VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 11.0s

Alternatives: 11

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. cancel-sign-sub-inv 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*r/ 99.8%

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

   7. *-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. Add Preprocessing

5. Final simplification 99.8%

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

Alternative 2: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-17} \lor \neg \left(x \leq 6.6 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x}{\tan B} \cdot \left(\frac{1}{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -4.5e-17) (not (<= x 6.6e-18)))
   (* (/ x (tan B)) (+ (/ 1.0 x) -1.0))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -4.5e-17) || !(x <= 6.6e-18)) {
		tmp = (x / tan(B)) * ((1.0 / x) + -1.0);
	} 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 <= (-4.5d-17)) .or. (.not. (x <= 6.6d-18))) then
        tmp = (x / tan(b)) * ((1.0d0 / x) + (-1.0d0))
    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 <= -4.5e-17) || !(x <= 6.6e-18)) {
		tmp = (x / Math.tan(B)) * ((1.0 / x) + -1.0);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -4.5e-17) or not (x <= 6.6e-18):
		tmp = (x / math.tan(B)) * ((1.0 / x) + -1.0)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -4.5e-17) || !(x <= 6.6e-18))
		tmp = Float64(Float64(x / tan(B)) * Float64(Float64(1.0 / x) + -1.0));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -4.5e-17) || ~((x <= 6.6e-18)))
		tmp = (x / tan(B)) * ((1.0 / x) + -1.0);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -4.5e-17], N[Not[LessEqual[x, 6.6e-18]], $MachinePrecision]], N[(N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.49999999999999978e-17 or 6.6000000000000003e-18 < 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 99.7%

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

      5. frac-sub 87.3%

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

      6. *-un-lft-identity 87.3%

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

      7. *-commutative 87.3%

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

      8. *-un-lft-identity 87.3%

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

   4. Applied egg-rr 87.3%

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

      1. expm1-log1p-u 41.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}\right)\right)} \]

      2. expm1-udef 41.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}\right)} - 1} \]

      3. div-inv 41.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{\tan B}{x} - \sin B\right) \cdot \frac{1}{\sin B \cdot \frac{\tan B}{x}}}\right)} - 1 \]

      4. *-commutative 41.4%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{\tan B}{x} - \sin B\right) \cdot \frac{1}{\color{blue}{\frac{\tan B}{x} \cdot \sin B}}\right)} - 1 \]

      5. associate-/r* 41.4%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{\tan B}{x} - \sin B\right) \cdot \color{blue}{\frac{\frac{1}{\frac{\tan B}{x}}}{\sin B}}\right)} - 1 \]

      6. clear-num 41.4%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{\tan B}{x} - \sin B\right) \cdot \frac{\color{blue}{\frac{x}{\tan B}}}{\sin B}\right)} - 1 \]

   6. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\tan B}{x} - \sin B\right) \cdot \frac{\frac{x}{\tan B}}{\sin B}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 41.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\tan B}{x} - \sin B\right) \cdot \frac{\frac{x}{\tan B}}{\sin B}\right)\right)} \]

      2. expm1-log1p 87.0%

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

      3. associate-*r/ 99.6%

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

      4. associate-*l/ 99.7%

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

      5. *-commutative 99.7%

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

      6. div-sub 99.7%

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

      7. *-inverses 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in B around 0 96.7%

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

   if -4.49999999999999978e-17 < x < 6.6000000000000003e-18

   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 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-17} \lor \neg \left(x \leq 6.6 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x}{\tan B} \cdot \left(\frac{1}{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-5} \lor \neg \left(x \leq 6.6 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.4e-5) (not (<= x 6.6e-18)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.4e-5) || !(x <= 6.6e-18)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.4d-5)) .or. (.not. (x <= 6.6d-18))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.4e-5) || !(x <= 6.6e-18)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.4e-5) or not (x <= 6.6e-18):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.4e-5) || !(x <= 6.6e-18))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.4e-5) || ~((x <= 6.6e-18)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.4e-5], N[Not[LessEqual[x, 6.6e-18]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999998e-5 or 6.6000000000000003e-18 < 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. cancel-sign-sub-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r/ 99.8%

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

      7. *-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. Add Preprocessing

   5. Taylor expanded in B around 0 96.9%

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

   if -1.39999999999999998e-5 < x < 6.6000000000000003e-18

   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 99.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-5} \lor \neg \left(x \leq 6.6 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.45) (not (<= x 1.0))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.45) || !(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 <= (-1.45d0)) .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 <= -1.45) || !(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 <= -1.45) 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 <= -1.45) || !(x <= 1.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.45) || ~((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, -1.45], 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 < -1.44999999999999996 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 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. associate-*l/ 98.4%

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

      3. *-commutative 98.4%

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

   5. Simplified 98.4%

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

      1. expm1-log1p-u 46.1%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos B \cdot \frac{x}{\sin B}\right)\right)} \]

      2. expm1-udef 46.1%

         \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\cos B \cdot \frac{x}{\sin B}\right)} - 1\right)} \]

      3. *-commutative 46.1%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sin B} \cdot \cos B}\right)} - 1\right) \]

      4. associate-/r/ 46.1%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}}\right)} - 1\right) \]

      5. tan-quot 46.1%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\tan B}}\right)} - 1\right) \]

   7. Applied egg-rr 46.1%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 46.1%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)} \]

      2. expm1-log1p 98.5%

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

   9. Simplified 98.5%

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

   if -1.44999999999999996 < 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 97.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.56:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.56)
   (-
    (+ (/ 1.0 B) (* B (+ 0.16666666666666666 (* x 0.3333333333333333))))
    (/ x B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.56) {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (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 <= 0.56d0) then
        tmp = ((1.0d0 / b) + (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0)))) - (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 <= 0.56) {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.56:
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.56)
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))) - Float64(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 <= 0.56)
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.56], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.56000000000000005

   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 69.6%

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

   if 0.56000000000000005 < B

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 66.5%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.56:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.3% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.24 \cdot 10^{-5} \lor \neg \left(x \leq 2.6 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.24e-5) (not (<= x 2.6e+27))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.24e-5) || !(x <= 2.6e+27)) {
		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 <= (-1.24d-5)) .or. (.not. (x <= 2.6d+27))) 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 <= -1.24e-5) || !(x <= 2.6e+27)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.24e-5) or not (x <= 2.6e+27):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.24e-5) || !(x <= 2.6e+27))
		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 <= -1.24e-5) || ~((x <= 2.6e+27)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.24e-5], N[Not[LessEqual[x, 2.6e+27]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.24e-5 or 2.60000000000000009e27 < 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 57.6%

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

   4. Taylor expanded in x around inf 57.2%

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

      1. neg-mul-1 57.2%

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

      2. distribute-neg-frac 57.2%

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

   6. Simplified 57.2%

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

   if -1.24e-5 < x < 2.60000000000000009e27

   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 47.1%

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

   4. Taylor expanded in x around 0 46.2%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.24 \cdot 10^{-5} \lor \neg \left(x \leq 2.6 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.4% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)\right) - \frac{x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (- (+ (/ 1.0 B) (* 0.3333333333333333 (* B x))) (/ x B)))

C

double code(double B, double x) {
	return ((1.0 / B) + (0.3333333333333333 * (B * x))) - (x / B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = ((1.0d0 / b) + (0.3333333333333333d0 * (b * x))) - (x / b)
end function

Java

public static double code(double B, double x) {
	return ((1.0 / B) + (0.3333333333333333 * (B * x))) - (x / B);
}

Python

def code(B, x):
	return ((1.0 / B) + (0.3333333333333333 * (B * x))) - (x / B)

Julia

function code(B, x)
	return Float64(Float64(Float64(1.0 / B) + Float64(0.3333333333333333 * Float64(B * x))) - Float64(x / B))
end

MATLAB

function tmp = code(B, x)
	tmp = ((1.0 / B) + (0.3333333333333333 * (B * x))) - (x / B);
end

Wolfram

code[B_, x_] := N[(N[(N[(1.0 / B), $MachinePrecision] + N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $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 70.4%

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

4. Taylor expanded in B around 0 52.3%

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

5. Final simplification 52.3%

   \[\leadsto \left(\frac{1}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)\right) - \frac{x}{B} \]
6. Add Preprocessing

Alternative 8: 51.4% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B)))

C

double code(double B, double x) {
	return (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / b)
end function

Java

public static double code(double B, double x) {
	return (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
}

Python

def code(B, x):
	return (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)

Julia

function code(B, x)
	return Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B))
end

MATLAB

function tmp = code(B, x)
	tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
end

Wolfram

code[B_, x_] := N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $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. Add Preprocessing

3. Taylor expanded in B around 0 70.4%

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

4. Taylor expanded in B around 0 52.3%

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

   1. associate--l+ 52.3%

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

   2. associate-*r* 52.3%

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

   3. div-sub 52.3%

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

6. Simplified 52.3%

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

7. Taylor expanded in B around 0 52.3%

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

8. Final simplification 52.3%

   \[\leadsto 0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B} \]
9. Add Preprocessing

Alternative 9: 51.0% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $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 51.8%

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

   1. div-sub 51.8%

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

5. Applied egg-rr 51.8%

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

6. Final simplification 51.8%

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

Alternative 10: 51.0% 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 51.8%

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

4. Final simplification 51.8%

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

Alternative 11: 26.8% 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 51.8%

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

4. Taylor expanded in x around 0 27.3%

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

5. Final simplification 27.3%

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

Reproduce

herbie shell --seed 2024014 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))