VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 6.6s

Alternatives: 8

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 99.8%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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. cancel-sign-sub-inv 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*r/ 99.9%

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

   7. *-rgt-identity 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+18} \lor \neg \left(x \leq 490000000000\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -5.5e+18) (not (<= x 490000000000.0)))
   (/ x (- (tan B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -5.5e+18) || !(x <= 490000000000.0)) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5.5d+18)) .or. (.not. (x <= 490000000000.0d0))) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -5.5e+18) || !(x <= 490000000000.0)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -5.5e+18) or not (x <= 490000000000.0):
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -5.5e+18) || !(x <= 490000000000.0))
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -5.5e+18) || ~((x <= 490000000000.0)))
		tmp = x / -tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -5.5e+18], N[Not[LessEqual[x, 490000000000.0]], $MachinePrecision]], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5e18 or 4.9e11 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-*r* 99.7%

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

      3. neg-mul-1 99.7%

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

   4. Simplified 99.7%

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

      1. associate-/l* 99.7%

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

      2. tan-quot 99.9%

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

      3. distribute-frac-neg 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. associate-*l/ 99.7%

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

      6. distribute-lft-neg-in 99.7%

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

      7. add-sqr-sqrt 54.6%

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

      8. associate-*r* 54.7%

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

      9. distribute-neg-frac 54.7%

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

      10. metadata-eval 54.7%

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

   6. Applied egg-rr 54.7%

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

      1. associate-*l* 54.6%

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

      2. frac-2neg 54.6%

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

      3. metadata-eval 54.6%

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

      4. add-sqr-sqrt 99.7%

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

      5. associate-*l/ 99.9%

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

      6. *-un-lft-identity 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -5.5e18 < x < 4.9e11

   1. Initial program 99.8%

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

   2. Taylor expanded in B around 0 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+18} \lor \neg \left(x \leq 490000000000\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 3: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.72 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.72) (not (<= x 1.0))) (/ x (- (tan B))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.72) || !(x <= 1.0)) {
		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 <= (-1.72d0)) .or. (.not. (x <= 1.0d0))) 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 <= -1.72) || !(x <= 1.0)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.72) or not (x <= 1.0):
		tmp = x / -math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.72) || !(x <= 1.0))
		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 <= -1.72) || ~((x <= 1.0)))
		tmp = x / -tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.72], N[Not[LessEqual[x, 1.0]], $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 < -1.71999999999999997 or 1 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. associate-*r* 98.7%

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

      3. neg-mul-1 98.7%

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

   4. Simplified 98.7%

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

      1. associate-/l* 98.7%

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

      2. tan-quot 98.9%

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

      3. distribute-frac-neg 98.9%

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

      4. *-un-lft-identity 98.9%

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

      5. associate-*l/ 98.7%

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

      6. distribute-lft-neg-in 98.7%

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

      7. add-sqr-sqrt 53.8%

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

      8. associate-*r* 53.9%

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

      9. distribute-neg-frac 53.9%

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

      10. metadata-eval 53.9%

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

   6. Applied egg-rr 53.9%

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

      1. associate-*l* 53.8%

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

      2. frac-2neg 53.8%

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

      3. metadata-eval 53.8%

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

      4. add-sqr-sqrt 98.7%

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

      5. associate-*l/ 98.9%

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

      6. *-un-lft-identity 98.9%

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

   8. Applied egg-rr 98.9%

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

   if -1.71999999999999997 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 105000:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 105000.0)
   (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 105000.0) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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 <= 105000.0d0) then
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((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 <= 105000.0) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 105000.0:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 105000.0)
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + 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 <= 105000.0)
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 105000.0], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 105000

   1. Initial program 99.8%

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

   2. Taylor expanded in B around 0 70.6%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   3. Step-by-step derivation

      1. associate--l+ 70.6%

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

      2. *-commutative 70.6%

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

      3. div-sub 70.6%

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

   4. Simplified 70.6%

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

   if 105000 < B

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 42.5%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 105000:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 51.6% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in B around 0 54.8%

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

   1. div-sub 54.8%

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

4. Applied egg-rr 54.8%

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

   1. frac-sub 40.8%

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

   2. associate-/r* 54.9%

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

   3. *-commutative 54.9%

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

   4. distribute-rgt-out-- 54.9%

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

6. Applied egg-rr 54.9%

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

7. Final simplification 54.9%

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

Alternative 6: 50.3% accurate, 25.8× 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(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -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 99.7%

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

   2. Taylor expanded in B around 0 50.3%

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

   3. Taylor expanded in x around inf 49.3%

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

      1. neg-mul-1 49.3%

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

      2. distribute-neg-frac 49.3%

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

   5. Simplified 49.3%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in B around 0 59.7%

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

   3. Taylor expanded in x around 0 58.9%

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

4. Final simplification 53.9%

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

Alternative 7: 51.5% 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 99.8%

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

2. Taylor expanded in B around 0 54.8%

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

3. Final simplification 54.8%

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

Alternative 8: 27.1% 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 99.8%

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

2. Taylor expanded in B around 0 54.8%

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

3. Taylor expanded in x around 0 29.9%

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

4. Final simplification 29.9%

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

Reproduce

herbie shell --seed 2023333 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))