.3%

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

Alternative 8: 48.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -17500000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -17500000.0) (not (<= x 1.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -17500000.0) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-17500000.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -17500000.0) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -17500000.0) or not (x <= 1.0):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -17500000.0) || !(x <= 1.0))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -17500000.0) || ~((x <= 1.0)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -17500000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e7 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 48.0%

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

   if -1.75e7 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.5%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -17500000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.0% accurate, 15.5× 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

real(8) function code(b, x)
    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. Add Preprocessing

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 48.3

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

5. Applied rewrites 48.3%

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

Alternative 10: 25.9% accurate, 19.4× 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

real(8) function code(b, x)
    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. Add Preprocessing

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 48.3

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

5. Applied rewrites 48.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 26.5%

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

Reproduce

herbie shell --seed 2024321 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))