VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 9.5s

Alternatives: 11

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. cancel-sign-sub-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*r/ 99.7%

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

   7. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.7%

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

   1. associate-/l* 99.6%

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

   2. un-div-inv 99.6%

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

   3. associate-/r* 99.6%

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

   4. frac-sub 99.6%

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

   5. *-un-lft-identity 99.6%

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

   6. un-div-inv 99.6%

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

   7. tan-quot 99.7%

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

6. Applied egg-rr 99.7%

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

   1. div-sub 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. times-frac 99.6%

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

   4. /-rgt-identity 99.6%

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

   5. times-frac 83.6%

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

   6. *-commutative 83.6%

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

   7. times-frac 99.7%

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

   8. *-inverses 99.7%

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

   9. *-lft-identity 99.7%

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

8. Simplified 99.7%

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

   1. sub-neg 99.7%

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

   2. div-inv 99.7%

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

   3. frac-times 99.7%

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

   4. metadata-eval 99.7%

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

10. Applied egg-rr 99.7%

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

    1. unsub-neg 99.7%

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

    2. associate-/r* 99.7%

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

    3. div-sub 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 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 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. cancel-sign-sub-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*r/ 99.7%

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

   7. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 3: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -420000 \lor \neg \left(x \leq 4 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -420000.0) (not (<= x 4e-10)))
   (/ (- 1.0 x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -420000.0) || !(x <= 4e-10)) {
		tmp = (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 <= (-420000.0d0)) .or. (.not. (x <= 4d-10))) then
        tmp = (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 <= -420000.0) || !(x <= 4e-10)) {
		tmp = (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 <= -420000.0) or not (x <= 4e-10):
		tmp = (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 <= -420000.0) || !(x <= 4e-10))
		tmp = Float64(Float64(1.0 - 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 <= -420000.0) || ~((x <= 4e-10)))
		tmp = (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, -420000.0], N[Not[LessEqual[x, 4e-10]], $MachinePrecision]], N[(N[(1.0 - 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 < -4.2e5 or 4.00000000000000015e-10 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. associate-/l* 99.6%

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

      2. un-div-inv 99.5%

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

      3. associate-/r* 99.6%

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

      4. frac-sub 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. un-div-inv 99.5%

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

      7. tan-quot 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-sub 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. times-frac 99.6%

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

      4. /-rgt-identity 99.6%

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

      5. times-frac 94.2%

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

      6. *-commutative 94.2%

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

      7. times-frac 99.7%

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

      8. *-inverses 99.7%

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

      9. *-lft-identity 99.7%

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

   8. Simplified 99.7%

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

      1. sub-neg 99.7%

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

      2. div-inv 99.7%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. unsub-neg 99.7%

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

       2. associate-/r* 99.7%

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

       3. div-sub 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in B around 0 99.2%

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

   if -4.2e5 < x < 4.00000000000000015e-10

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

      2. +-commutative 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. div-sub 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -420000 \lor \neg \left(x \leq 4 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -420000:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -420000.0)
   (- (/ 1.0 B) (/ x (tan B)))
   (if (<= x 4e-10) (/ (- 1.0 x) (sin B)) (/ (- 1.0 x) (tan B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -420000.0) {
		tmp = (1.0 / B) - (x / tan(B));
	} else if (x <= 4e-10) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = (1.0 - x) / tan(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 <= (-420000.0d0)) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else if (x <= 4d-10) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = (1.0d0 - x) / tan(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -420000.0) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else if (x <= 4e-10) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = (1.0 - x) / Math.tan(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -420000.0:
		tmp = (1.0 / B) - (x / math.tan(B))
	elif x <= 4e-10:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = (1.0 - x) / math.tan(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -420000.0)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	elseif (x <= 4e-10)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(1.0 - x) / tan(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -420000.0)
		tmp = (1.0 / B) - (x / tan(B));
	elseif (x <= 4e-10)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = (1.0 - x) / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -420000.0], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-10], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2e5

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r/ 99.6%

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

      7. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 99.3%

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

   if -4.2e5 < x < 4.00000000000000015e-10

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

      2. +-commutative 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. div-sub 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 99.4%

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

   if 4.00000000000000015e-10 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

      2. +-commutative 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. associate-/l* 99.7%

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

      2. un-div-inv 99.6%

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

      3. associate-/r* 99.6%

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

      4. frac-sub 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. un-div-inv 99.5%

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

      7. tan-quot 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-sub 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. times-frac 99.7%

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

      4. /-rgt-identity 99.7%

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

      5. times-frac 94.9%

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

      6. *-commutative 94.9%

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

      7. times-frac 99.7%

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

      8. *-inverses 99.7%

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

      9. *-lft-identity 99.7%

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

   8. Simplified 99.7%

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

      1. sub-neg 99.7%

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

      2. div-inv 99.7%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. unsub-neg 99.7%

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

       2. associate-/r* 99.7%

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

       3. div-sub 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in B around 0 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -420000:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]

Alternative 5: 74.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -0.021 \lor \neg \left(B \leq 0.13\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + x \cdot \left(B \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -0.021) (not (<= B 0.13)))
   (/ 1.0 (sin B))
   (+ (/ (- 1.0 x) B) (* x (* B 0.3333333333333333)))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -0.021) || !(B <= 0.13)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = ((1.0 - x) / B) + (x * (B * 0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((b <= (-0.021d0)) .or. (.not. (b <= 0.13d0))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = ((1.0d0 - x) / b) + (x * (b * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -0.021) || !(B <= 0.13)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = ((1.0 - x) / B) + (x * (B * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -0.021) or not (B <= 0.13):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = ((1.0 - x) / B) + (x * (B * 0.3333333333333333))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -0.021) || !(B <= 0.13))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(Float64(1.0 - x) / B) + Float64(x * Float64(B * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -0.021) || ~((B <= 0.13)))
		tmp = 1.0 / sin(B);
	else
		tmp = ((1.0 - x) / B) + (x * (B * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -0.021], N[Not[LessEqual[B, 0.13]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -0.0210000000000000013 or 0.13 < B

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. distribute-lft-neg-in 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 41.2%

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

   if -0.0210000000000000013 < B < 0.13

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-neg-in 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 99.9%

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

   5. Taylor expanded in B around 0 99.9%

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

      1. neg-mul-1 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. sub-neg 99.9%

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

      6. div-sub 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B} + x \cdot \left(B \cdot 0.3333333333333333\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.021 \lor \neg \left(B \leq 0.13\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + x \cdot \left(B \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 6: 76.8% 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 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. cancel-sign-sub-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*r/ 99.7%

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

   7. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. mul-1-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. div-sub 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in B around 0 74.5%

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

9. Final simplification 74.5%

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

Alternative 7: 51.5% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. distribute-lft-neg-in 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in B around 0 81.9%

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

5. Taylor expanded in B around 0 55.4%

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

   1. neg-mul-1 55.4%

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

   2. +-commutative 55.4%

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

   3. associate-+r+ 55.4%

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

   4. +-commutative 55.4%

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

   5. sub-neg 55.4%

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

   6. div-sub 55.4%

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

   7. *-commutative 55.4%

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

   8. *-commutative 55.4%

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

   9. associate-*l* 55.4%

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

7. Simplified 55.4%

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

8. Final simplification 55.4%

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

Alternative 8: 51.3% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{B} + B \cdot 0.16666666666666666 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = ((1.0d0 - x) / b) + (b * 0.16666666666666666d0)
end function

Java

public static double code(double B, double x) {
	return ((1.0 - x) / B) + (B * 0.16666666666666666);
}

Python

def code(B, x):
	return ((1.0 - x) / B) + (B * 0.16666666666666666)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. distribute-lft-neg-in 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in B around 0 55.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right)} \]
5. Step-by-step derivation

   1. +-commutative 55.2%

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

   2. mul-1-neg 55.2%

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

   3. sub-neg 55.2%

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

   4. associate--l+ 55.2%

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

   5. *-commutative 55.2%

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

   6. div-sub 55.3%

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

6. Simplified 55.3%

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

7. Taylor expanded in x around 0 55.3%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot B} + \frac{1 - x}{B} \]
8. Step-by-step derivation

   1. *-commutative 55.3%

      \[\leadsto \color{blue}{B \cdot 0.16666666666666666} + \frac{1 - x}{B} \]

9. Simplified 55.3%

   \[\leadsto \color{blue}{B \cdot 0.16666666666666666} + \frac{1 - x}{B} \]

10. Final simplification 55.3%

    \[\leadsto \frac{1 - x}{B} + B \cdot 0.16666666666666666 \]

Alternative 9: 49.7% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.1 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.1e+15))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.1e+15)) {
		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 <= (-1.0d0)) .or. (.not. (x <= 1.1d+15))) 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 <= -1.0) || !(x <= 1.1e+15)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.1e+15):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.1e+15))
		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 <= -1.0) || ~((x <= 1.1e+15)))
		tmp = -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, 1.1e+15]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.1e15 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 51.8%

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

      1. neg-mul-1 51.8%

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

      2. sub-neg 51.8%

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

   6. Simplified 51.8%

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

   7. Taylor expanded in x around inf 51.4%

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

      1. neg-mul-1 51.4%

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

      2. distribute-neg-frac 51.4%

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

   9. Simplified 51.4%

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

   if -1 < x < 1.1e15

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 59.3%

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

      1. neg-mul-1 59.3%

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

      2. sub-neg 59.3%

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

   6. Simplified 59.3%

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

   7. Taylor expanded in x around 0 57.8%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.1 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 10: 51.1% 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 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. distribute-lft-neg-in 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in B around 0 55.2%

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

   1. neg-mul-1 55.2%

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

   2. sub-neg 55.2%

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

6. Simplified 55.2%

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

7. Final simplification 55.2%

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

Alternative 11: 26.2% 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 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. distribute-lft-neg-in 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in B around 0 55.2%

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

   1. neg-mul-1 55.2%

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

   2. sub-neg 55.2%

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

6. Simplified 55.2%

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

7. Taylor expanded in x around 0 27.7%

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

8. Final simplification 27.7%

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

Reproduce

herbie shell --seed 2023311 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))