VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.0s

Alternatives: 7

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. Final simplification 99.8%

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

Alternative 2: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.6) (not (<= x 1.0))) (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.6) || !(x <= 1.0)) {
		tmp = -1.0 / (tan(B) / x);
	} 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 <= (-1.6d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-1.0d0) / (tan(b) / x)
    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 <= -1.6) || !(x <= 1.0)) {
		tmp = -1.0 / (Math.tan(B) / x);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.6) or not (x <= 1.0):
		tmp = -1.0 / (math.tan(B) / x)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.6], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 1 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. div-inv 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.6%

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

      5. frac-sub 90.0%

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

      6. *-un-lft-identity 90.0%

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

      7. *-commutative 90.0%

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

      8. *-un-lft-identity 90.0%

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

   4. Applied egg-rr 90.0%

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

      1. associate-/r* 99.6%

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

      2. div-sub 99.6%

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

      3. *-inverses 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in B around 0 99.0%

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

   8. Taylor expanded in x around inf 97.8%

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

   if -1.6000000000000001 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.025 \lor \neg \left(x \leq 3.2 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.025) (not (<= x 3.2e-32)))
   (/ (- 1.0 x) (tan B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.025) || !(x <= 3.2e-32)) {
		tmp = (1.0 - x) / tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.025d0)) .or. (.not. (x <= 3.2d-32))) then
        tmp = (1.0d0 - x) / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -0.025) || !(x <= 3.2e-32)) {
		tmp = (1.0 - x) / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.025) or not (x <= 3.2e-32):
		tmp = (1.0 - x) / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.025) || !(x <= 3.2e-32))
		tmp = Float64(Float64(1.0 - x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.025) || ~((x <= 3.2e-32)))
		tmp = (1.0 - x) / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.025], N[Not[LessEqual[x, 3.2e-32]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.025000000000000001 or 3.2000000000000002e-32 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. div-inv 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.6%

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

      5. frac-sub 88.6%

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

      6. *-un-lft-identity 88.6%

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

      7. *-commutative 88.6%

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

      8. *-un-lft-identity 88.6%

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

   4. Applied egg-rr 88.6%

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

      1. associate-/r* 99.6%

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

      2. div-sub 99.6%

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

      3. *-inverses 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in B around 0 99.1%

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

      1. div-sub 88.6%

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

      2. clear-num 88.8%

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

      3. sub-neg 88.8%

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

      4. div-inv 87.4%

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

      5. clear-num 87.4%

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

   9. Applied egg-rr 87.4%

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

       1. neg-mul-1 87.4%

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

       2. distribute-rgt-in 99.2%

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

       3. associate-*l/ 99.3%

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

       4. distribute-rgt-in 99.3%

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

       5. lft-mult-inverse 99.3%

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

       6. mul-1-neg 99.3%

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

       7. sub-neg 99.3%

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

   11. Simplified 99.3%

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

   if -0.025000000000000001 < x < 3.2000000000000002e-32

   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 98.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.025 \lor \neg \left(x \leq 3.2 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 7200:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 7200.0) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 7200.0) {
		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 <= 7200.0d0) 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 <= 7200.0) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 7200.0:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 7200.0)
		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 <= 7200.0)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 7200.0], 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 < 7200

   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 64.9%

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

   if 7200 < B

   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 0 48.2%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7200:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 52.4%

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

   4. Taylor expanded in x around inf 51.2%

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

      1. neg-mul-1 51.2%

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

      2. distribute-neg-frac 51.2%

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

   6. Simplified 51.2%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 54.0%

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

   4. Taylor expanded in x around 0 52.7%

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

4. Final simplification 51.9%

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

Alternative 6: 51.7% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 53.2%

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

4. Final simplification 53.2%

   \[\leadsto \frac{1 - x}{B} \]
5. Add Preprocessing

Alternative 7: 27.3% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 53.2%

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

4. Taylor expanded in x around 0 26.2%

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

5. Final simplification 26.2%

   \[\leadsto \frac{1}{B} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))