VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.6s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \lor \neg \left(x \leq 0.0017\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.6) (not (<= x 0.0017)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.6) || !(x <= 0.0017)) {
		tmp = (1.0 / B) - (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 <= (-2.6d0)) .or. (.not. (x <= 0.0017d0))) then
        tmp = (1.0d0 / b) - (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 <= -2.6) || !(x <= 0.0017)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.6) or not (x <= 0.0017):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.6) || !(x <= 0.0017))
		tmp = Float64(Float64(1.0 / B) - 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 <= -2.6) || ~((x <= 0.0017)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.6], N[Not[LessEqual[x, 0.0017]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009 or 0.00169999999999999991 < x

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 97.9%

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

   if -2.60000000000000009 < x < 0.00169999999999999991

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-inverses 99.8%

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

      4. *-commutative 99.8%

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

      5. *-lft-identity 99.8%

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

      6. *-lft-identity 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-inverses 99.8%

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

      10. times-frac 78.8%

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

      11. *-commutative 78.8%

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

      12. associate-*l* 78.8%

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

      13. associate-/r* 99.8%

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

      14. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. 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 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \lor \neg \left(x \leq 0.0017\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 3: 74.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -0.028 \lor \neg \left(B \leq 0.011\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -0.028) (not (<= B 0.011)))
   (/ 1.0 (sin B))
   (+ (* B (* x 0.3333333333333333)) (/ (- 1.0 x) B))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -0.028) || !(B <= 0.011)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (B * (x * 0.3333333333333333)) + ((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 <= (-0.028d0)) .or. (.not. (b <= 0.011d0))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (b * (x * 0.3333333333333333d0)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -0.028) || !(B <= 0.011)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (B * (x * 0.3333333333333333)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -0.028) or not (B <= 0.011):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (B * (x * 0.3333333333333333)) + ((1.0 - x) / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -0.028) || !(B <= 0.011))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(B * Float64(x * 0.3333333333333333)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -0.028) || ~((B <= 0.011)))
		tmp = 1.0 / sin(B);
	else
		tmp = (B * (x * 0.3333333333333333)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -0.028], N[Not[LessEqual[B, 0.011]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -0.0280000000000000006 or 0.010999999999999999 < B

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 52.3%

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

   if -0.0280000000000000006 < B < 0.010999999999999999

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 99.8%

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

   5. Taylor expanded in B around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. mul-1-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. div-sub 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.028 \lor \neg \left(B \leq 0.011\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 4: 76.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $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. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. mul-1-neg 99.7%

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

   2. unsub-neg 99.7%

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

   3. *-inverses 99.7%

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

   4. *-commutative 99.7%

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

   5. *-lft-identity 99.7%

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

   6. *-lft-identity 99.7%

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

   7. times-frac 99.7%

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

   8. metadata-eval 99.7%

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

   9. *-inverses 99.7%

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

   10. times-frac 86.7%

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

   11. *-commutative 86.7%

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

   12. associate-*l* 86.7%

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

   13. associate-/r* 99.7%

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

   14. div-sub 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in B around 0 76.3%

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

9. Final simplification 76.3%

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

Alternative 5: 52.2% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 72.4%

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

5. Taylor expanded in B around 0 49.0%

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

   1. +-commutative 49.0%

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

   2. associate-+l+ 49.0%

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

   3. *-commutative 49.0%

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

   4. associate-*l* 49.0%

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

   5. mul-1-neg 49.0%

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

   6. sub-neg 49.0%

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

   7. div-sub 49.0%

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

7. Simplified 49.0%

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

8. Final simplification 49.0%

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

Alternative 6: 50.9% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105000 \lor \neg \left(x \leq 7800\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -105000.0) (not (<= x 7800.0))) (/ (- x) B) (/ (+ 1.0 x) B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -105000.0) || !(x <= 7800.0)) {
		tmp = -x / B;
	} else {
		tmp = (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 ((x <= (-105000.0d0)) .or. (.not. (x <= 7800.0d0))) then
        tmp = -x / b
    else
        tmp = (1.0d0 + x) / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -105000.0) || !(x <= 7800.0)) {
		tmp = -x / B;
	} else {
		tmp = (1.0 + x) / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -105000.0) or not (x <= 7800.0):
		tmp = -x / B
	else:
		tmp = (1.0 + x) / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -105000.0) || !(x <= 7800.0))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 + x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -105000.0) || ~((x <= 7800.0)))
		tmp = -x / B;
	else
		tmp = (1.0 + x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -105000.0], N[Not[LessEqual[x, 7800.0]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -105000 or 7800 < x

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in B around 0 98.6%

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

   5. Taylor expanded in B around 0 50.5%

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

      1. associate-*r/ 50.5%

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

      2. mul-1-neg 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in x around inf 49.7%

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

      1. associate-*r/ 49.7%

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

      2. mul-1-neg 49.7%

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

   10. Simplified 49.7%

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

   if -105000 < x < 7800

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 47.3%

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

   5. Taylor expanded in B around 0 46.4%

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

      1. associate-*r/ 46.4%

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

      2. mul-1-neg 46.4%

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

   7. Simplified 46.4%

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

      1. expm1-log1p-u 22.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{B} + \frac{1}{B}\right)\right)} \]

      2. expm1-udef 22.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{B} + \frac{1}{B}\right)} - 1} \]

      3. +-commutative 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{B} + \frac{-x}{B}}\right)} - 1 \]

      4. *-un-lft-identity 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{1 \cdot \frac{1}{B}} + \frac{-x}{B}\right)} - 1 \]

      5. div-inv 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(1 \cdot \frac{1}{B} + \color{blue}{\left(-x\right) \cdot \frac{1}{B}}\right)} - 1 \]

      6. distribute-rgt-out 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{B} \cdot \left(1 + \left(-x\right)\right)}\right)} - 1 \]

      7. add-sqr-sqrt 10.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{B} \cdot \left(1 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)\right)} - 1 \]

      8. sqrt-unprod 21.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{B} \cdot \left(1 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)\right)} - 1 \]

      9. sqr-neg 21.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{B} \cdot \left(1 + \sqrt{\color{blue}{x \cdot x}}\right)\right)} - 1 \]

      10. sqrt-unprod 10.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{B} \cdot \left(1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right)} - 1 \]

      11. add-sqr-sqrt 21.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{B} \cdot \left(1 + \color{blue}{x}\right)\right)} - 1 \]

   9. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{B} \cdot \left(1 + x\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 21.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{B} \cdot \left(1 + x\right)\right)\right)} \]

       2. expm1-log1p 45.4%

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

       3. associate-*l/ 45.4%

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

       4. *-lft-identity 45.4%

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

   11. Simplified 45.4%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105000 \lor \neg \left(x \leq 7800\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \]

Alternative 7: 50.9% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105000 \lor \neg \left(x \leq 7800\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -105000.0) (not (<= x 7800.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -105000.0) || !(x <= 7800.0)) {
		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 <= (-105000.0d0)) .or. (.not. (x <= 7800.0d0))) 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 <= -105000.0) || !(x <= 7800.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -105000.0) or not (x <= 7800.0):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -105000.0) || !(x <= 7800.0))
		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 <= -105000.0) || ~((x <= 7800.0)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -105000.0], N[Not[LessEqual[x, 7800.0]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -105000 or 7800 < x

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in B around 0 98.6%

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

   5. Taylor expanded in B around 0 50.5%

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

      1. associate-*r/ 50.5%

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

      2. mul-1-neg 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in x around inf 49.7%

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

      1. associate-*r/ 49.7%

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

      2. mul-1-neg 49.7%

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

   10. Simplified 49.7%

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

   if -105000 < x < 7800

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 47.3%

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

   5. Taylor expanded in B around 0 46.4%

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

      1. associate-*r/ 46.4%

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

      2. mul-1-neg 46.4%

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

   7. Simplified 46.4%

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

   8. Taylor expanded in x around 0 45.4%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105000 \lor \neg \left(x \leq 7800\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 8: 51.8% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 48.4%

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

   1. mul-1-neg 48.4%

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

   2. sub-neg 48.4%

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

6. Simplified 48.4%

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

7. Final simplification 48.4%

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

Alternative 9: 27.5% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 72.4%

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

5. Taylor expanded in B around 0 48.4%

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

   1. associate-*r/ 48.4%

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

   2. mul-1-neg 48.4%

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

7. Simplified 48.4%

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

8. Taylor expanded in x around 0 24.6%

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

9. Final simplification 24.6%

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

Reproduce

herbie shell --seed 2023230 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))