VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.5s

Alternatives: 8

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. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*r/ 99.8%

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

   10. *-rgt-identity 99.8%

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

   11. tan-neg 99.8%

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

   12. distribute-neg-frac2 99.8%

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

   13. distribute-neg-frac 99.8%

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

   14. 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 2: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000225:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{elif}\;x \leq 0.00022:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -0.000225)
   (/ (- 1.0 x) (tan B))
   (if (<= x 0.00022) (/ 1.0 (sin B)) (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -0.000225) {
		tmp = (1.0 - x) / tan(B);
	} else if (x <= 0.00022) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.000225d0)) then
        tmp = (1.0d0 - x) / tan(b)
    else if (x <= 0.00022d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -0.000225) {
		tmp = (1.0 - x) / Math.tan(B);
	} else if (x <= 0.00022) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -0.000225:
		tmp = (1.0 - x) / math.tan(B)
	elif x <= 0.00022:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -0.000225)
		tmp = Float64(Float64(1.0 - x) / tan(B));
	elseif (x <= 0.00022)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -0.000225)
		tmp = (1.0 - x) / tan(B);
	elseif (x <= 0.00022)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -0.000225], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00022], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2499999999999999e-4

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. div-inv 99.7%

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

      3. sub-neg 99.7%

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

      4. clear-num 99.6%

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

      5. frac-sub 80.7%

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

      6. *-un-lft-identity 80.7%

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

      7. *-commutative 80.7%

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

      8. *-un-lft-identity 80.7%

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

   4. Applied egg-rr 80.7%

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

      1. associate-/r* 99.6%

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

      2. div-sub 99.6%

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

      3. *-inverses 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in B around 0 97.0%

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

      1. div-sub 97.0%

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

      2. associate-/r/ 97.0%

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

      3. sub-neg 97.0%

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

      4. div-inv 97.0%

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

      5. clear-num 97.0%

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

      6. associate-*l/ 97.1%

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

      7. *-un-lft-identity 97.1%

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

      8. distribute-neg-frac2 97.1%

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

   9. Applied egg-rr 97.1%

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

       1. distribute-frac-neg2 97.1%

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

       2. neg-mul-1 97.1%

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

       3. distribute-rgt-in 97.1%

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

       4. associate-*l/ 97.1%

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

       5. distribute-rgt-in 97.1%

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

       6. lft-mult-inverse 97.1%

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

       7. neg-mul-1 97.1%

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

       8. sub-neg 97.1%

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

   11. Simplified 97.1%

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

   if -2.2499999999999999e-4 < x < 2.20000000000000008e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 98.2%

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

   if 2.20000000000000008e-4 < x

   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. *-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. tan-neg 99.8%

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

      7. cancel-sign-sub-inv 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.7%

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

      10. *-rgt-identity 99.7%

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

      11. tan-neg 99.7%

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

      12. distribute-neg-frac2 99.7%

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

      13. distribute-neg-frac 99.7%

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

      14. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.8%

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

      2. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in B around 0 99.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000225:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{elif}\;x \leq 0.00022:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000225 \lor \neg \left(x \leq 0.00022\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.000225) (not (<= x 0.00022)))
   (/ (- 1.0 x) (tan B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.000225) || !(x <= 0.00022)) {
		tmp = (1.0 - x) / tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.000225d0)) .or. (.not. (x <= 0.00022d0))) then
        tmp = (1.0d0 - x) / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -0.000225) || !(x <= 0.00022)) {
		tmp = (1.0 - x) / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.000225) or not (x <= 0.00022):
		tmp = (1.0 - x) / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.000225) || !(x <= 0.00022))
		tmp = Float64(Float64(1.0 - x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.000225) || ~((x <= 0.00022)))
		tmp = (1.0 - x) / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.000225], N[Not[LessEqual[x, 0.00022]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2499999999999999e-4 or 2.20000000000000008e-4 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. div-inv 99.7%

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

      3. sub-neg 99.7%

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

      4. clear-num 99.6%

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

      5. frac-sub 85.2%

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

      6. *-un-lft-identity 85.2%

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

      7. *-commutative 85.2%

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

      8. *-un-lft-identity 85.2%

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

   4. Applied egg-rr 85.2%

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

      1. associate-/r* 99.6%

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

      2. div-sub 99.6%

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

      3. *-inverses 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in B around 0 98.1%

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

      1. div-sub 86.5%

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

      2. associate-/r/ 86.6%

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

      3. sub-neg 86.6%

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

      4. div-inv 84.5%

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

      5. clear-num 84.5%

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

      6. associate-*l/ 84.6%

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

      7. *-un-lft-identity 84.6%

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

      8. distribute-neg-frac2 84.6%

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

   9. Applied egg-rr 84.6%

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

       1. distribute-frac-neg2 84.6%

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

       2. neg-mul-1 84.6%

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

       3. distribute-rgt-in 98.2%

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

       4. associate-*l/ 98.2%

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

       5. distribute-rgt-in 98.2%

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

       6. lft-mult-inverse 98.2%

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

       7. neg-mul-1 98.2%

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

       8. sub-neg 98.2%

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

   11. Simplified 98.2%

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

   if -2.2499999999999999e-4 < x < 2.20000000000000008e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000225 \lor \neg \left(x \leq 0.00022\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 99.7%

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

   3. Taylor expanded in x around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. associate-*l/ 97.5%

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

      3. *-commutative 97.5%

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

   5. Simplified 97.5%

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

      1. add-sqr-sqrt 51.1%

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

      2. sqrt-unprod 27.4%

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

      3. sqr-neg 27.4%

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

      4. sqrt-unprod 0.3%

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

      5. add-sqr-sqrt 0.5%

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

      6. neg-sub0 0.5%

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

      7. sub-neg 0.5%

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

      8. add-sqr-sqrt 0.3%

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

      9. sqrt-unprod 27.4%

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

      10. sqr-neg 27.4%

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

      11. sqrt-unprod 51.1%

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

      12. add-sqr-sqrt 97.5%

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

      13. associate-*r/ 97.3%

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

      14. clear-num 97.2%

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

      15. /-rgt-identity 97.2%

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

      16. div-inv 97.2%

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

      17. metadata-eval 97.2%

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

   7. Applied egg-rr 97.5%

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

      1. +-lft-identity 97.5%

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

   9. Simplified 97.5%

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

   if -1.19999999999999996 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.7%

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

4. Final simplification 97.5%

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

Alternative 5: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.8:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 3.8) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 3.8) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 3.8d0) then
        tmp = (1.0d0 - x) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 3.8) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 3.8:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 3.8)
		tmp = Float64(Float64(1.0 - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 3.8)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 3.8], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.7999999999999998

   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 66.4%

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

   if 3.7999999999999998 < B

   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 0 47.8%

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

Alternative 6: 50.6% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.0], 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 < -1 or 1 < x

   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 57.1%

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

   4. Taylor expanded in x around inf 56.4%

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

      1. neg-mul-1 56.4%

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

      2. distribute-neg-frac 56.4%

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

   6. Simplified 56.4%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 48.4%

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

   4. Taylor expanded in x around 0 47.8%

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

4. Final simplification 52.7%

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

Alternative 7: 51.6% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 53.4%

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

Alternative 8: 27.3% 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 53.4%

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

4. Taylor expanded in x around 0 22.2%

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

Reproduce

herbie shell --seed 2024112 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))