VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.2s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;x \leq 1.85:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x} + -1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -4.5e+15)
   (/ (* (- x) (cos B)) (sin B))
   (if (<= x 1.85)
     (- (/ 1.0 (sin B)) (/ x B))
     (* x (/ (+ (/ 1.0 x) -1.0) (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -4.5e+15) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (x <= 1.85) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = x * (((1.0 / 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 <= (-4.5d+15)) then
        tmp = (-x * cos(b)) / sin(b)
    else if (x <= 1.85d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = x * (((1.0d0 / x) + (-1.0d0)) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -4.5e+15) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (x <= 1.85) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = x * (((1.0 / x) + -1.0) / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -4.5e+15:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif x <= 1.85:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = x * (((1.0 / x) + -1.0) / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -4.5e+15)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (x <= 1.85)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(x * Float64(Float64(Float64(1.0 / x) + -1.0) / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -4.5e+15)
		tmp = (-x * cos(B)) / sin(B);
	elseif (x <= 1.85)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = x * (((1.0 / x) + -1.0) / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -4.5e+15], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.85], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5e15

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

      1. un-div-inv 99.7%

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

      2. add-sqr-sqrt 99.3%

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

      3. sqrt-unprod 70.3%

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

      4. sqr-neg 70.3%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 0.5%

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

      7. frac-2neg 0.5%

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

      8. div-inv 0.5%

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

      9. fma-def 0.5%

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

      10. add-sqr-sqrt 0.5%

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

      11. sqrt-unprod 0.4%

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

      12. sqr-neg 0.4%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. mul-1-neg 99.6%

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

   8. Simplified 99.6%

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

   if -4.5e15 < x < 1.8500000000000001

   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 B around 0 97.2%

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

   if 1.8500000000000001 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. div-inv 99.8%

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

      4. clear-num 99.5%

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

      5. frac-sub 84.0%

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

      6. *-un-lft-identity 84.0%

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

      7. *-commutative 84.0%

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

      8. *-un-lft-identity 84.0%

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

   5. Applied egg-rr 84.0%

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

      1. associate-/r* 99.5%

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

      2. associate-/r/ 99.6%

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

      3. div-sub 99.6%

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

      4. *-inverses 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 99.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;x \leq 1.85:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x} + -1}{\tan B}\\ \end{array} \]

Alternative 3: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+15} \lor \neg \left(x \leq 1.35\right):\\ \;\;\;\;x \cdot \frac{\frac{1}{x} + -1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -4.5e+15) (not (<= x 1.35)))
   (* x (/ (+ (/ 1.0 x) -1.0) (tan B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -4.5e+15) || !(x <= 1.35)) {
		tmp = x * (((1.0 / x) + -1.0) / 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 <= (-4.5d+15)) .or. (.not. (x <= 1.35d0))) then
        tmp = x * (((1.0d0 / x) + (-1.0d0)) / 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 <= -4.5e+15) || !(x <= 1.35)) {
		tmp = x * (((1.0 / x) + -1.0) / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -4.5e+15) or not (x <= 1.35):
		tmp = x * (((1.0 / x) + -1.0) / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -4.5e+15) || !(x <= 1.35))
		tmp = Float64(x * Float64(Float64(Float64(1.0 / x) + -1.0) / 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 <= -4.5e+15) || ~((x <= 1.35)))
		tmp = x * (((1.0 / x) + -1.0) / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -4.5e+15], N[Not[LessEqual[x, 1.35]], $MachinePrecision]], N[(x * N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $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 < -4.5e15 or 1.3500000000000001 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. div-inv 99.7%

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

      4. clear-num 99.5%

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

      5. frac-sub 87.0%

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

      6. *-un-lft-identity 87.0%

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

      7. *-commutative 87.0%

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

      8. *-un-lft-identity 87.0%

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

   5. Applied egg-rr 87.0%

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

      1. associate-/r* 99.5%

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

      2. associate-/r/ 99.6%

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

      3. div-sub 99.6%

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

      4. *-inverses 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 99.6%

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

   if -4.5e15 < x < 1.3500000000000001

   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 B around 0 97.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+15} \lor \neg \left(x \leq 1.35\right):\\ \;\;\;\;x \cdot \frac{\frac{1}{x} + -1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 4: 75.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0215:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} - \frac{x}{B}\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -0.0215)
   (+ (* B 0.16666666666666666) (- (/ 1.0 B) (/ x B)))
   (if (<= x 1.15e-5)
     (/ 1.0 (sin B))
     (+ (* B (* x 0.3333333333333333)) (/ (- 1.0 x) B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -0.0215) {
		tmp = (B * 0.16666666666666666) + ((1.0 / B) - (x / B));
	} else if (x <= 1.15e-5) {
		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 (x <= (-0.0215d0)) then
        tmp = (b * 0.16666666666666666d0) + ((1.0d0 / b) - (x / b))
    else if (x <= 1.15d-5) 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 (x <= -0.0215) {
		tmp = (B * 0.16666666666666666) + ((1.0 / B) - (x / B));
	} else if (x <= 1.15e-5) {
		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 x <= -0.0215:
		tmp = (B * 0.16666666666666666) + ((1.0 / B) - (x / B))
	elif x <= 1.15e-5:
		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 (x <= -0.0215)
		tmp = Float64(Float64(B * 0.16666666666666666) + Float64(Float64(1.0 / B) - Float64(x / B)));
	elseif (x <= 1.15e-5)
		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 (x <= -0.0215)
		tmp = (B * 0.16666666666666666) + ((1.0 / B) - (x / B));
	elseif (x <= 1.15e-5)
		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[LessEqual[x, -0.0215], N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e-5], 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 3 regimes

2. if x < -0.021499999999999998

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 50.7%

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

   5. Taylor expanded in B around 0 51.0%

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

   if -0.021499999999999998 < x < 1.15e-5

   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 x around 0 97.0%

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

   if 1.15e-5 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 48.5%

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

   5. Taylor expanded in x around inf 48.5%

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

      1. associate-*r* 48.5%

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

      2. *-commutative 48.5%

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

   7. Simplified 48.5%

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

      1. associate--l+ 48.5%

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

      2. *-commutative 48.5%

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

      3. sub-div 48.5%

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

   9. Applied egg-rr 48.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0215:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} - \frac{x}{B}\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 5: 75.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $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. 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 B around 0 75.3%

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

5. Final simplification 75.3%

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

Alternative 6: 52.6% 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. +-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 B around 0 51.4%

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

5. Taylor expanded in x around inf 51.8%

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

   1. associate-*r* 51.8%

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

   2. *-commutative 51.8%

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

7. Simplified 51.8%

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

   1. associate--l+ 51.8%

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

   2. *-commutative 51.8%

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

   3. sub-div 51.8%

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

9. Applied egg-rr 51.8%

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

10. Final simplification 51.8%

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

Alternative 7: 51.2% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0195 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.0195) (not (<= x 1.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.0195) || !(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 <= (-0.0195d0)) .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 <= -0.0195) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.0195) 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 <= -0.0195) || !(x <= 1.0))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.0195) || ~((x <= 1.0)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.0195], 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 < -0.0195 or 1 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. sub-neg 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in x around inf 47.0%

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

      1. neg-mul-1 47.0%

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

      2. distribute-neg-frac 47.0%

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

   9. Simplified 47.0%

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

   if -0.0195 < x < 1

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   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 53.5%

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

      1. mul-1-neg 53.5%

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

      2. sub-neg 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in x around 0 53.4%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0195 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 8: 52.3% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 51.3%

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

   1. mul-1-neg 51.3%

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

   2. sub-neg 51.3%

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

6. Simplified 51.3%

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

7. Final simplification 51.3%

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

Alternative 9: 3.2% accurate, 70.0× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (* B 0.16666666666666666))

C

double code(double B, double x) {
	return B * 0.16666666666666666;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = b * 0.16666666666666666d0
end function

Java

public static double code(double B, double x) {
	return B * 0.16666666666666666;
}

Python

def code(B, x):
	return B * 0.16666666666666666

Julia

function code(B, x)
	return Float64(B * 0.16666666666666666)
end

MATLAB

function tmp = code(B, x)
	tmp = B * 0.16666666666666666;
end

Wolfram

code[B_, x_] := N[(B * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

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 63.6%

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

   1. add-log-exp 26.0%

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

   2. *-un-lft-identity 26.0%

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

   3. log-prod 26.0%

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

   4. metadata-eval 26.0%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\left(-x\right) \cdot \frac{1}{\tan B}}\right)\right) + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]

   5. add-log-exp 63.6%

      \[\leadsto \left(0 + \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}}\right) + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]

   6. add-sqr-sqrt 33.0%

      \[\leadsto \left(0 + \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \frac{1}{\tan B}\right) + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]

   7. sqrt-unprod 42.6%

      \[\leadsto \left(0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \frac{1}{\tan B}\right) + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]

   8. sqr-neg 42.6%

      \[\leadsto \left(0 + \sqrt{\color{blue}{x \cdot x}} \cdot \frac{1}{\tan B}\right) + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]

   9. sqrt-unprod 14.8%

      \[\leadsto \left(0 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\tan B}\right) + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]

   10. add-sqr-sqrt 29.2%

       \[\leadsto \left(0 + \color{blue}{x} \cdot \frac{1}{\tan B}\right) + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]

   11. div-inv 29.2%

       \[\leadsto \left(0 + \color{blue}{\frac{x}{\tan B}}\right) + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]

6. Applied egg-rr 29.2%

   \[\leadsto \color{blue}{\left(0 + \frac{x}{\tan B}\right)} + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]
7. Step-by-step derivation

   1. +-lft-identity 29.2%

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

8. Simplified 29.2%

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

9. Taylor expanded in B around inf 3.3%

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

    1. *-commutative 3.3%

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

11. Simplified 3.3%

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

12. Final simplification 3.3%

    \[\leadsto B \cdot 0.16666666666666666 \]

Alternative 10: 27.2% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 51.3%

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

   1. mul-1-neg 51.3%

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

   2. sub-neg 51.3%

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

6. Simplified 51.3%

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

7. Taylor expanded in x around 0 30.3%

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

8. Final simplification 30.3%

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

Reproduce

herbie shell --seed 2023175 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))