VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.0s

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. 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. Final simplification 99.8%

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

Alternative 2: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+19} \lor \neg \left(x \leq 230000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -6.8e+19) (not (<= x 230000000.0)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -6.8e+19) || !(x <= 230000000.0)) {
		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 <= (-6.8d+19)) .or. (.not. (x <= 230000000.0d0))) 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 <= -6.8e+19) || !(x <= 230000000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -6.8e+19) or not (x <= 230000000.0):
		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 <= -6.8e+19) || !(x <= 230000000.0))
		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 <= -6.8e+19) || ~((x <= 230000000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -6.8e+19], N[Not[LessEqual[x, 230000000.0]], $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 < -6.8e19 or 2.3e8 < 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. add-sqr-sqrt 57.8%

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

      2. sqrt-unprod 55.9%

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

      3. sqr-neg 55.9%

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

      4. sqrt-unprod 0.1%

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

      5. add-sqr-sqrt 0.4%

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

      6. div-inv 0.4%

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

      7. frac-2neg 0.4%

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

      8. frac-add 0.4%

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

      9. *-un-lft-identity 0.4%

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

   5. Applied egg-rr 0.4%

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

      1. associate-/r* 0.4%

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

   7. Simplified 0.4%

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

      1. div-sub 0.4%

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

      2. add-sqr-sqrt 0.3%

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

      3. sqrt-unprod 0.4%

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

      4. sqr-neg 0.4%

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

      5. sqrt-unprod 0.1%

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

      6. add-sqr-sqrt 0.4%

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

      7. add-sqr-sqrt 0.3%

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

      8. sqrt-unprod 0.6%

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

      9. sqr-neg 0.6%

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

      10. sqrt-unprod 0.1%

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

      11. add-sqr-sqrt 0.4%

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

      12. add-sqr-sqrt 0.3%

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

      13. sqrt-unprod 39.7%

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

      14. sqr-neg 39.7%

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

      15. sqrt-unprod 41.5%

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

      16. add-sqr-sqrt 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. div-sub 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around inf 99.7%

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

       1. neg-mul-1 99.7%

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

   14. Simplified 99.7%

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

   if -6.8e19 < x < 2.3e8

   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 x around 0 99.8%

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

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}} \]
   6. 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. sub-neg 99.8%

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

      4. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0 97.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+19} \lor \neg \left(x \leq 230000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3.0) (not (<= x 1.0)))
   (/ (- 1.0 x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3.0) || !(x <= 1.0)) {
		tmp = (1.0 - x) / tan(B);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (1.0d0 - x) / tan(b)
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -3.0) || !(x <= 1.0)) {
		tmp = (1.0 - x) / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3.0) or not (x <= 1.0):
		tmp = (1.0 - x) / math.tan(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -3.0) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 - x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -3.0) || ~((x <= 1.0)))
		tmp = (1.0 - x) / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3 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. add-sqr-sqrt 57.4%

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

      2. sqrt-unprod 55.6%

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

      3. sqr-neg 55.6%

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

      4. sqrt-unprod 0.1%

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

      5. add-sqr-sqrt 0.4%

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

      6. div-inv 0.4%

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

      7. frac-2neg 0.4%

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

      8. frac-add 0.4%

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

      9. *-un-lft-identity 0.4%

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

   5. Applied egg-rr 0.4%

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

      1. associate-/r* 0.4%

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

   7. Simplified 0.4%

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

      1. div-sub 0.4%

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

      2. add-sqr-sqrt 0.3%

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

      3. sqrt-unprod 0.4%

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

      4. sqr-neg 0.4%

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

      5. sqrt-unprod 0.1%

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

      6. add-sqr-sqrt 0.4%

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

      7. add-sqr-sqrt 0.3%

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

      8. sqrt-unprod 0.7%

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

      9. sqr-neg 0.7%

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

      10. sqrt-unprod 0.1%

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

      11. add-sqr-sqrt 0.4%

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

      12. add-sqr-sqrt 0.3%

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

      13. sqrt-unprod 38.8%

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

      14. sqr-neg 38.8%

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

      15. sqrt-unprod 41.9%

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

      16. add-sqr-sqrt 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. div-sub 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in B around 0 99.2%

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

   if -3 < 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. 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 x around 0 99.8%

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

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}} \]
   6. 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. sub-neg 99.8%

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

      4. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0 98.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.2) (not (<= x 1.0))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.2) || !(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 <= (-1.2d0)) .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 <= -1.2) || !(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 <= -1.2) 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 <= -1.2) || !(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 <= -1.2) || ~((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, -1.2], 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 < -1.19999999999999996 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. add-sqr-sqrt 57.4%

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

      2. sqrt-unprod 55.6%

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

      3. sqr-neg 55.6%

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

      4. sqrt-unprod 0.1%

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

      5. add-sqr-sqrt 0.4%

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

      6. div-inv 0.4%

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

      7. frac-2neg 0.4%

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

      8. frac-add 0.4%

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

      9. *-un-lft-identity 0.4%

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

   5. Applied egg-rr 0.4%

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

      1. associate-/r* 0.4%

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

   7. Simplified 0.4%

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

      1. div-sub 0.4%

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

      2. add-sqr-sqrt 0.3%

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

      3. sqrt-unprod 0.4%

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

      4. sqr-neg 0.4%

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

      5. sqrt-unprod 0.1%

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

      6. add-sqr-sqrt 0.4%

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

      7. add-sqr-sqrt 0.3%

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

      8. sqrt-unprod 0.7%

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

      9. sqr-neg 0.7%

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

      10. sqrt-unprod 0.1%

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

      11. add-sqr-sqrt 0.4%

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

      12. add-sqr-sqrt 0.3%

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

      13. sqrt-unprod 38.8%

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

      14. sqr-neg 38.8%

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

      15. sqrt-unprod 41.9%

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

      16. add-sqr-sqrt 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. div-sub 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around inf 98.9%

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

       1. neg-mul-1 98.9%

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

   14. Simplified 98.9%

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

   if -1.19999999999999996 < 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 96.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 74.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -0.008 \lor \neg \left(B \leq 0.009\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -0.008) (not (<= B 0.009)))
   (/ 1.0 (sin B))
   (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -0.008) || !(B <= 0.009)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - 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 ((b <= (-0.008d0)) .or. (.not. (b <= 0.009d0))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -0.008) || !(B <= 0.009)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -0.008) or not (B <= 0.009):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -0.008) || !(B <= 0.009))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -0.008) || ~((B <= 0.009)))
		tmp = 1.0 / sin(B);
	else
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -0.008], N[Not[LessEqual[B, 0.009]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -0.0080000000000000002 or 0.00899999999999999932 < B

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 53.9%

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

   if -0.0080000000000000002 < B < 0.00899999999999999932

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-neg-in 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--l+ 100.0%

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

      5. *-commutative 100.0%

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

      6. div-sub 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.008 \lor \neg \left(B \leq 0.009\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 6: 52.4% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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 55.5%

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

   1. +-commutative 55.5%

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

   2. mul-1-neg 55.5%

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

   3. sub-neg 55.5%

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

   4. associate--l+ 55.5%

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

   5. *-commutative 55.5%

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

   6. div-sub 55.5%

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

6. Simplified 55.5%

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

7. Taylor expanded in x around inf 55.6%

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

8. Final simplification 55.6%

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

Alternative 7: 51.1% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -580 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -580.0) (not (<= x 1.0))) (- (/ x B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -580.0) || !(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 <= (-580.0d0)) .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 <= -580.0) || !(x <= 1.0)) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -580.0) 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 <= -580.0) || !(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 <= -580.0) || ~((x <= 1.0)))
		tmp = -(x / B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -580.0], 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 < -580 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. Taylor expanded in B around 0 59.3%

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

      1. neg-mul-1 59.3%

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

      2. sub-neg 59.3%

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

   6. Simplified 59.3%

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

   7. Taylor expanded in x around inf 59.1%

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

      1. neg-mul-1 59.1%

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

      2. distribute-neg-frac 59.1%

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

   9. Simplified 59.1%

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

   if -580 < 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 B around 0 51.2%

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

      1. neg-mul-1 51.2%

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

      2. sub-neg 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in x around 0 49.6%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -580 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 8: 52.1% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 55.2%

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

   1. neg-mul-1 55.2%

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

   2. sub-neg 55.2%

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

6. Simplified 55.2%

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

7. Final simplification 55.2%

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

Alternative 9: 3.2% 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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 65.0%

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

5. Taylor expanded in B around inf 3.2%

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

   1. *-commutative 3.2%

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

7. Simplified 3.2%

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

8. Final simplification 3.2%

   \[\leadsto B \cdot 0.16666666666666666 \]

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%

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

   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 55.2%

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

   1. neg-mul-1 55.2%

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

   2. sub-neg 55.2%

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

6. Simplified 55.2%

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

7. Taylor expanded in x around 0 26.6%

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

8. Final simplification 26.6%

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

Reproduce

herbie shell --seed 2023332 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))