VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.7s

Alternatives: 9

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/ 99.8%

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

   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. Final simplification 99.8%

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

Alternative 2: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.65) (not (<= x 1.0)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.65) || !(x <= 1.0)) {
		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 <= (-1.65d0)) .or. (.not. (x <= 1.0d0))) 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 <= -1.65) || !(x <= 1.0)) {
		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 <= -1.65) or not (x <= 1.0):
		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 <= -1.65) || !(x <= 1.0))
		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 <= -1.65) || ~((x <= 1.0)))
		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, -1.65], N[Not[LessEqual[x, 1.0]], $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 < -1.6499999999999999 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/ 99.8%

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

      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 98.2%

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

   if -1.6499999999999999 < 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. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      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 x around 0 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-inverses 99.8%

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

      4. *-commutative 99.8%

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

      5. *-lft-identity 99.8%

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

      6. *-lft-identity 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-inverses 99.8%

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

      10. times-frac 79.9%

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

      11. associate-/r* 99.8%

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

      12. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0 99.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 3: 74.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00044 \lor \neg \left(x \leq 3.2 \cdot 10^{-13}\right):\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.00044) (not (<= x 3.2e-13)))
   (+ (* B 0.16666666666666666) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.00044) || !(x <= 3.2e-13)) {
		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 <= (-0.00044d0)) .or. (.not. (x <= 3.2d-13))) 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 <= -0.00044) || !(x <= 3.2e-13)) {
		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 <= -0.00044) or not (x <= 3.2e-13):
		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 <= -0.00044) || !(x <= 3.2e-13))
		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 <= -0.00044) || ~((x <= 3.2e-13)))
		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, -0.00044], N[Not[LessEqual[x, 3.2e-13]], $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 < -4.40000000000000016e-4 or 3.2e-13 < 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. Step-by-step derivation

      1. expm1-log1p-u 99.8%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin B\right)\right)}} - \frac{x}{\tan B} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin B\right)\right)}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. expm1-udef 62.8%

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

      2. log1p-udef 62.8%

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

      3. add-exp-log 62.8%

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

      4. +-commutative 62.8%

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

   7. Applied egg-rr 62.8%

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

   8. Taylor expanded in B around 0 19.6%

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

   9. Taylor expanded in B around 0 56.7%

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

       1. associate--l+ 56.7%

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

       2. *-commutative 56.7%

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

       3. div-sub 56.8%

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

   11. Simplified 56.8%

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

   if -4.40000000000000016e-4 < x < 3.2e-13

   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 99.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00044 \lor \neg \left(x \leq 3.2 \cdot 10^{-13}\right):\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 76.7% 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. +-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/ 99.8%

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

   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 x around 0 99.7%

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

5. Taylor expanded in x around 0 99.7%

   \[\leadsto \color{blue}{\frac{1}{\sin B} + -1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
6. 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. unsub-neg 99.7%

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

   3. *-inverses 99.7%

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

   4. *-commutative 99.7%

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

   5. *-lft-identity 99.7%

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

   6. *-lft-identity 99.7%

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

   7. times-frac 99.7%

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

   8. metadata-eval 99.7%

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

   9. *-inverses 99.7%

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

   10. times-frac 85.3%

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

   11. associate-/r* 99.7%

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

   12. div-sub 99.7%

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

7. Simplified 99.8%

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

8. Taylor expanded in B around 0 79.2%

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

9. Final simplification 79.2%

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

Alternative 5: 51.5% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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 72.1%

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

5. Taylor expanded in B around 0 51.2%

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

   1. +-commutative 51.2%

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

   2. associate-+l+ 51.2%

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

   3. *-commutative 51.2%

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

   4. associate-*l* 51.2%

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

   5. mul-1-neg 51.2%

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

   6. sub-neg 51.2%

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

   7. div-sub 51.3%

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

7. Simplified 51.3%

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

8. Final simplification 51.3%

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

Alternative 6: 51.4% 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. +-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/ 99.8%

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

   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. Step-by-step derivation

   1. expm1-log1p-u 99.7%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin B\right)\right)}} - \frac{x}{\tan B} \]

5. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin B\right)\right)}} - \frac{x}{\tan B} \]
6. Step-by-step derivation

   1. expm1-udef 61.0%

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

   2. log1p-udef 61.0%

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

   3. add-exp-log 61.0%

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

   4. +-commutative 61.0%

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

7. Applied egg-rr 61.0%

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

8. Taylor expanded in B around 0 39.2%

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

9. Taylor expanded in B around 0 51.1%

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

    1. associate--l+ 51.1%

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

    2. *-commutative 51.1%

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

    3. div-sub 51.1%

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

11. Simplified 51.1%

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

12. Final simplification 51.1%

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

Alternative 7: 50.1% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 80\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 80.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 80.0)) {
		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 <= 80.0d0))) 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 <= 80.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 80.0):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 80.0))
		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 <= 80.0)))
		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, 80.0]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 80 < 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 98.6%

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

   5. Taylor expanded in B around 0 55.7%

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

      1. associate-*r/ 55.7%

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

      2. mul-1-neg 55.7%

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

   7. Simplified 55.7%

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

   8. Taylor expanded in x around inf 55.4%

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

      1. associate-*r/ 55.4%

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

      2. mul-1-neg 55.4%

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

   10. Simplified 55.4%

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

   if -1 < x < 80

   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 46.7%

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

   5. Taylor expanded in B around 0 46.1%

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

      1. associate-*r/ 46.1%

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

      2. mul-1-neg 46.1%

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

   7. Simplified 46.1%

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

   8. Taylor expanded in x around 0 44.8%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 80\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 8: 51.2% 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 50.8%

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

   1. mul-1-neg 50.8%

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

   2. sub-neg 50.8%

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

6. Simplified 50.8%

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

7. Final simplification 50.8%

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

Alternative 9: 27.0% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. 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 72.1%

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

5. Taylor expanded in B around 0 50.8%

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

   1. associate-*r/ 50.8%

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

   2. mul-1-neg 50.8%

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

7. Simplified 50.8%

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

8. Taylor expanded in x around 0 24.2%

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

9. Final simplification 24.2%

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

Reproduce

herbie shell --seed 2023256 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))