VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.4s

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. distribute-lft-neg-in 99.8%

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

   2. +-commutative 99.8%

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

   3. cancel-sign-sub-inv 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -195000 \lor \neg \left(x \leq 45000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -195000.0) (not (<= x 45000.0)))
   (/ (- x) (tan B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -195000.0) || !(x <= 45000.0)) {
		tmp = -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 <= (-195000.0d0)) .or. (.not. (x <= 45000.0d0))) then
        tmp = -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 <= -195000.0) || !(x <= 45000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -195000.0) or not (x <= 45000.0):
		tmp = -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 <= -195000.0) || !(x <= 45000.0))
		tmp = Float64(Float64(-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 <= -195000.0) || ~((x <= 45000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -195000.0], N[Not[LessEqual[x, 45000.0]], $MachinePrecision]], N[((-x) / 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 < -195000 or 45000 < 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. 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. Step-by-step derivation

      1. distribute-rgt-neg-out 99.7%

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

      2. div-inv 99.8%

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

      3. clear-num 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in B around inf 99.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}} \]
   7. 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. associate-*r/ 99.7%

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

      4. sub-neg 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 98.7%

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

       1. mul-1-neg 98.7%

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

       2. distribute-frac-neg 98.7%

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

       3. distribute-rgt-neg-in 98.7%

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

   11. Simplified 98.7%

       \[\leadsto \color{blue}{\frac{x \cdot \left(-\cos B\right)}{\sin B}} \]
   12. Step-by-step derivation

       1. associate-/l* 98.6%

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

       2. add-sqr-sqrt 24.6%

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

       3. sqrt-unprod 25.0%

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

       4. sqr-neg 25.0%

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

       5. sqrt-unprod 0.2%

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

       6. add-sqr-sqrt 0.4%

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

       7. tan-quot 0.4%

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

       8. frac-2neg 0.4%

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

       9. div-inv 0.4%

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

       10. neg-mul-1 0.4%

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

       11. metadata-eval 0.4%

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

       12. tan-quot 0.4%

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

       13. times-frac 0.4%

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

       14. neg-mul-1 0.4%

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

       15. add-sqr-sqrt 0.1%

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

       16. sqrt-unprod 73.9%

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

       17. sqr-neg 73.9%

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

       18. sqrt-unprod 73.7%

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

       19. *-un-lft-identity 73.7%

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

       20. add-sqr-sqrt 98.5%

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

       21. tan-quot 98.7%

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

   13. Applied egg-rr 98.7%

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

       1. associate-*r/ 98.8%

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

       2. *-rgt-identity 98.8%

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

   15. Simplified 98.8%

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

   if -195000 < x < 45000

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. cancel-sign-sub-inv 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 97.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -195000 \lor \neg \left(x \leq 45000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8000000 \lor \neg \left(x \leq 13800\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -8000000.0) (not (<= x 13800.0)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -8000000.0) || !(x <= 13800.0)) {
		tmp = -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 <= (-8000000.0d0)) .or. (.not. (x <= 13800.0d0))) then
        tmp = -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 <= -8000000.0) || !(x <= 13800.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -8000000.0) or not (x <= 13800.0):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -8000000.0) || !(x <= 13800.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -8000000.0) || ~((x <= 13800.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -8000000.0], N[Not[LessEqual[x, 13800.0]], $MachinePrecision]], N[((-x) / 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 < -8e6 or 13800 < 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. 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. Step-by-step derivation

      1. distribute-rgt-neg-out 99.7%

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

      2. div-inv 99.8%

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

      3. clear-num 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in B around inf 99.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}} \]
   7. 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. associate-*r/ 99.7%

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

      4. sub-neg 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 98.7%

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

       1. mul-1-neg 98.7%

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

       2. distribute-frac-neg 98.7%

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

       3. distribute-rgt-neg-in 98.7%

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

   11. Simplified 98.7%

       \[\leadsto \color{blue}{\frac{x \cdot \left(-\cos B\right)}{\sin B}} \]
   12. Step-by-step derivation

       1. associate-/l* 98.6%

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

       2. add-sqr-sqrt 24.6%

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

       3. sqrt-unprod 25.0%

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

       4. sqr-neg 25.0%

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

       5. sqrt-unprod 0.2%

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

       6. add-sqr-sqrt 0.4%

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

       7. tan-quot 0.4%

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

       8. frac-2neg 0.4%

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

       9. div-inv 0.4%

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

       10. neg-mul-1 0.4%

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

       11. metadata-eval 0.4%

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

       12. tan-quot 0.4%

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

       13. times-frac 0.4%

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

       14. neg-mul-1 0.4%

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

       15. add-sqr-sqrt 0.1%

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

       16. sqrt-unprod 73.9%

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

       17. sqr-neg 73.9%

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

       18. sqrt-unprod 73.7%

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

       19. *-un-lft-identity 73.7%

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

       20. add-sqr-sqrt 98.5%

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

       21. tan-quot 98.7%

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

   13. Applied egg-rr 98.7%

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

       1. associate-*r/ 98.8%

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

       2. *-rgt-identity 98.8%

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

   15. Simplified 98.8%

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

   if -8e6 < x < 13800

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-rgt-neg-out 99.8%

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

      2. div-inv 99.8%

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

      3. clear-num 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in B around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. sub-neg 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around inf 99.8%

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

       1. div-sub 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in B around 0 97.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8000000 \lor \neg \left(x \leq 13800\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 4: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.52) (not (<= x 1.0))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.52) || !(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 <= (-1.52d0)) .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 <= -1.52) || !(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 <= -1.52) 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 <= -1.52) || !(x <= 1.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.52) || ~((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, -1.52], 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 < -1.52 or 1 < 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. 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. Step-by-step derivation

      1. distribute-rgt-neg-out 99.7%

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

      2. div-inv 99.8%

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

      3. clear-num 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in B around inf 99.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}} \]
   7. 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. associate-*r/ 99.7%

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

      4. sub-neg 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 97.6%

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

       1. mul-1-neg 97.6%

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

       2. distribute-frac-neg 97.6%

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

       3. distribute-rgt-neg-in 97.6%

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

   11. Simplified 97.6%

       \[\leadsto \color{blue}{\frac{x \cdot \left(-\cos B\right)}{\sin B}} \]
   12. Step-by-step derivation

       1. associate-/l* 97.6%

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

       2. add-sqr-sqrt 24.3%

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

       3. sqrt-unprod 24.6%

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

       4. sqr-neg 24.6%

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

       5. sqrt-unprod 0.2%

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

       6. add-sqr-sqrt 0.4%

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

       7. tan-quot 0.4%

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

       8. frac-2neg 0.4%

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

       9. div-inv 0.4%

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

       10. neg-mul-1 0.4%

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

       11. metadata-eval 0.4%

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

       12. tan-quot 0.4%

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

       13. times-frac 0.4%

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

       14. neg-mul-1 0.4%

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

       15. add-sqr-sqrt 0.1%

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

       16. sqrt-unprod 73.2%

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

       17. sqr-neg 73.2%

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

       18. sqrt-unprod 73.0%

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

       19. *-un-lft-identity 73.0%

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

       20. add-sqr-sqrt 97.5%

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

       21. tan-quot 97.6%

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

   13. Applied egg-rr 97.6%

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

       1. associate-*r/ 97.7%

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

       2. *-rgt-identity 97.7%

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

   15. Simplified 97.7%

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

   if -1.52 < x < 1

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 95.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-10} \lor \neg \left(x \leq 1.05 \cdot 10^{-5}\right):\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.25e-10) (not (<= x 1.05e-5)))
   (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.25e-10) || !(x <= 1.05e-5)) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - 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 ((x <= (-2.25d-10)) .or. (.not. (x <= 1.05d-5))) then
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - 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 ((x <= -2.25e-10) || !(x <= 1.05e-5)) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.25e-10) or not (x <= 1.05e-5):
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.25e-10) || !(x <= 1.05e-5))
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -2.25e-10) || ~((x <= 1.05e-5)))
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.25e-10], N[Not[LessEqual[x, 1.05e-5]], $MachinePrecision]], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.25e-10 or 1.04999999999999994e-5 < 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. 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 55.7%

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

      1. +-commutative 55.7%

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

      2. mul-1-neg 55.7%

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

      3. sub-neg 55.7%

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

      4. associate--l+ 55.7%

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

      5. *-commutative 55.7%

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

      6. div-sub 55.8%

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

   6. Simplified 55.8%

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

   if -2.25e-10 < x < 1.04999999999999994e-5

   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 x around 0 99.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-10} \lor \neg \left(x \leq 1.05 \cdot 10^{-5}\right):\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 51.7% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B)))

C

double code(double B, double x) {
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
end function

Java

public static double code(double B, double x) {
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
}

Python

def code(B, x):
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)

Julia

function code(B, x)
	return Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B))
end

MATLAB

function tmp = code(B, x)
	tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
end

Wolfram

code[B_, x_] := N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. distribute-lft-neg-in 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-in 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in B around 0 53.2%

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

   1. +-commutative 53.2%

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

   2. mul-1-neg 53.2%

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

   3. sub-neg 53.2%

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

   4. associate--l+ 53.2%

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

   5. *-commutative 53.2%

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

   6. div-sub 53.2%

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

6. Simplified 53.2%

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

7. Final simplification 53.2%

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

Alternative 7: 51.7% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. distribute-lft-neg-in 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-in 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in B around 0 65.1%

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

5. Taylor expanded in B around 0 53.2%

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

6. Taylor expanded in B around 0 53.3%

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

7. Final simplification 53.3%

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

Alternative 8: 51.6% 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 99.8%

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

   1. distribute-lft-neg-in 99.8%

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

   2. +-commutative 99.8%

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

   3. cancel-sign-sub-inv 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in B around 0 74.0%

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

5. Taylor expanded in B around 0 52.9%

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

   1. associate--l+ 52.9%

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

   2. *-commutative 52.9%

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

   3. div-sub 52.9%

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

7. Simplified 52.9%

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

8. Final simplification 52.9%

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

Alternative 9: 49.0% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.1 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 3.1e-39))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 3.1e-39)) {
		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 <= 3.1d-39))) 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 <= 3.1e-39)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 3.1e-39):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 3.1e-39))
		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 <= 3.1e-39)))
		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, 3.1e-39]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 3.0999999999999997e-39 < 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/ 99.8%

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

      7. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 55.1%

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

      1. clear-num 55.1%

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

      2. frac-sub 45.6%

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

      3. *-un-lft-identity 45.6%

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

   6. Applied egg-rr 45.6%

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

      1. *-rgt-identity 45.6%

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

   8. Simplified 45.6%

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

   9. Taylor expanded in x around inf 52.2%

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

       1. associate-*r/ 52.2%

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

       2. neg-mul-1 52.2%

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

   11. Simplified 52.2%

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

   if -1 < x < 3.0999999999999997e-39

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 52.0%

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

      1. neg-mul-1 52.0%

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

      2. sub-neg 52.0%

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

   6. Simplified 52.0%

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

   7. Taylor expanded in x around 0 49.7%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.1 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 10: 51.5% 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 99.8%

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

   1. distribute-lft-neg-in 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-in 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in B around 0 52.8%

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

   1. neg-mul-1 52.8%

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

   2. sub-neg 52.8%

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

6. Simplified 52.8%

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

7. Final simplification 52.8%

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

Alternative 11: 3.1% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. distribute-lft-neg-in 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-in 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

3. Simplified 99.8%

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

   1. distribute-rgt-neg-out 99.8%

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

   2. div-inv 99.8%

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

   3. clear-num 99.8%

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

   4. distribute-neg-frac 99.8%

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

   5. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in B around 0 70.4%

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

7. Taylor expanded in B around inf 3.2%

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

   1. *-commutative 3.2%

      \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]

9. Simplified 3.2%

   \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]

10. Final simplification 3.2%

    \[\leadsto B \cdot 0.16666666666666666 \]

Alternative 12: 26.9% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. distribute-lft-neg-in 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-in 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in B around 0 52.8%

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

   1. neg-mul-1 52.8%

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

   2. sub-neg 52.8%

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

6. Simplified 52.8%

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

7. Taylor expanded in x around 0 23.4%

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

8. Final simplification 23.4%

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

Reproduce

herbie shell --seed 2023293 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))