VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.0s

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 99.7%

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

   1. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.7%

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

   4. *-rgt-identity 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. Final simplification 99.7%

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

Alternative 2: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -180 \lor \neg \left(x \leq 980000000\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -180.0) (not (<= x 980000000.0)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -180.0) || !(x <= 980000000.0)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 - x) / 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 <= (-180.0d0)) .or. (.not. (x <= 980000000.0d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -180.0) || !(x <= 980000000.0)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -180.0) or not (x <= 980000000.0):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -180.0) || !(x <= 980000000.0))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -180.0) || ~((x <= 980000000.0)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -180.0], N[Not[LessEqual[x, 980000000.0]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -180 or 9.8e8 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 99.6%

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

   if -180 < x < 9.8e8

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

      1. un-div-inv 99.7%

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

      2. neg-mul-1 99.7%

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

      3. associate-/l* 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in B around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. div-sub 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180 \lor \neg \left(x \leq 980000000\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 3: 74.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{-9} \lor \neg \left(x \leq 7 \cdot 10^{-6}\right):\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.52e-9) (not (<= x 7e-6)))
   (+ (* B 0.16666666666666666) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.52e-9) || !(x <= 7e-6)) {
		tmp = (B * 0.16666666666666666) + ((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 ((x <= (-1.52d-9)) .or. (.not. (x <= 7d-6))) then
        tmp = (b * 0.16666666666666666d0) + ((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 ((x <= -1.52e-9) || !(x <= 7e-6)) {
		tmp = (B * 0.16666666666666666) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.52e-9) or not (x <= 7e-6):
		tmp = (B * 0.16666666666666666) + ((1.0 - x) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.52e-9) || !(x <= 7e-6))
		tmp = Float64(Float64(B * 0.16666666666666666) + 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 ((x <= -1.52e-9) || ~((x <= 7e-6)))
		tmp = (B * 0.16666666666666666) + ((1.0 - x) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.52e-9], N[Not[LessEqual[x, 7e-6]], $MachinePrecision]], N[(N[(B * 0.16666666666666666), $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 x < -1.51999999999999992e-9 or 6.99999999999999989e-6 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 55.1%

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

      1. +-commutative 55.1%

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

      2. mul-1-neg 55.1%

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

      3. sub-neg 55.1%

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

      4. associate--l+ 55.1%

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

      5. *-commutative 55.1%

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

      6. *-commutative 55.1%

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

      7. div-sub 55.1%

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

   6. Simplified 55.1%

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

   7. Taylor expanded in x around 0 55.5%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot B} + \frac{1 - x}{B} \]
   8. Step-by-step derivation

      1. *-commutative 55.5%

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

   9. Simplified 55.5%

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

   if -1.51999999999999992e-9 < x < 6.99999999999999989e-6

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{-9} \lor \neg \left(x \leq 7 \cdot 10^{-6}\right):\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 77.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

   1. un-div-inv 99.7%

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

   2. neg-mul-1 99.7%

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

   3. associate-/l* 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in B around inf 99.7%

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

   1. mul-1-neg 99.7%

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

   2. sub-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. div-sub 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in B around 0 77.4%

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

10. Final simplification 77.4%

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

Alternative 5: 51.9% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 + \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (* B 0.16666666666666666) (/ (- 1.0 x) B)))

C

double code(double B, double x) {
	return (B * 0.16666666666666666) + ((1.0 - x) / B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (b * 0.16666666666666666d0) + ((1.0d0 - x) / b)
end function

Java

public static double code(double B, double x) {
	return (B * 0.16666666666666666) + ((1.0 - x) / B);
}

Python

def code(B, x):
	return (B * 0.16666666666666666) + ((1.0 - x) / B)

Julia

function code(B, x)
	return Float64(Float64(B * 0.16666666666666666) + Float64(Float64(1.0 - x) / B))
end

MATLAB

function tmp = code(B, x)
	tmp = (B * 0.16666666666666666) + ((1.0 - x) / B);
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 54.4%

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

   1. +-commutative 54.4%

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

   2. mul-1-neg 54.4%

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

   3. sub-neg 54.4%

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

   4. associate--l+ 54.4%

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

   5. *-commutative 54.4%

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

   6. *-commutative 54.4%

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

   7. div-sub 54.4%

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

6. Simplified 54.4%

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

7. Taylor expanded in x around 0 54.6%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot B} + \frac{1 - x}{B} \]
8. Step-by-step derivation

   1. *-commutative 54.6%

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

9. Simplified 54.6%

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

10. Final simplification 54.6%

    \[\leadsto B \cdot 0.16666666666666666 + \frac{1 - x}{B} \]

Alternative 6: 50.6% 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 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 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. Step-by-step derivation

      1. add-sqr-sqrt 41.1%

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

      2. sqrt-unprod 86.1%

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

      3. inv-pow 86.1%

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

      4. inv-pow 86.1%

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

      5. pow-prod-up 86.1%

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

      6. metadata-eval 86.1%

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

   5. Applied egg-rr 86.1%

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

   6. Taylor expanded in B around 0 39.9%

      \[\leadsto \sqrt{{\sin B}^{-2}} - \color{blue}{\frac{x}{B}} \]

   7. Taylor expanded in x around inf 52.3%

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

      1. neg-mul-1 52.3%

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

      2. distribute-neg-frac 52.3%

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

   9. Simplified 52.3%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 55.0%

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

      1. mul-1-neg 55.0%

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

      2. sub-neg 55.0%

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

   6. Simplified 55.0%

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

   7. Taylor expanded in x around 0 52.6%

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

4. Final simplification 52.4%

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

Alternative 7: 51.7% 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. Step-by-step derivation

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 54.1%

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

   1. mul-1-neg 54.1%

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

   2. sub-neg 54.1%

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

6. Simplified 54.1%

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

7. Final simplification 54.1%

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

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 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 54.1%

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

   1. mul-1-neg 54.1%

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

   2. sub-neg 54.1%

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

6. Simplified 54.1%

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

7. Taylor expanded in x around 0 27.4%

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

8. Final simplification 27.4%

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

Reproduce

herbie shell --seed 2023189 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))