VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 13.2s

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. 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: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-\tan B}\\ \mathbf{if}\;x \leq -650000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 490000000000:\\ \;\;\;\;\frac{x}{B} \cdot \left(-1 + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ x (- (tan B)))))
   (if (<= x -650000000.0)
     t_0
     (if (<= x 7.4e-6)
       (/ 1.0 (sin B))
       (if (<= x 490000000000.0) (* (/ x B) (+ -1.0 (/ 1.0 x))) t_0)))))

C

double code(double B, double x) {
	double t_0 = x / -tan(B);
	double tmp;
	if (x <= -650000000.0) {
		tmp = t_0;
	} else if (x <= 7.4e-6) {
		tmp = 1.0 / sin(B);
	} else if (x <= 490000000000.0) {
		tmp = (x / B) * (-1.0 + (1.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / -tan(b)
    if (x <= (-650000000.0d0)) then
        tmp = t_0
    else if (x <= 7.4d-6) then
        tmp = 1.0d0 / sin(b)
    else if (x <= 490000000000.0d0) then
        tmp = (x / b) * ((-1.0d0) + (1.0d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = x / -Math.tan(B);
	double tmp;
	if (x <= -650000000.0) {
		tmp = t_0;
	} else if (x <= 7.4e-6) {
		tmp = 1.0 / Math.sin(B);
	} else if (x <= 490000000000.0) {
		tmp = (x / B) * (-1.0 + (1.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = x / -math.tan(B)
	tmp = 0
	if x <= -650000000.0:
		tmp = t_0
	elif x <= 7.4e-6:
		tmp = 1.0 / math.sin(B)
	elif x <= 490000000000.0:
		tmp = (x / B) * (-1.0 + (1.0 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(x / Float64(-tan(B)))
	tmp = 0.0
	if (x <= -650000000.0)
		tmp = t_0;
	elseif (x <= 7.4e-6)
		tmp = Float64(1.0 / sin(B));
	elseif (x <= 490000000000.0)
		tmp = Float64(Float64(x / B) * Float64(-1.0 + Float64(1.0 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = x / -tan(B);
	tmp = 0.0;
	if (x <= -650000000.0)
		tmp = t_0;
	elseif (x <= 7.4e-6)
		tmp = 1.0 / sin(B);
	elseif (x <= 490000000000.0)
		tmp = (x / B) * (-1.0 + (1.0 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[x, -650000000.0], t$95$0, If[LessEqual[x, 7.4e-6], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 490000000000.0], N[(N[(x / B), $MachinePrecision] * N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e8 or 4.9e11 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. associate-/l* 99.3%

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

      3. distribute-lft-neg-in 99.3%

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

   5. Simplified 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. clear-num 99.3%

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

      3. tan-quot 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. *-lft-identity 99.3%

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

   9. Simplified 99.3%

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

       1. distribute-lft-neg-out 99.3%

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

       2. neg-sub0 99.3%

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

       3. un-div-inv 99.6%

          \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   11. Applied egg-rr 99.6%

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

       1. neg-sub0 99.6%

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

       2. distribute-frac-neg2 99.6%

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

   13. Simplified 99.6%

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

   if -6.5e8 < x < 7.4000000000000003e-6

   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 95.3%

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

   if 7.4000000000000003e-6 < x < 4.9e11

   1. Initial program 99.1%

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

   3. Taylor expanded in B around 0 84.7%

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

   4. Taylor expanded in x around inf 84.8%

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

      1. sub-neg 84.8%

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

      2. associate-/r* 84.4%

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

      3. distribute-neg-frac 84.4%

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

      4. metadata-eval 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in B around 0 83.7%

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

      1. *-commutative 83.7%

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

      2. associate-/l* 84.0%

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

      3. sub-neg 84.0%

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

      4. metadata-eval 84.0%

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

      5. +-commutative 84.0%

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

   9. Simplified 84.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -650000000:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 490000000000:\\ \;\;\;\;\frac{x}{B} \cdot \left(-1 + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-5} \lor \neg \left(x \leq 6 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.2e-5) (not (<= x 6e-29)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.2e-5) || !(x <= 6e-29)) {
		tmp = (1.0 / B) - (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 <= (-1.2d-5)) .or. (.not. (x <= 6d-29))) then
        tmp = (1.0d0 / b) - (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 <= -1.2e-5) || !(x <= 6e-29)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.2e-5) or not (x <= 6e-29):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.2e-5) || !(x <= 6e-29))
		tmp = Float64(Float64(1.0 / B) - 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 <= -1.2e-5) || ~((x <= 6e-29)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.2e-5], N[Not[LessEqual[x, 6e-29]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-5 or 6.0000000000000005e-29 < 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%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-frac-neg2 99.5%

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

      6. tan-neg 99.5%

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

      7. cancel-sign-sub-inv 99.5%

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

      8. *-commutative 99.5%

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

      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. inv-pow 99.7%

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

      2. add-sqr-sqrt 45.0%

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

      3. unpow-prod-down 45.0%

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

   6. Applied egg-rr 45.0%

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

      1. pow-sqr 45.0%

         \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{\left(2 \cdot -1\right)}} - \frac{x}{\tan B} \]

      2. metadata-eval 45.0%

         \[\leadsto {\left(\sqrt{\sin B}\right)}^{\color{blue}{-2}} - \frac{x}{\tan B} \]

   8. Simplified 45.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{-2}} - \frac{x}{\tan B} \]

   9. Taylor expanded in B around 0 95.8%

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

   if -1.2e-5 < x < 6.0000000000000005e-29

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-5} \lor \neg \left(x \leq 6 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3200000000000:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -3200000000000.0)
   (/ x (- (tan B)))
   (if (<= x 2.15e-13)
     (- (/ 1.0 (sin B)) (/ x B))
     (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -3200000000000.0) {
		tmp = x / -tan(B);
	} else if (x <= 2.15e-13) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / 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 <= (-3200000000000.0d0)) then
        tmp = x / -tan(b)
    else if (x <= 2.15d-13) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -3200000000000.0) {
		tmp = x / -Math.tan(B);
	} else if (x <= 2.15e-13) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -3200000000000.0:
		tmp = x / -math.tan(B)
	elif x <= 2.15e-13:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -3200000000000.0)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (x <= 2.15e-13)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -3200000000000.0)
		tmp = x / -tan(B);
	elseif (x <= 2.15e-13)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -3200000000000.0], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 2.15e-13], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2e12

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-/l* 99.4%

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

      3. distribute-lft-neg-in 99.4%

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

   5. Simplified 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. *-lft-identity 99.4%

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

   9. Simplified 99.4%

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

       1. distribute-lft-neg-out 99.4%

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

       2. neg-sub0 99.4%

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

       3. un-div-inv 99.7%

          \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   11. Applied egg-rr 99.7%

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

       1. neg-sub0 99.7%

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

       2. distribute-frac-neg2 99.7%

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

   13. Simplified 99.7%

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

   if -3.2e12 < x < 2.1499999999999999e-13

   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/ 99.8%

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

      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. Taylor expanded in B around 0 98.7%

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

   if 2.1499999999999999e-13 < 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%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-frac-neg2 99.5%

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

      6. tan-neg 99.5%

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

      7. cancel-sign-sub-inv 99.5%

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

      8. *-commutative 99.5%

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

      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. inv-pow 99.7%

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

      2. add-sqr-sqrt 43.3%

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

      3. unpow-prod-down 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. pow-sqr 43.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{\left(2 \cdot -1\right)}} - \frac{x}{\tan B} \]

      2. metadata-eval 43.3%

         \[\leadsto {\left(\sqrt{\sin B}\right)}^{\color{blue}{-2}} - \frac{x}{\tan B} \]

   8. Simplified 43.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{-2}} - \frac{x}{\tan B} \]

   9. Taylor expanded in B around 0 96.3%

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

Alternative 5: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 2.1e-5) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 2.1e-5) {
		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 <= 2.1d-5) 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 <= 2.1e-5) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 2.1e-5:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 2.1e-5)
		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 <= 2.1e-5)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 2.1e-5], 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 < 2.09999999999999988e-5

   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 65.5%

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

   if 2.09999999999999988e-5 < 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 57.6%

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

Alternative 6: 48.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+40} \lor \neg \left(x \leq 3.3 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3e+40) (not (<= x 3.3e-19))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3e+40) || !(x <= 3.3e-19)) {
		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 <= (-3d+40)) .or. (.not. (x <= 3.3d-19))) 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 <= -3e+40) || !(x <= 3.3e-19)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3e+40) or not (x <= 3.3e-19):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -3e+40) || !(x <= 3.3e-19))
		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 <= -3e+40) || ~((x <= 3.3e-19)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3e+40], N[Not[LessEqual[x, 3.3e-19]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0000000000000002e40 or 3.2999999999999998e-19 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 57.7%

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

   4. Taylor expanded in x around inf 51.9%

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

      1. neg-mul-1 51.9%

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

      2. distribute-frac-neg2 51.9%

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

   6. Simplified 51.9%

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

   if -3.0000000000000002e40 < x < 3.2999999999999998e-19

   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/ 99.8%

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

      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. Step-by-step derivation

      1. inv-pow 99.8%

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

      2. add-sqr-sqrt 48.9%

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

      3. unpow-prod-down 48.7%

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

   6. Applied egg-rr 48.7%

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

      1. pow-sqr 48.9%

         \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{\left(2 \cdot -1\right)}} - \frac{x}{\tan B} \]

      2. metadata-eval 48.9%

         \[\leadsto {\left(\sqrt{\sin B}\right)}^{\color{blue}{-2}} - \frac{x}{\tan B} \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{-2}} - \frac{x}{\tan B} \]

   9. Taylor expanded in B around 0 46.1%

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

   10. Taylor expanded in x around 0 38.3%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+40} \lor \neg \left(x \leq 3.3 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.2% 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 48.5%

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

Alternative 8: 26.9% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-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. Step-by-step derivation

   1. inv-pow 99.8%

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

   2. add-sqr-sqrt 46.6%

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

   3. unpow-prod-down 46.5%

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

6. Applied egg-rr 46.5%

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

   1. pow-sqr 46.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{\left(2 \cdot -1\right)}} - \frac{x}{\tan B} \]

   2. metadata-eval 46.6%

      \[\leadsto {\left(\sqrt{\sin B}\right)}^{\color{blue}{-2}} - \frac{x}{\tan B} \]

8. Simplified 46.6%

   \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{-2}} - \frac{x}{\tan B} \]

9. Taylor expanded in B around 0 70.0%

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

10. Taylor expanded in x around 0 22.5%

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

Alternative 9: 2.6% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 77.7%

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

4. Taylor expanded in x around inf 67.6%

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

   1. sub-neg 67.6%

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

   2. associate-/r* 67.7%

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

   3. distribute-neg-frac 67.7%

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

   4. metadata-eval 67.7%

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

6. Simplified 67.7%

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

7. Taylor expanded in B around 0 48.5%

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

   1. *-commutative 48.5%

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

   2. associate-/l* 48.4%

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

   3. sub-neg 48.4%

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

   4. metadata-eval 48.4%

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

   5. +-commutative 48.4%

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

9. Simplified 48.4%

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

    1. associate-*r/ 48.5%

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

    2. frac-2neg 48.5%

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

    3. *-commutative 48.5%

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

    4. add-sqr-sqrt 26.5%

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

    5. sqrt-unprod 22.2%

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

    6. sqr-neg 22.2%

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

    7. sqrt-unprod 1.2%

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

    8. add-sqr-sqrt 2.2%

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

11. Applied egg-rr 2.2%

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

    1. distribute-lft-neg-in 2.2%

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

    2. associate-/l* 2.1%

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

13. Simplified 2.1%

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

14. Taylor expanded in x around 0 3.0%

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

Reproduce

herbie shell --seed 2024105 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))