VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.9s

Alternatives: 10

Speedup: 0.7×

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 99.8%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.45) (not (<= x 1.25)))
   (+ (/ -1.0 (/ (tan B) x)) (pow B -1.0))
   (+ (- (/ x B)) (pow (sin B) -1.0))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.45) || !(x <= 1.25)) {
		tmp = (-1.0 / (tan(B) / x)) + pow(B, -1.0);
	} else {
		tmp = -(x / B) + pow(sin(B), -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 <= (-1.45d0)) .or. (.not. (x <= 1.25d0))) then
        tmp = ((-1.0d0) / (tan(b) / x)) + (b ** (-1.0d0))
    else
        tmp = -(x / b) + (sin(b) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.45) || !(x <= 1.25)) {
		tmp = (-1.0 / (Math.tan(B) / x)) + Math.pow(B, -1.0);
	} else {
		tmp = -(x / B) + Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.45) or not (x <= 1.25):
		tmp = (-1.0 / (math.tan(B) / x)) + math.pow(B, -1.0)
	else:
		tmp = -(x / B) + math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.45) || !(x <= 1.25))
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + (B ^ -1.0));
	else
		tmp = Float64(Float64(-Float64(x / B)) + (sin(B) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.45) || ~((x <= 1.25)))
		tmp = (-1.0 / (tan(B) / x)) + (B ^ -1.0);
	else
		tmp = -(x / B) + (sin(B) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.45], N[Not[LessEqual[x, 1.25]], $MachinePrecision]], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 1.25 < x

   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-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 98.4

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

   9. Applied rewrites 98.4%

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

   if -1.44999999999999996 < x < 1.25

   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

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

      1. lower-/.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;\frac{-1}{\tan B} \cdot x + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.45) (not (<= x 0.58)))
   (+ (* (/ -1.0 (tan B)) x) (pow B -1.0))
   (+ (- (/ x B)) (pow (sin B) -1.0))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.45) || !(x <= 0.58)) {
		tmp = ((-1.0 / tan(B)) * x) + pow(B, -1.0);
	} else {
		tmp = -(x / B) + pow(sin(B), -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 <= (-1.45d0)) .or. (.not. (x <= 0.58d0))) then
        tmp = (((-1.0d0) / tan(b)) * x) + (b ** (-1.0d0))
    else
        tmp = -(x / b) + (sin(b) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.45) || !(x <= 0.58)) {
		tmp = ((-1.0 / Math.tan(B)) * x) + Math.pow(B, -1.0);
	} else {
		tmp = -(x / B) + Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.45) or not (x <= 0.58):
		tmp = ((-1.0 / math.tan(B)) * x) + math.pow(B, -1.0)
	else:
		tmp = -(x / B) + math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.45) || !(x <= 0.58))
		tmp = Float64(Float64(Float64(-1.0 / tan(B)) * x) + (B ^ -1.0));
	else
		tmp = Float64(Float64(-Float64(x / B)) + (sin(B) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.45) || ~((x <= 0.58)))
		tmp = ((-1.0 / tan(B)) * x) + (B ^ -1.0);
	else
		tmp = -(x / B) + (sin(B) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.45], N[Not[LessEqual[x, 0.58]], $MachinePrecision]], N[(N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 0.57999999999999996 < x

   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-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 98.3

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

   9. Applied rewrites 98.3%

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

   if -1.44999999999999996 < x < 0.57999999999999996

   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 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;\frac{-1}{\tan B} \cdot x + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.45) (not (<= x 1.25)))
   (+ (/ (- x) (tan B)) (pow B -1.0))
   (+ (- (/ x B)) (pow (sin B) -1.0))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.45) || !(x <= 1.25)) {
		tmp = (-x / tan(B)) + pow(B, -1.0);
	} else {
		tmp = -(x / B) + pow(sin(B), -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 <= (-1.45d0)) .or. (.not. (x <= 1.25d0))) then
        tmp = (-x / tan(b)) + (b ** (-1.0d0))
    else
        tmp = -(x / b) + (sin(b) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.45) || !(x <= 1.25)) {
		tmp = (-x / Math.tan(B)) + Math.pow(B, -1.0);
	} else {
		tmp = -(x / B) + Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.45) or not (x <= 1.25):
		tmp = (-x / math.tan(B)) + math.pow(B, -1.0)
	else:
		tmp = -(x / B) + math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.45) || !(x <= 1.25))
		tmp = Float64(Float64(Float64(-x) / tan(B)) + (B ^ -1.0));
	else
		tmp = Float64(Float64(-Float64(x / B)) + (sin(B) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.45) || ~((x <= 1.25)))
		tmp = (-x / tan(B)) + (B ^ -1.0);
	else
		tmp = -(x / B) + (sin(B) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.45], N[Not[LessEqual[x, 1.25]], $MachinePrecision]], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 1.25 < x

   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-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-cos.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 98.4

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

   9. Applied rewrites 98.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r/ N/A

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

       4. lift-sin.f64 N/A

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

       5. lift-cos.f64 N/A

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

       6. tan-quot N/A

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

       7. lift-tan.f64 N/A

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

       8. lower-/.f64 98.5

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

   11. Applied rewrites 98.5%

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

   if -1.44999999999999996 < x < 1.25

   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

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

      1. lower-/.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+15} \lor \neg \left(x \leq 5.1 \cdot 10^{+36}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + 0.16666666666666666 \cdot B\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -5.2e+15) (not (<= x 5.1e+36)))
   (+ (* x (/ -1.0 (tan B))) (* 0.16666666666666666 B))
   (+ (- (/ x B)) (pow (sin B) -1.0))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -5.2e+15) || !(x <= 5.1e+36)) {
		tmp = (x * (-1.0 / tan(B))) + (0.16666666666666666 * B);
	} else {
		tmp = -(x / B) + pow(sin(B), -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 <= (-5.2d+15)) .or. (.not. (x <= 5.1d+36))) then
        tmp = (x * ((-1.0d0) / tan(b))) + (0.16666666666666666d0 * b)
    else
        tmp = -(x / b) + (sin(b) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -5.2e+15) || !(x <= 5.1e+36)) {
		tmp = (x * (-1.0 / Math.tan(B))) + (0.16666666666666666 * B);
	} else {
		tmp = -(x / B) + Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -5.2e+15) or not (x <= 5.1e+36):
		tmp = (x * (-1.0 / math.tan(B))) + (0.16666666666666666 * B)
	else:
		tmp = -(x / B) + math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -5.2e+15) || !(x <= 5.1e+36))
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(0.16666666666666666 * B));
	else
		tmp = Float64(Float64(-Float64(x / B)) + (sin(B) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -5.2e+15) || ~((x <= 5.1e+36)))
		tmp = (x * (-1.0 / tan(B))) + (0.16666666666666666 * B);
	else
		tmp = -(x / B) + (sin(B) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -5.2e+15], N[Not[LessEqual[x, 5.1e+36]], $MachinePrecision]], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * B), $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2e15 or 5.09999999999999973e36 < x

   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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 77.9%

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

   if -5.2e15 < x < 5.09999999999999973e36

   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 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+15} \lor \neg \left(x \leq 5.1 \cdot 10^{+36}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + 0.16666666666666666 \cdot B\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.48:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), B \cdot B, 0.019444444444444445\right)\right), B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + 0.16666666666666666 \cdot B\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.48)
   (/
    (fma
     (fma
      (fma
       0.022222222222222223
       x
       (fma
        (fma x 0.0021164021164021165 0.00205026455026455)
        (* B B)
        0.019444444444444445))
      (* B B)
      (fma 0.3333333333333333 x 0.16666666666666666))
     (* B B)
     (- 1.0 x))
    B)
   (+ (* x (/ -1.0 (tan B))) (* 0.16666666666666666 B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.48) {
		tmp = fma(fma(fma(0.022222222222222223, x, fma(fma(x, 0.0021164021164021165, 0.00205026455026455), (B * B), 0.019444444444444445)), (B * B), fma(0.3333333333333333, x, 0.16666666666666666)), (B * B), (1.0 - x)) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (0.16666666666666666 * B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.48)
		tmp = Float64(fma(fma(fma(0.022222222222222223, x, fma(fma(x, 0.0021164021164021165, 0.00205026455026455), Float64(B * B), 0.019444444444444445)), Float64(B * B), fma(0.3333333333333333, x, 0.16666666666666666)), Float64(B * B), Float64(1.0 - x)) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(0.16666666666666666 * B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.48], N[(N[(N[(N[(0.022222222222222223 * x + N[(N[(x * 0.0021164021164021165 + 0.00205026455026455), $MachinePrecision] * N[(B * B), $MachinePrecision] + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.47999999999999998

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{31}{15120} + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   7. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), B \cdot B, 0.019444444444444445\right)\right), B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B}} \]

   if 0.47999999999999998 < B

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 18.0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 29.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + 0.16666666666666666 \cdot \color{blue}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.48:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), B \cdot B, 0.019444444444444445\right)\right), B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + 0.16666666666666666 \cdot B\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -850 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -850.0) (not (<= x 1.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -850.0) || !(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 <= (-850.0d0)) .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 <= -850.0) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -850.0) 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 <= -850.0) || !(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 <= -850.0) || ~((x <= 1.0)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -850.0], 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 < -850 or 1 < x

   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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.0%

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

   if -850 < 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 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.3%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -850 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.2% 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 99.8%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 56.8

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

5. Applied rewrites 56.8%

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

Alternative 9: 27.4% 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 99.8%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 56.8

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

5. Applied rewrites 56.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 29.6%

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

Alternative 10: 3.2% accurate, 38.8× speedup

Math

\[\begin{array}{l} \\ 0.16666666666666666 \cdot B \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (* 0.16666666666666666 B))

C

double code(double B, double x) {
	return 0.16666666666666666 * B;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 0.16666666666666666d0 * b
end function

Java

public static double code(double B, double x) {
	return 0.16666666666666666 * B;
}

Python

def code(B, x):
	return 0.16666666666666666 * B

Julia

function code(B, x)
	return Float64(0.16666666666666666 * B)
end

MATLAB

function tmp = code(B, x)
	tmp = 0.16666666666666666 * B;
end

Wolfram

code[B_, x_] := N[(0.16666666666666666 * B), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 56.5

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

7. Applied rewrites 56.5%

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

8. Taylor expanded in B around inf

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

10. Applied rewrites 2.7%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right) \cdot \color{blue}{B} \]

11. Taylor expanded in x around 0

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

13. Applied rewrites 3.2%

    \[\leadsto 0.16666666666666666 \cdot B \]

14. Final simplification 3.2%

    \[\leadsto 0.16666666666666666 \cdot B \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))