VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.5s

Alternatives: 10

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 99.7%

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

   1. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -180000000000:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;x \leq 29000000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -180000000000.0)
   (/ (* x (- (cos B))) (sin B))
   (if (<= x 29000000.0) (/ (- 1.0 x) (sin B)) (/ (- x) (tan B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -180000000000.0) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (x <= 29000000.0) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-180000000000.0d0)) then
        tmp = (x * -cos(b)) / sin(b)
    else if (x <= 29000000.0d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = -x / tan(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -180000000000.0) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (x <= 29000000.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = -x / Math.tan(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -180000000000.0:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif x <= 29000000.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = -x / math.tan(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -180000000000.0)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (x <= 29000000.0)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -180000000000.0)
		tmp = (x * -cos(B)) / sin(B);
	elseif (x <= 29000000.0)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = -x / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -180000000000.0], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 29000000.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8e11

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. neg-mul-1 99.5%

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

      3. distribute-lft-neg-in 99.5%

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

   7. Simplified 99.5%

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

   if -1.8e11 < x < 2.9e7

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-*r/ 99.9%

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

      4. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. sub-div 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in B around 0 97.7%

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

   if 2.9e7 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-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 x around 0 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. neg-mul-1 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

   7. Simplified 99.6%

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

      1. associate-/l* 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. add-sqr-sqrt 72.3%

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

      4. sqrt-unprod 72.6%

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

      5. sqr-neg 72.6%

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

      6. sqrt-unprod 0.2%

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

      7. add-sqr-sqrt 0.5%

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

      8. associate-/l* 0.5%

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

      9. *-commutative 0.5%

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

      10. associate-/l* 0.5%

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

      11. add-sqr-sqrt 0.2%

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

      12. sqrt-unprod 72.8%

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

      13. sqr-neg 72.8%

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

      14. sqrt-unprod 72.4%

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

      15. add-sqr-sqrt 99.7%

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

      16. tan-quot 99.8%

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

      17. div-inv 99.6%

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

      18. *-commutative 99.6%

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

      19. neg-mul-1 99.6%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180000000000:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;x \leq 29000000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -180000000000 \lor \neg \left(x \leq 4500000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -180000000000.0) (not (<= x 4500000.0)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -180000000000.0) || !(x <= 4500000.0)) {
		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 <= (-180000000000.0d0)) .or. (.not. (x <= 4500000.0d0))) 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 <= -180000000000.0) || !(x <= 4500000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -180000000000.0) or not (x <= 4500000.0):
		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 <= -180000000000.0) || !(x <= 4500000.0))
		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 <= -180000000000.0) || ~((x <= 4500000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -180000000000.0], N[Not[LessEqual[x, 4500000.0]], $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 < -1.8e11 or 4.5e6 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. neg-mul-1 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

   7. Simplified 99.6%

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

      1. associate-/l* 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. add-sqr-sqrt 70.4%

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

      4. sqrt-unprod 70.6%

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

      5. sqr-neg 70.6%

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

      6. sqrt-unprod 0.2%

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

      7. add-sqr-sqrt 0.4%

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

      8. associate-/l* 0.4%

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

      9. *-commutative 0.4%

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

      10. associate-/l* 0.4%

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

      11. add-sqr-sqrt 0.2%

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

      12. sqrt-unprod 70.9%

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

      13. sqr-neg 70.9%

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

      14. sqrt-unprod 70.6%

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

      15. add-sqr-sqrt 99.6%

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

      16. tan-quot 99.6%

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

      17. div-inv 99.5%

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

      18. *-commutative 99.5%

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

      19. neg-mul-1 99.5%

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

   9. Applied egg-rr 99.6%

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

   if -1.8e11 < x < 4.5e6

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-*r/ 99.9%

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

      4. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. sub-div 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in B around 0 97.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180000000000 \lor \neg \left(x \leq 4500000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 4: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 2.7))) (/ (- x) (sin B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.7)) {
		tmp = -x / sin(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 <= (-1.05d0)) .or. (.not. (x <= 2.7d0))) then
        tmp = -x / sin(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 <= -1.05) || !(x <= 2.7)) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.05) or not (x <= 2.7):
		tmp = -x / math.sin(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 2.7))
		tmp = Float64(Float64(-x) / sin(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 2.7)))
		tmp = -x / sin(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 2.7]], $MachinePrecision]], N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.7000000000000002 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. neg-mul-1 97.9%

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

      3. distribute-lft-neg-in 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in B around 0 51.2%

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

      1. neg-mul-1 51.2%

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

   10. Simplified 51.2%

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

   if -1.05000000000000004 < x < 2.7000000000000002

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 97.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.85) (not (<= x 1.0))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.85) || !(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 <= (-1.85d0)) .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 <= -1.85) || !(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 <= -1.85) 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 <= -1.85) || !(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 <= -1.85) || ~((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, -1.85], 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 < -1.8500000000000001 or 1 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. neg-mul-1 97.9%

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

      3. distribute-lft-neg-in 97.9%

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

   7. Simplified 97.9%

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

      1. associate-/l* 97.7%

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

      2. distribute-frac-neg 97.7%

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

      3. add-sqr-sqrt 69.8%

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

      4. sqrt-unprod 70.0%

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

      5. sqr-neg 70.0%

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

      6. sqrt-unprod 0.2%

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

      7. add-sqr-sqrt 0.4%

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

      8. associate-/l* 0.4%

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

      9. *-commutative 0.4%

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

      10. associate-/l* 0.4%

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

      11. add-sqr-sqrt 0.2%

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

      12. sqrt-unprod 70.2%

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

      13. sqr-neg 70.2%

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

      14. sqrt-unprod 69.9%

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

      15. add-sqr-sqrt 97.9%

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

      16. tan-quot 98.0%

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

      17. div-inv 97.8%

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

      18. *-commutative 97.8%

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

      19. neg-mul-1 97.8%

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

   9. Applied egg-rr 98.0%

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

   if -1.8500000000000001 < x < 1

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 97.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 74.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-6} \lor \neg \left(x \leq 2.55 \cdot 10^{-5}\right):\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -4.8e-6) (not (<= x 2.55e-5)))
   (+ (* B 0.16666666666666666) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -4.8e-6) || !(x <= 2.55e-5)) {
		tmp = (B * 0.16666666666666666) + ((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 <= (-4.8d-6)) .or. (.not. (x <= 2.55d-5))) then
        tmp = (b * 0.16666666666666666d0) + ((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 <= -4.8e-6) || !(x <= 2.55e-5)) {
		tmp = (B * 0.16666666666666666) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -4.8e-6) or not (x <= 2.55e-5):
		tmp = (B * 0.16666666666666666) + ((1.0 - x) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -4.8e-6) || !(x <= 2.55e-5))
		tmp = Float64(Float64(B * 0.16666666666666666) + 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 <= -4.8e-6) || ~((x <= 2.55e-5)))
		tmp = (B * 0.16666666666666666) + ((1.0 - x) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -4.8e-6], N[Not[LessEqual[x, 2.55e-5]], $MachinePrecision]], N[(N[(B * 0.16666666666666666), $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 < -4.7999999999999998e-6 or 2.54999999999999998e-5 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 48.9%

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

      1. +-commutative 48.9%

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

      2. mul-1-neg 48.9%

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

      3. sub-neg 48.9%

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

      4. associate--l+ 48.9%

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

      5. *-commutative 48.9%

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

      6. *-commutative 48.9%

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

      7. div-sub 48.9%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in x around 0 49.7%

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

      1. *-commutative 49.7%

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

   9. Simplified 49.7%

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

   if -4.7999999999999998e-6 < x < 2.54999999999999998e-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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 98.7%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-6} \lor \neg \left(x \leq 2.55 \cdot 10^{-5}\right):\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 7: 51.8% accurate, 23.3× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	return (B * 0.16666666666666666) + ((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) + ((1.0d0 - x) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 50.9%

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

   1. +-commutative 50.9%

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

   2. mul-1-neg 50.9%

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

   3. sub-neg 50.9%

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

   4. associate--l+ 50.9%

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

   5. *-commutative 50.9%

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

   6. *-commutative 50.9%

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

   7. div-sub 50.9%

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

6. Simplified 50.9%

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

7. Taylor expanded in x around 0 51.3%

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

   1. *-commutative 51.3%

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

9. Simplified 51.3%

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

10. Final simplification 51.3%

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

Alternative 8: 50.5% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -750.0) (not (<= x 1.0))) (- (/ x B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -750.0) || !(x <= 1.0)) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-750.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = -(x / b)
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -750.0) || !(x <= 1.0)) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -750.0) or not (x <= 1.0):
		tmp = -(x / B)
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -750.0) || !(x <= 1.0))
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -750.0) || ~((x <= 1.0)))
		tmp = -(x / B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -750.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], (-N[(x / B), $MachinePrecision]), N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -750 or 1 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 49.0%

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

      1. mul-1-neg 49.0%

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

      2. sub-neg 49.0%

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

   6. Simplified 49.0%

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

   7. Taylor expanded in x around inf 48.3%

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

      1. neg-mul-1 48.3%

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

      2. distribute-neg-frac 48.3%

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

   9. Simplified 48.3%

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

   if -750 < x < 1

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 54.1%

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

      1. mul-1-neg 54.1%

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

      2. sub-neg 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in x around 0 52.4%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 9: 51.6% 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. Step-by-step derivation

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 51.3%

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

   1. mul-1-neg 51.3%

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

   2. sub-neg 51.3%

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

6. Simplified 51.3%

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

7. Final simplification 51.3%

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

Alternative 10: 27.1% 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. Step-by-step derivation

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 51.3%

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

   1. mul-1-neg 51.3%

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

   2. sub-neg 51.3%

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

6. Simplified 51.3%

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

7. Taylor expanded in x around 0 24.5%

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

8. Final simplification 24.5%

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

Reproduce

herbie shell --seed 2023182 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))