VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 17.4s

Alternatives: 9

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sin B}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \left(\color{blue}{x} \cdot \frac{1}{\tan B}\right)\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(x \cdot \frac{1}{\tan B}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x \cdot 1}{\color{blue}{\tan B}}\right)\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{\tan \color{blue}{B}}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{\tan B}\right)\right) \]

   9. tan-lowering-tan.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

3. Simplified 99.8%

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

Alternative 2: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-5} \lor \neg \left(x \leq 5.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.05e-5) (not (<= x 5.8e-7)))
   (/ (- 1.0 x) (tan B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.05e-5) || !(x <= 5.8e-7)) {
		tmp = (1.0 - x) / tan(B);
	} else {
		tmp = 1.0 / 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 <= (-1.05d-5)) .or. (.not. (x <= 5.8d-7))) then
        tmp = (1.0d0 - x) / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.05e-5) || !(x <= 5.8e-7)) {
		tmp = (1.0 - x) / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.05e-5) or not (x <= 5.8e-7):
		tmp = (1.0 - x) / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.05e-5) || !(x <= 5.8e-7))
		tmp = Float64(Float64(1.0 - x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.05e-5) || ~((x <= 5.8e-7)))
		tmp = (1.0 - x) / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.05e-5], N[Not[LessEqual[x, 5.8e-7]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04999999999999994e-5 or 5.7999999999999995e-7 < 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. +-commutative N/A

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

      2. div-inv N/A

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

      3. sub-neg N/A

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

      4. frac-sub N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot \tan B - \sin B \cdot x}{\sin B}\right), \color{blue}{\tan B}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot \tan B - \sin B \cdot x\right), \sin B\right), \tan \color{blue}{B}\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\tan B - \sin B \cdot x\right), \sin B\right), \tan B\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\tan B, \left(\sin B \cdot x\right)\right), \sin B\right), \tan B\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \left(\sin B \cdot x\right)\right), \sin B\right), \tan B\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \left(x \cdot \sin B\right)\right), \sin B\right), \tan B\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \sin B\right)\right), \sin B\right), \tan B\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right)\right), \sin B\right), \tan B\right) \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \tan B\right) \]

      15. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{tan.f64}\left(B\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - x\right)}, \mathsf{tan.f64}\left(B\right)\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 97.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{tan.f64}\left(\color{blue}{B}\right)\right) \]

   7. Simplified 97.8%

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

   if -1.04999999999999994e-5 < x < 5.7999999999999995e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 99.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-5} \lor \neg \left(x \leq 5.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -28 \lor \neg \left(x \leq 126000\right):\\ \;\;\;\;\frac{0 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -28.0) (not (<= x 126000.0)))
   (/ (- 0.0 x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -28.0) || !(x <= 126000.0)) {
		tmp = (0.0 - 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 <= (-28.0d0)) .or. (.not. (x <= 126000.0d0))) then
        tmp = (0.0d0 - 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 <= -28.0) || !(x <= 126000.0)) {
		tmp = (0.0 - x) / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -28.0) or not (x <= 126000.0):
		tmp = (0.0 - x) / math.tan(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -28.0) || !(x <= 126000.0))
		tmp = Float64(Float64(0.0 - 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 <= -28.0) || ~((x <= 126000.0)))
		tmp = (0.0 - x) / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -28.0], N[Not[LessEqual[x, 126000.0]], $MachinePrecision]], N[(N[(0.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -28 or 126000 < 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. +-commutative N/A

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

      2. div-inv N/A

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

      3. sub-neg N/A

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

      4. frac-sub N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot \tan B - \sin B \cdot x}{\sin B}\right), \color{blue}{\tan B}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot \tan B - \sin B \cdot x\right), \sin B\right), \tan \color{blue}{B}\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\tan B - \sin B \cdot x\right), \sin B\right), \tan B\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\tan B, \left(\sin B \cdot x\right)\right), \sin B\right), \tan B\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \left(\sin B \cdot x\right)\right), \sin B\right), \tan B\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \left(x \cdot \sin B\right)\right), \sin B\right), \tan B\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \sin B\right)\right), \sin B\right), \tan B\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right)\right), \sin B\right), \tan B\right) \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \tan B\right) \]

      15. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{tan.f64}\left(B\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, \mathsf{tan.f64}\left(B\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \mathsf{tan.f64}\left(\color{blue}{B}\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), \mathsf{tan.f64}\left(\color{blue}{B}\right)\right) \]

      3. --lowering--.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{tan.f64}\left(\color{blue}{B}\right)\right) \]

   7. Simplified 97.6%

      \[\leadsto \frac{\color{blue}{0 - x}}{\tan B} \]

   if -28 < x < 126000

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sin B}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \left(\color{blue}{x} \cdot \frac{1}{\tan B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x \cdot 1}{\color{blue}{\tan B}}\right)\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{\tan \color{blue}{B}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{\tan B}\right)\right) \]

      9. tan-lowering-tan.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   3. Simplified 99.9%

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

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

      1. associate-*l/ N/A

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

      2. sub-div N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

      7. sin-lowering-sin.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \color{blue}{x}\right), \mathsf{sin.f64}\left(B\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 99.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -28 \lor \neg \left(x \leq 126000\right):\\ \;\;\;\;\frac{0 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{0 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.28) (not (<= x 1.0))) (/ (- 0.0 x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.28) || !(x <= 1.0)) {
		tmp = (0.0 - x) / tan(B);
	} else {
		tmp = 1.0 / 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 <= (-1.28d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (0.0d0 - x) / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.28) || !(x <= 1.0)) {
		tmp = (0.0 - x) / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.28) or not (x <= 1.0):
		tmp = (0.0 - x) / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.28) || !(x <= 1.0))
		tmp = Float64(Float64(0.0 - x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.28) || ~((x <= 1.0)))
		tmp = (0.0 - x) / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.28], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(0.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.28000000000000003 or 1 < 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. +-commutative N/A

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

      2. div-inv N/A

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

      3. sub-neg N/A

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

      4. frac-sub N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot \tan B - \sin B \cdot x}{\sin B}\right), \color{blue}{\tan B}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot \tan B - \sin B \cdot x\right), \sin B\right), \tan \color{blue}{B}\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\tan B - \sin B \cdot x\right), \sin B\right), \tan B\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\tan B, \left(\sin B \cdot x\right)\right), \sin B\right), \tan B\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \left(\sin B \cdot x\right)\right), \sin B\right), \tan B\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \left(x \cdot \sin B\right)\right), \sin B\right), \tan B\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \sin B\right)\right), \sin B\right), \tan B\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right)\right), \sin B\right), \tan B\right) \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \tan B\right) \]

      15. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{tan.f64}\left(B\right), \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{tan.f64}\left(B\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, \mathsf{tan.f64}\left(B\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \mathsf{tan.f64}\left(\color{blue}{B}\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), \mathsf{tan.f64}\left(\color{blue}{B}\right)\right) \]

      3. --lowering--.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{tan.f64}\left(\color{blue}{B}\right)\right) \]

   7. Simplified 97.0%

      \[\leadsto \frac{\color{blue}{0 - x}}{\tan B} \]

   if -1.28000000000000003 < 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 x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 98.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{0 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.72:\\ \;\;\;\;\frac{\left(1 - x\right) + \left(B \cdot B\right) \cdot \left(x \cdot 0.3333333333333333 + \left(0.16666666666666666 + B \cdot \left(B \cdot \left(0.019444444444444445 + \left(x \cdot 0.022222222222222223 + \left(B \cdot B\right) \cdot \left(x \cdot 0.0021164021164021165 + 0.00205026455026455\right)\right)\right)\right)\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.72)
   (/
    (+
     (- 1.0 x)
     (*
      (* B B)
      (+
       (* x 0.3333333333333333)
       (+
        0.16666666666666666
        (*
         B
         (*
          B
          (+
           0.019444444444444445
           (+
            (* x 0.022222222222222223)
            (*
             (* B B)
             (+ (* x 0.0021164021164021165) 0.00205026455026455))))))))))
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.72) {
		tmp = ((1.0 - x) + ((B * B) * ((x * 0.3333333333333333) + (0.16666666666666666 + (B * (B * (0.019444444444444445 + ((x * 0.022222222222222223) + ((B * B) * ((x * 0.0021164021164021165) + 0.00205026455026455)))))))))) / B;
	} else {
		tmp = 1.0 / 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 (b <= 0.72d0) then
        tmp = ((1.0d0 - x) + ((b * b) * ((x * 0.3333333333333333d0) + (0.16666666666666666d0 + (b * (b * (0.019444444444444445d0 + ((x * 0.022222222222222223d0) + ((b * b) * ((x * 0.0021164021164021165d0) + 0.00205026455026455d0)))))))))) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 0.72) {
		tmp = ((1.0 - x) + ((B * B) * ((x * 0.3333333333333333) + (0.16666666666666666 + (B * (B * (0.019444444444444445 + ((x * 0.022222222222222223) + ((B * B) * ((x * 0.0021164021164021165) + 0.00205026455026455)))))))))) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.72:
		tmp = ((1.0 - x) + ((B * B) * ((x * 0.3333333333333333) + (0.16666666666666666 + (B * (B * (0.019444444444444445 + ((x * 0.022222222222222223) + ((B * B) * ((x * 0.0021164021164021165) + 0.00205026455026455)))))))))) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.72)
		tmp = Float64(Float64(Float64(1.0 - x) + Float64(Float64(B * B) * Float64(Float64(x * 0.3333333333333333) + Float64(0.16666666666666666 + Float64(B * Float64(B * Float64(0.019444444444444445 + Float64(Float64(x * 0.022222222222222223) + Float64(Float64(B * B) * Float64(Float64(x * 0.0021164021164021165) + 0.00205026455026455)))))))))) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 0.72)
		tmp = ((1.0 - x) + ((B * B) * ((x * 0.3333333333333333) + (0.16666666666666666 + (B * (B * (0.019444444444444445 + ((x * 0.022222222222222223) + ((B * B) * ((x * 0.0021164021164021165) + 0.00205026455026455)))))))))) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.72], N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(B * B), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] + N[(0.16666666666666666 + N[(B * N[(B * N[(0.019444444444444445 + N[(N[(x * 0.022222222222222223), $MachinePrecision] + N[(N[(B * B), $MachinePrecision] * N[(N[(x * 0.0021164021164021165), $MachinePrecision] + 0.00205026455026455), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.71999999999999997

   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

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

   5. Simplified 65.6%

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

   if 0.71999999999999997 < B

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 51.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 51.6%

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

Alternative 6: 50.8% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ (- 0.0 x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (0.0 - 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 <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (0.0d0 - 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 <= -1.0) || !(x <= 1.0)) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (0.0 - x) / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 45.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), B\right) \]

   5. Simplified 45.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 44.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 44.9%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-lowering-neg.f64 44.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(x\right), B\right) \]

   10. Applied egg-rr 44.9%

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

   if -1 < 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 x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 98.5%

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

   6. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 51.9%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

   8. Simplified 51.9%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.9% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = ((1.0d0 - x) + ((b * b) * (x * 0.3333333333333333d0))) / b
end function

Java

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

Python

def code(B, x):
	return ((1.0 - x) + ((B * B) * (x * 0.3333333333333333))) / B

Julia

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

MATLAB

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

Wolfram

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

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right) - x\right), B\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + \left(1 - x\right)\right), B\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 - x\right) + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), B\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(1 - x\right), \left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right)\right), B\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right)\right), B\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right)\right), B\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right)\right), B\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right)\right), B\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{3} \cdot x + \frac{1}{6}\right)\right)\right), B\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot x\right), \frac{1}{6}\right)\right)\right), B\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\left(x \cdot \frac{1}{3}\right), \frac{1}{6}\right)\right)\right), B\right) \]

   13. *-lowering-*.f64 49.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{1}{6}\right)\right)\right), B\right) \]

5. Simplified 49.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \color{blue}{\left(\frac{1}{3} \cdot x\right)}\right)\right), B\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(x \cdot \frac{1}{3}\right)\right)\right), B\right) \]

   2. *-lowering-*.f64 49.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), B\right) \]

8. Simplified 49.8%

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

Alternative 8: 51.9% accurate, 42.0× 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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - x\right), \color{blue}{B}\right) \]

   2. --lowering--.f64 49.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), B\right) \]

5. Simplified 49.4%

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

Alternative 9: 28.0% accurate, 70.0× 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 x around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

   2. sin-lowering-sin.f64 54.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

5. Simplified 54.5%

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

6. Taylor expanded in B around 0

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

   1. /-lowering-/.f64 29.3%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

8. Simplified 29.3%

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

Reproduce

herbie shell --seed 2024125 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))