VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.1s

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. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. associate-*l/ 99.7%

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

   9. *-lft-identity 99.7%

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

   10. tan-neg 99.7%

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

   11. distribute-neg-frac2 99.7%

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

   12. distribute-neg-frac 99.7%

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

   13. 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. Add Preprocessing

Alternative 2: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-7} \lor \neg \left(x \leq 8.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -4.1e-7) (not (<= x 8.5e-21)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -4.1e-7) || !(x <= 8.5e-21)) {
		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 <= (-4.1d-7)) .or. (.not. (x <= 8.5d-21))) 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 <= -4.1e-7) || !(x <= 8.5e-21)) {
		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 <= -4.1e-7) or not (x <= 8.5e-21):
		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 <= -4.1e-7) || !(x <= 8.5e-21))
		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 <= -4.1e-7) || ~((x <= 8.5e-21)))
		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, -4.1e-7], N[Not[LessEqual[x, 8.5e-21]], $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 < -4.0999999999999999e-7 or 8.4999999999999993e-21 < 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%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-frac-neg2 99.6%

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

      6. tan-neg 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. associate-*l/ 99.7%

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

      9. *-lft-identity 99.7%

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

      10. tan-neg 99.7%

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

      11. distribute-neg-frac2 99.7%

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

      12. distribute-neg-frac 99.7%

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

      13. 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. Taylor expanded in B around 0 97.9%

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

   if -4.0999999999999999e-7 < x < 8.4999999999999993e-21

   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 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-7} \lor \neg \left(x \leq 8.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= x -9.2e-5)
     (* t_0 (+ -1.0 (/ 1.0 x)))
     (if (<= x 8.5e-21) (/ 1.0 (sin B)) (- (/ 1.0 B) t_0)))))

C

double code(double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (x <= -9.2e-5) {
		tmp = t_0 * (-1.0 + (1.0 / x));
	} else if (x <= 8.5e-21) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (1.0 / B) - 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 <= (-9.2d-5)) then
        tmp = t_0 * ((-1.0d0) + (1.0d0 / x))
    else if (x <= 8.5d-21) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 / b) - 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 <= -9.2e-5) {
		tmp = t_0 * (-1.0 + (1.0 / x));
	} else if (x <= 8.5e-21) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if x <= -9.2e-5:
		tmp = t_0 * (-1.0 + (1.0 / x))
	elif x <= 8.5e-21:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (x <= -9.2e-5)
		tmp = Float64(t_0 * Float64(-1.0 + Float64(1.0 / x)));
	elseif (x <= 8.5e-21)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (x <= -9.2e-5)
		tmp = t_0 * (-1.0 + (1.0 / x));
	elseif (x <= 8.5e-21)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 / B) - 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, -9.2e-5], N[(t$95$0 * N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-21], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.20000000000000001e-5

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. div-inv 99.6%

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

      3. sub-neg 99.6%

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

      4. clear-num 99.5%

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

      5. frac-sub 87.7%

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

      6. *-un-lft-identity 87.7%

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

      7. *-commutative 87.7%

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

      8. *-un-lft-identity 87.7%

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

   4. Applied egg-rr 87.7%

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

      1. associate-/r* 99.5%

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

      2. associate-/r/ 99.5%

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

      3. div-sub 99.5%

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

      4. *-inverses 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around 0 97.0%

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

      1. *-commutative 97.0%

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

      2. clear-num 97.0%

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

      3. un-div-inv 97.1%

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

      4. sub-neg 97.1%

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

      5. metadata-eval 97.1%

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

   9. Applied egg-rr 97.1%

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

       1. associate-/r/ 97.1%

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

       2. +-commutative 97.1%

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

   11. Simplified 97.1%

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

   if -9.20000000000000001e-5 < x < 8.4999999999999993e-21

   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 99.8%

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

   if 8.4999999999999993e-21 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. remove-double-neg 99.7%

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

      5. distribute-frac-neg2 99.7%

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

      6. tan-neg 99.7%

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

      7. cancel-sign-sub-inv 99.7%

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

      8. associate-*l/ 99.8%

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

      9. *-lft-identity 99.8%

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

      10. tan-neg 99.8%

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

      11. distribute-neg-frac2 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0 98.7%

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

Alternative 4: 97.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.08) (not (<= x 1.05))) (/ x (- (tan B))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.08) || !(x <= 1.05)) {
		tmp = x / -tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.08d0)) .or. (.not. (x <= 1.05d0))) then
        tmp = x / -tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.08) || !(x <= 1.05)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.08) or not (x <= 1.05):
		tmp = x / -math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.08) || !(x <= 1.05))
		tmp = Float64(x / Float64(-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 <= -1.08) || ~((x <= 1.05)))
		tmp = x / -tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.08], N[Not[LessEqual[x, 1.05]], $MachinePrecision]], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0800000000000001 or 1.05000000000000004 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 96.8%

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

      1. mul-1-neg 96.8%

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

      2. div-inv 96.6%

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

      3. associate-*l* 96.7%

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

      4. div-inv 96.8%

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

      5. clear-num 96.7%

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

      6. tan-quot 96.8%

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

      7. un-div-inv 96.9%

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

      8. distribute-neg-frac 96.9%

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

   5. Applied egg-rr 96.9%

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

   if -1.0800000000000001 < x < 1.05000000000000004

   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 99.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;\frac{-1 - x}{\tan B}\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.6)
   (/ (- -1.0 x) (tan B))
   (if (<= x 1.3) (/ 1.0 (sin B)) (/ x (- (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.6) {
		tmp = (-1.0 - x) / tan(B);
	} else if (x <= 1.3) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = 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 <= (-1.6d0)) then
        tmp = ((-1.0d0) - x) / tan(b)
    else if (x <= 1.3d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = x / -tan(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.6) {
		tmp = (-1.0 - x) / Math.tan(B);
	} else if (x <= 1.3) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.6:
		tmp = (-1.0 - x) / math.tan(B)
	elif x <= 1.3:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.6)
		tmp = Float64(Float64(-1.0 - x) / tan(B));
	elseif (x <= 1.3)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.6)
		tmp = (-1.0 - x) / tan(B);
	elseif (x <= 1.3)
		tmp = 1.0 / sin(B);
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.6], N[(N[(-1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. div-inv 99.6%

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

      3. sub-neg 99.6%

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

      4. clear-num 99.5%

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

      5. frac-sub 87.7%

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

      6. *-un-lft-identity 87.7%

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

      7. *-commutative 87.7%

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

      8. *-un-lft-identity 87.7%

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

   4. Applied egg-rr 87.7%

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

      1. associate-/r* 99.5%

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

      2. associate-/r/ 99.5%

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

      3. div-sub 99.5%

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

      4. *-inverses 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around 0 97.0%

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

      1. frac-2neg 97.0%

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

      2. associate-*l/ 97.1%

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

      3. distribute-lft-neg-in 97.1%

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

      4. distribute-rgt-neg-in 97.1%

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

      5. add-sqr-sqrt 96.8%

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

      6. sqrt-unprod 63.8%

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

      7. sqr-neg 63.8%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 0.4%

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

      10. *-commutative 0.4%

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

      11. sub-neg 0.4%

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

      12. metadata-eval 0.4%

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

      13. distribute-lft-in 0.4%

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

      14. div-inv 0.4%

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

      15. *-inverses 0.4%

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

      16. *-commutative 0.4%

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

      17. neg-mul-1 0.4%

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

      18. add-sqr-sqrt 0.4%

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

      19. sqrt-unprod 0.4%

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

      20. sqr-neg 0.4%

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

      21. sqrt-unprod 0.0%

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

      22. add-sqr-sqrt 95.1%

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

   9. Applied egg-rr 95.1%

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

       1. distribute-frac-neg2 95.1%

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

       2. distribute-neg-frac 95.1%

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

       3. neg-sub0 95.1%

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

       4. associate--r+ 95.1%

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

       5. metadata-eval 95.1%

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

   11. Simplified 95.1%

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

   if -1.6000000000000001 < x < 1.30000000000000004

   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 99.0%

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

   if 1.30000000000000004 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. div-inv 98.2%

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

      3. associate-*l* 98.4%

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

      4. div-inv 98.4%

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

      5. clear-num 98.3%

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

      6. tan-quot 98.4%

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

      7. un-div-inv 98.6%

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

      8. distribute-neg-frac 98.6%

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

   5. Applied egg-rr 98.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;\frac{-1 - x}{\tan B}\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 360:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 360.0) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 360.0) {
		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 <= 360.0d0) 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 <= 360.0) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 360.0:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 360.0)
		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 <= 360.0)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 360.0], 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 < 360

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 67.0%

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

   if 360 < B

   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 0 50.4%

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

Alternative 7: 49.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-21} \lor \neg \left(x \leq 3.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -6.2e-21) (not (<= x 3.2e-5))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -6.2e-21) || !(x <= 3.2e-5)) {
		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 <= (-6.2d-21)) .or. (.not. (x <= 3.2d-5))) 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 <= -6.2e-21) || !(x <= 3.2e-5)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -6.2e-21) or not (x <= 3.2e-5):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -6.2e-21) || !(x <= 3.2e-5))
		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 <= -6.2e-21) || ~((x <= 3.2e-5)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -6.2e-21], N[Not[LessEqual[x, 3.2e-5]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.1999999999999997e-21 or 3.19999999999999986e-5 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 51.6%

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

   4. Taylor expanded in x around inf 50.0%

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

      1. neg-mul-1 50.0%

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

   6. Simplified 50.0%

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

   if -6.1999999999999997e-21 < x < 3.19999999999999986e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 51.0%

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

   4. Taylor expanded in x around 0 50.7%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-21} \lor \neg \left(x \leq 3.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.8% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.8e-28)
   (/ (- -1.0 x) B)
   (if (<= x 3.2e-5) (/ 1.0 B) (/ x (- B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.8e-28) {
		tmp = (-1.0 - x) / B;
	} else if (x <= 3.2e-5) {
		tmp = 1.0 / B;
	} else {
		tmp = 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 <= (-1.8d-28)) then
        tmp = ((-1.0d0) - x) / b
    else if (x <= 3.2d-5) then
        tmp = 1.0d0 / b
    else
        tmp = x / -b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.8e-28) {
		tmp = (-1.0 - x) / B;
	} else if (x <= 3.2e-5) {
		tmp = 1.0 / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.8e-28:
		tmp = (-1.0 - x) / B
	elif x <= 3.2e-5:
		tmp = 1.0 / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.8e-28)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (x <= 3.2e-5)
		tmp = Float64(1.0 / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.8e-28)
		tmp = (-1.0 - x) / B;
	elseif (x <= 3.2e-5)
		tmp = 1.0 / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.8e-28], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[x, 3.2e-5], N[(1.0 / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e-28

   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 45.0%

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

   4. Taylor expanded in x around inf 45.0%

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

      1. frac-2neg 45.0%

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

      2. div-inv 45.0%

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

      3. distribute-lft-neg-in 45.0%

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

      4. add-sqr-sqrt 44.9%

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

      5. sqrt-unprod 39.8%

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

      6. sqr-neg 39.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 2.0%

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

      9. sub-neg 2.0%

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

      10. metadata-eval 2.0%

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

      11. distribute-lft-in 2.0%

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

      12. div-inv 2.0%

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

      13. *-inverses 2.0%

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

      14. *-commutative 2.0%

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

      15. neg-mul-1 2.0%

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

      16. add-sqr-sqrt 2.0%

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

      17. sqrt-unprod 2.2%

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

      18. sqr-neg 2.2%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 43.2%

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

   6. Applied egg-rr 43.2%

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

      1. associate-*r/ 43.2%

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

      2. *-rgt-identity 43.2%

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

      3. distribute-neg-frac2 43.2%

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

      4. distribute-neg-frac 43.2%

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

      5. neg-sub0 43.2%

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

      6. associate--r+ 43.2%

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

      7. metadata-eval 43.2%

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

   8. Simplified 43.2%

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

   if -1.7999999999999999e-28 < x < 3.19999999999999986e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 51.9%

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

   4. Taylor expanded in x around 0 51.6%

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

   if 3.19999999999999986e-5 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 56.7%

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

   4. Taylor expanded in x around inf 55.5%

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

      1. neg-mul-1 55.5%

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

   6. Simplified 55.5%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% 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. Add Preprocessing

3. Taylor expanded in B around 0 51.4%

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

Alternative 10: 26.0% 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 51.4%

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

4. Taylor expanded in x around 0 23.8%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))