VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 9.2s

Alternatives: 12

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. div-inv 99.7%

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

   2. clear-num 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*r/ 99.7%

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

   10. *-rgt-identity 99.7%

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

   11. tan-neg 99.7%

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

   12. distribute-neg-frac2 99.7%

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

   13. distribute-neg-frac 99.7%

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

   14. remove-double-neg 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. Add Preprocessing

5. Final simplification 99.7%

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

Alternative 3: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \lor \neg \left(x \leq 7.5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.7) (not (<= x 7.5e-7)))
   (/ (- 1.0 x) (tan B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.7) || !(x <= 7.5e-7)) {
		tmp = (1.0 - x) / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / 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 <= (-2.7d0)) .or. (.not. (x <= 7.5d-7))) then
        tmp = (1.0d0 - x) / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -2.7) || !(x <= 7.5e-7)) {
		tmp = (1.0 - x) / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.7) or not (x <= 7.5e-7):
		tmp = (1.0 - x) / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.7) || !(x <= 7.5e-7))
		tmp = Float64(Float64(1.0 - x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -2.7) || ~((x <= 7.5e-7)))
		tmp = (1.0 - x) / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.7], N[Not[LessEqual[x, 7.5e-7]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002 or 7.5000000000000002e-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 99.6%

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

      2. div-inv 99.7%

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

      3. sub-neg 99.7%

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

      4. clear-num 99.7%

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

      5. frac-sub 89.2%

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

      6. *-un-lft-identity 89.2%

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

      7. *-commutative 89.2%

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

      8. *-un-lft-identity 89.2%

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

   4. Applied egg-rr 89.2%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in B around 0 98.3%

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

      1. associate-*l/ 98.4%

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

      2. clear-num 98.4%

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

      3. *-commutative 98.4%

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

   9. Applied egg-rr 98.4%

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

       1. *-rgt-identity 98.4%

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

       2. *-commutative 98.4%

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

       3. times-frac 98.2%

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

       4. associate-/l/ 98.3%

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

       5. remove-double-div 98.4%

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

       6. associate-/r/ 98.3%

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

       7. associate-*l/ 98.4%

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

       8. metadata-eval 98.4%

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

       9. sub-neg 98.4%

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

       10. sub-neg 98.4%

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

       11. metadata-eval 98.4%

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

       12. distribute-rgt-in 98.4%

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

       13. neg-mul-1 98.4%

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

       14. unsub-neg 98.4%

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

       15. lft-mult-inverse 98.4%

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

   11. Simplified 98.4%

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

   if -2.7000000000000002 < x < 7.5000000000000002e-7

   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 99.3%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. sub-neg 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \lor \neg \left(x \leq 7.5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-12} \lor \neg \left(x \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -9.5e-12) (not (<= x 4e-8)))
   (/ (- 1.0 x) (tan B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -9.5e-12) || !(x <= 4e-8)) {
		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 <= (-9.5d-12)) .or. (.not. (x <= 4d-8))) 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 <= -9.5e-12) || !(x <= 4e-8)) {
		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 <= -9.5e-12) or not (x <= 4e-8):
		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 <= -9.5e-12) || !(x <= 4e-8))
		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 <= -9.5e-12) || ~((x <= 4e-8)))
		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, -9.5e-12], N[Not[LessEqual[x, 4e-8]], $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 < -9.4999999999999995e-12 or 4.0000000000000001e-8 < 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 99.6%

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

      2. div-inv 99.7%

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

      3. sub-neg 99.7%

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

      4. clear-num 99.7%

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

      5. frac-sub 88.2%

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

      6. *-un-lft-identity 88.2%

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

      7. *-commutative 88.2%

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

      8. *-un-lft-identity 88.2%

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

   4. Applied egg-rr 88.2%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in B around 0 96.9%

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

      1. associate-*l/ 97.6%

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

      2. clear-num 97.6%

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

      3. *-commutative 97.6%

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

   9. Applied egg-rr 97.6%

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

       1. *-rgt-identity 97.6%

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

       2. *-commutative 97.6%

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

       3. times-frac 97.1%

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

       4. associate-/l/ 97.2%

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

       5. remove-double-div 97.3%

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

       6. associate-/r/ 97.6%

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

       7. associate-*l/ 97.6%

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

       8. metadata-eval 97.6%

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

       9. sub-neg 97.6%

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

       10. sub-neg 97.6%

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

       11. metadata-eval 97.6%

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

       12. distribute-rgt-in 97.6%

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

       13. neg-mul-1 97.6%

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

       14. unsub-neg 97.6%

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

       15. lft-mult-inverse 97.6%

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

   11. Simplified 97.6%

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

   if -9.4999999999999995e-12 < x < 4.0000000000000001e-8

   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 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-12} \lor \neg \left(x \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.35)
   (/ x (- (tan B)))
   (if (<= x 1.0) (/ 1.0 (sin B)) (* x (/ -1.0 (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.35) {
		tmp = x / -tan(B);
	} else if (x <= 1.0) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = x * (-1.0 / 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 <= (-1.35d0)) then
        tmp = x / -tan(b)
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = x * ((-1.0d0) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.35) {
		tmp = x / -Math.tan(B);
	} else if (x <= 1.0) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = x * (-1.0 / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.35:
		tmp = x / -math.tan(B)
	elif x <= 1.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = x * (-1.0 / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.35)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (x <= 1.0)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.35)
		tmp = x / -tan(B);
	elseif (x <= 1.0)
		tmp = 1.0 / sin(B);
	else
		tmp = x * (-1.0 / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.35], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. associate-/l* 96.7%

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

      3. distribute-rgt-neg-in 96.7%

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

      4. distribute-neg-frac2 96.7%

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

   5. Simplified 96.7%

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

      1. *-commutative 96.7%

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

      2. distribute-frac-neg2 96.7%

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

      3. clear-num 96.5%

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

      4. tan-quot 96.6%

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

      5. distribute-lft-neg-in 96.6%

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

      6. associate-/r/ 96.7%

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

      7. add-sqr-sqrt 42.1%

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

      8. sqrt-unprod 28.5%

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

      9. sqr-neg 28.5%

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

      10. sqrt-unprod 0.3%

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

      11. add-sqr-sqrt 0.5%

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

      12. clear-num 0.5%

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

      13. frac-2neg 0.5%

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

      14. tan-quot 0.5%

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

      15. distribute-frac-neg 0.5%

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

      16. add-sqr-sqrt 0.2%

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

      17. sqrt-unprod 47.6%

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

      18. sqr-neg 47.6%

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

      19. sqrt-unprod 48.8%

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

      20. add-sqr-sqrt 96.6%

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

   7. Applied egg-rr 96.7%

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

   if -1.3500000000000001 < x < 1

   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 98.0%

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

   if 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. associate-/l* 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

      4. distribute-neg-frac2 97.1%

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

   5. Simplified 97.1%

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

      1. distribute-frac-neg2 97.1%

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

      2. clear-num 97.0%

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

      3. tan-quot 97.1%

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

      4. neg-sub0 97.1%

         \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]

   7. Applied egg-rr 97.1%

      \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 97.1%

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

      2. distribute-neg-frac 97.1%

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

      3. metadata-eval 97.1%

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

   9. Simplified 97.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.3)
   (/ x (- (tan B)))
   (if (<= x 1.1) (/ 1.0 (sin B)) (/ -1.0 (/ (tan B) x)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = x / -tan(B);
	} else if (x <= 1.1) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = -1.0 / (tan(B) / x);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = x / -tan(b)
    else if (x <= 1.1d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (-1.0d0) / (tan(b) / x)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = x / -Math.tan(B);
	} else if (x <= 1.1) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = -1.0 / (Math.tan(B) / x);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.3:
		tmp = x / -math.tan(B)
	elif x <= 1.1:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = -1.0 / (math.tan(B) / x)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (x <= 1.1)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(-1.0 / Float64(tan(B) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = x / -tan(B);
	elseif (x <= 1.1)
		tmp = 1.0 / sin(B);
	else
		tmp = -1.0 / (tan(B) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.3], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 1.1], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. associate-/l* 96.7%

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

      3. distribute-rgt-neg-in 96.7%

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

      4. distribute-neg-frac2 96.7%

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

   5. Simplified 96.7%

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

      1. *-commutative 96.7%

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

      2. distribute-frac-neg2 96.7%

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

      3. clear-num 96.5%

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

      4. tan-quot 96.6%

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

      5. distribute-lft-neg-in 96.6%

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

      6. associate-/r/ 96.7%

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

      7. add-sqr-sqrt 42.1%

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

      8. sqrt-unprod 28.5%

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

      9. sqr-neg 28.5%

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

      10. sqrt-unprod 0.3%

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

      11. add-sqr-sqrt 0.5%

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

      12. clear-num 0.5%

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

      13. frac-2neg 0.5%

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

      14. tan-quot 0.5%

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

      15. distribute-frac-neg 0.5%

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

      16. add-sqr-sqrt 0.2%

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

      17. sqrt-unprod 47.6%

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

      18. sqr-neg 47.6%

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

      19. sqrt-unprod 48.8%

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

      20. add-sqr-sqrt 96.6%

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

   7. Applied egg-rr 96.7%

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

   if -1.30000000000000004 < x < 1.1000000000000001

   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 98.0%

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

   if 1.1000000000000001 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. associate-/l* 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

      4. distribute-neg-frac2 97.1%

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

   5. Simplified 97.1%

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

      1. *-commutative 97.1%

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

      2. distribute-frac-neg2 97.1%

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

      3. clear-num 97.0%

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

      4. tan-quot 97.1%

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

      5. distribute-lft-neg-in 97.1%

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

      6. associate-/r/ 97.1%

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

      7. distribute-neg-frac 97.1%

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

      8. metadata-eval 97.1%

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

   7. Applied egg-rr 97.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.36) (not (<= x 1.0))) (/ x (- (tan B))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.36) || !(x <= 1.0)) {
		tmp = 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 <= (-1.36d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 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 <= -1.36) || !(x <= 1.0)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.36) or not (x <= 1.0):
		tmp = x / -math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.36) || !(x <= 1.0))
		tmp = Float64(x / Float64(-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 <= -1.36) || ~((x <= 1.0)))
		tmp = x / -tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.36], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3600000000000001 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 96.8%

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

      1. mul-1-neg 96.8%

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

      2. associate-/l* 96.9%

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

      3. distribute-rgt-neg-in 96.9%

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

      4. distribute-neg-frac2 96.9%

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

   5. Simplified 96.9%

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

      1. *-commutative 96.9%

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

      2. distribute-frac-neg2 96.9%

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

      3. clear-num 96.8%

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

      4. tan-quot 96.9%

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

      5. distribute-lft-neg-in 96.9%

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

      6. associate-/r/ 96.9%

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

      7. add-sqr-sqrt 43.7%

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

      8. sqrt-unprod 27.0%

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

      9. sqr-neg 27.0%

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

      10. sqrt-unprod 0.3%

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

      11. add-sqr-sqrt 0.5%

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

      12. clear-num 0.5%

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

      13. frac-2neg 0.5%

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

      14. tan-quot 0.5%

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

      15. distribute-frac-neg 0.5%

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

      16. add-sqr-sqrt 0.2%

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

      17. sqrt-unprod 51.1%

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

      18. sqr-neg 51.1%

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

      19. sqrt-unprod 53.1%

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

      20. add-sqr-sqrt 96.8%

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

   7. Applied egg-rr 96.9%

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

   if -1.3600000000000001 < x < 1

   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 98.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.36 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 8.5e-6) (- (/ 1.0 B) (/ x B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 8.5e-6) {
		tmp = (1.0 / B) - (x / 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 <= 8.5d-6) then
        tmp = (1.0d0 / b) - (x / 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 <= 8.5e-6) {
		tmp = (1.0 / B) - (x / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 8.5e-6:
		tmp = (1.0 / B) - (x / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 8.5e-6)
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 8.5e-6)
		tmp = (1.0 / B) - (x / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 8.5e-6], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.4999999999999999e-6

   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 67.1%

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

      1. div-sub 67.1%

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

   5. Applied egg-rr 67.1%

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

   if 8.4999999999999999e-6 < B

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 46.9%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1850\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1850.0))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1850.0)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1850.0d0))) then
        tmp = x / -b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1850.0)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1850.0):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1850.0))
		tmp = Float64(x / Float64(-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 <= 1850.0)))
		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, 1850.0]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1850 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 42.9%

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

   4. Taylor expanded in x around inf 41.5%

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

      1. associate-*r/ 41.5%

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

      2. neg-mul-1 41.5%

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

   6. Simplified 41.5%

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

   if -1 < x < 1850

   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 57.1%

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

   4. Taylor expanded in x around 0 55.8%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1850\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.5% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 50.2%

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

   1. div-sub 50.2%

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

5. Applied egg-rr 50.2%

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

6. Final simplification 50.2%

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

Alternative 11: 50.5% 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 99.7%

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

3. Taylor expanded in B around 0 50.2%

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

4. Final simplification 50.2%

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

Alternative 12: 26.1% 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 99.7%

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

3. Taylor expanded in B around 0 50.2%

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

4. Taylor expanded in x around 0 30.4%

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

5. Final simplification 30.4%

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

Reproduce

herbie shell --seed 2024075 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))