VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 9.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around inf 99.8%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -750000000 \lor \neg \left(x \leq 450000\right):\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -750000000.0) (not (<= x 450000.0)))
   (* (/ x (sin B)) (- (cos B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -750000000.0) || !(x <= 450000.0)) {
		tmp = (x / sin(B)) * -cos(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-750000000.0d0)) .or. (.not. (x <= 450000.0d0))) then
        tmp = (x / sin(b)) * -cos(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -750000000.0) || !(x <= 450000.0)) {
		tmp = (x / Math.sin(B)) * -Math.cos(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -750000000.0) or not (x <= 450000.0):
		tmp = (x / math.sin(B)) * -math.cos(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -750000000.0) || !(x <= 450000.0))
		tmp = Float64(Float64(x / sin(B)) * Float64(-cos(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -750000000.0) || ~((x <= 450000.0)))
		tmp = (x / sin(B)) * -cos(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -750000000.0], N[Not[LessEqual[x, 450000.0]], $MachinePrecision]], N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * (-N[Cos[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5e8 or 4.5e5 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   if -7.5e8 < x < 4.5e5

   1. Initial program 99.8%

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

   3. Taylor expanded in B around inf 99.9%

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

   4. Taylor expanded in B around 0 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -750000000 \lor \neg \left(x \leq 450000\right):\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around inf 99.8%

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

4. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. mul-1-neg 99.8%

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

   3. associate-/l* 99.7%

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

   4. distribute-lft-neg-in 99.7%

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

   5. cancel-sign-sub-inv 99.7%

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

   6. associate-/l* 99.8%

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

   7. div-sub 99.8%

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

6. Simplified 99.8%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. associate-*l/ 99.8%

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

   9. *-lft-identity 99.8%

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

   10. tan-neg 99.8%

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

   11. distribute-neg-frac2 99.8%

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

   12. distribute-neg-frac 99.8%

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

   13. remove-double-neg 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. Add Preprocessing
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -600000000 \lor \neg \left(x \leq 132000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -600000000.0) (not (<= x 132000000.0)))
   (/ (- x) (tan B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -600000000.0) || !(x <= 132000000.0)) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-600000000.0d0)) .or. (.not. (x <= 132000000.0d0))) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -600000000.0) || !(x <= 132000000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -600000000.0) or not (x <= 132000000.0):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -600000000.0) || !(x <= 132000000.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -600000000.0) || ~((x <= 132000000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -600000000.0], N[Not[LessEqual[x, 132000000.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6e8 or 1.32e8 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. associate-/l* 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

      4. distribute-neg-frac2 99.3%

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

   5. Simplified 99.3%

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

      1. distribute-frac-neg2 99.3%

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

      2. neg-sub0 99.3%

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

      3. clear-num 99.3%

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

      4. quot-tan 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. neg-sub0 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

       1. clear-num 99.3%

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

       2. tan-quot 99.3%

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

       3. associate-/l/ 99.3%

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

       4. neg-mul-1 99.3%

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

       5. add-sqr-sqrt 22.3%

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

       6. sqrt-unprod 22.5%

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

       7. sqr-neg 22.5%

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

       8. sqrt-unprod 0.2%

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

       9. add-sqr-sqrt 0.4%

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

       10. associate-/r/ 0.4%

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

       11. add-sqr-sqrt 0.2%

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

       12. sqrt-unprod 22.5%

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

       13. sqr-neg 22.5%

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

       14. sqrt-unprod 22.3%

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

       15. add-sqr-sqrt 99.2%

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

       16. associate-*l* 99.3%

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

       17. div-inv 99.5%

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

   11. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   12. Step-by-step derivation

       1. neg-sub0 99.4%

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

       2. distribute-frac-neg 99.4%

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

   13. Simplified 99.4%

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

   if -6e8 < x < 1.32e8

   1. Initial program 99.8%

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

   3. Taylor expanded in B around inf 99.9%

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

   4. Taylor expanded in B around 0 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -600000000 \lor \neg \left(x \leq 132000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -240000000 \lor \neg \left(x \leq 2500000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -240000000.0) (not (<= x 2500000000.0)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -240000000.0) || !(x <= 2500000000.0)) {
		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 <= (-240000000.0d0)) .or. (.not. (x <= 2500000000.0d0))) 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 <= -240000000.0) || !(x <= 2500000000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -240000000.0) or not (x <= 2500000000.0):
		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 <= -240000000.0) || !(x <= 2500000000.0))
		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 <= -240000000.0) || ~((x <= 2500000000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -240000000.0], N[Not[LessEqual[x, 2500000000.0]], $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 < -2.4e8 or 2.5e9 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. associate-/l* 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

      4. distribute-neg-frac2 99.3%

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

   5. Simplified 99.3%

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

      1. distribute-frac-neg2 99.3%

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

      2. neg-sub0 99.3%

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

      3. clear-num 99.3%

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

      4. quot-tan 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. neg-sub0 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

       1. clear-num 99.3%

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

       2. tan-quot 99.3%

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

       3. associate-/l/ 99.3%

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

       4. neg-mul-1 99.3%

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

       5. add-sqr-sqrt 22.3%

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

       6. sqrt-unprod 22.5%

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

       7. sqr-neg 22.5%

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

       8. sqrt-unprod 0.2%

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

       9. add-sqr-sqrt 0.4%

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

       10. associate-/r/ 0.4%

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

       11. add-sqr-sqrt 0.2%

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

       12. sqrt-unprod 22.5%

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

       13. sqr-neg 22.5%

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

       14. sqrt-unprod 22.3%

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

       15. add-sqr-sqrt 99.2%

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

       16. associate-*l* 99.3%

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

       17. div-inv 99.5%

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

   11. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   12. Step-by-step derivation

       1. neg-sub0 99.4%

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

       2. distribute-frac-neg 99.4%

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

   13. Simplified 99.4%

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

   if -2.4e8 < x < 2.5e9

   1. Initial program 99.8%

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

   3. Taylor expanded in B around inf 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. distribute-lft-neg-in 99.8%

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

      5. cancel-sign-sub-inv 99.8%

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

      6. associate-/l* 99.9%

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

      7. div-sub 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in B around 0 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -240000000 \lor \neg \left(x \leq 2500000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 1.8× 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 99.6%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. associate-/l* 99.0%

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

      3. distribute-rgt-neg-in 99.0%

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

      4. distribute-neg-frac2 99.0%

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

   5. Simplified 99.0%

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

      1. distribute-frac-neg2 99.0%

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

      2. neg-sub0 99.0%

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

      3. clear-num 99.0%

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

      4. quot-tan 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. neg-sub0 99.0%

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

      2. distribute-neg-frac 99.0%

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

      3. metadata-eval 99.0%

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

   9. Simplified 99.0%

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

       1. clear-num 99.0%

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

       2. tan-quot 99.0%

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

       3. associate-/l/ 99.0%

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

       4. neg-mul-1 99.0%

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

       5. add-sqr-sqrt 22.1%

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

       6. sqrt-unprod 22.4%

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

       7. sqr-neg 22.4%

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

       8. sqrt-unprod 0.2%

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

       9. add-sqr-sqrt 0.4%

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

       10. associate-/r/ 0.4%

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

       11. add-sqr-sqrt 0.2%

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

       12. sqrt-unprod 22.4%

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

       13. sqr-neg 22.4%

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

       14. sqrt-unprod 22.1%

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

       15. add-sqr-sqrt 98.9%

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

       16. associate-*l* 99.0%

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

       17. div-inv 99.2%

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

   11. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   12. Step-by-step derivation

       1. neg-sub0 99.1%

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

       2. distribute-frac-neg 99.1%

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

   13. Simplified 99.1%

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

   if -1.19999999999999996 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.9%

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

4. Final simplification 98.0%

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

Alternative 8: 63.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 9e-5) (- (/ 1.0 B) (/ x B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 9e-5) {
		tmp = (1.0 / B) - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 9d-5) then
        tmp = (1.0d0 / b) - (x / b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 9e-5) {
		tmp = (1.0 / B) - (x / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 9e-5:
		tmp = (1.0 / B) - (x / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 9e-5)
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 9e-5)
		tmp = (1.0 / B) - (x / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 9e-5], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9.00000000000000057e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 73.8%

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

      1. div-sub 73.8%

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

   5. Applied egg-rr 73.8%

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

   if 9.00000000000000057e-5 < B

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 44.3%

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

Alternative 9: 50.9% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-5} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3.2e-5) (not (<= x 1.0))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3.2e-5) || !(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 <= (-3.2d-5)) .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 <= -3.2e-5) || !(x <= 1.0)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3.2e-5) 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 <= -3.2e-5) || !(x <= 1.0))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -3.2e-5) || ~((x <= 1.0)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3.2e-5], 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 < -3.19999999999999986e-5 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 58.6%

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

   4. Taylor expanded in x around inf 58.3%

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

      1. neg-mul-1 58.3%

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

   6. Simplified 58.3%

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

   if -3.19999999999999986e-5 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in B around 0 58.6%

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

   4. Taylor expanded in x around 0 56.9%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-5} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.9% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 58.6%

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

   1. div-sub 58.6%

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

5. Applied egg-rr 58.6%

   \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
6. Add Preprocessing

Alternative 11: 51.9% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 58.6%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
4. Add Preprocessing

Alternative 12: 27.8% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 58.6%

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

4. Taylor expanded in x around 0 29.7%

   \[\leadsto \color{blue}{\frac{1}{B}} \]
5. Add Preprocessing

Alternative 13: 2.6% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 58.6%

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

   1. frac-2neg 58.6%

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

   2. div-inv 58.5%

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

   3. add-sqr-sqrt 26.7%

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

   4. sqrt-unprod 20.4%

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

   5. sqr-neg 20.4%

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

   6. sqrt-unprod 1.0%

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

   7. add-sqr-sqrt 1.8%

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

5. Applied egg-rr 1.8%

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

6. Taylor expanded in x around 0 2.5%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))