VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 21.2s

Alternatives: 10

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: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4600000000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 52:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -4600000000000.0)
   (/ (- x) (tan B))
   (if (<= x 52.0) (- (/ 1.0 (sin B)) (/ x B)) (/ (* x (- (cos B))) (sin B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -4600000000000.0) {
		tmp = -x / tan(B);
	} else if (x <= 52.0) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (x * -cos(B)) / 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 <= (-4600000000000.0d0)) then
        tmp = -x / tan(b)
    else if (x <= 52.0d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (x * -cos(b)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -4600000000000.0) {
		tmp = -x / Math.tan(B);
	} else if (x <= 52.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -4600000000000.0:
		tmp = -x / math.tan(B)
	elif x <= 52.0:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (x * -math.cos(B)) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -4600000000000.0)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (x <= 52.0)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -4600000000000.0)
		tmp = -x / tan(B);
	elseif (x <= 52.0)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (x * -cos(B)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -4600000000000.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 52.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e12

   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/ 99.9%

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

      4. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. 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 inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. associate-*l/ 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   7. Simplified 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. neg-sub0 99.4%

         \[\leadsto \color{blue}{0 - \frac{\cos B}{\sin B} \cdot x} \]

      3. clear-num 99.3%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \cdot x \]

      4. associate-*l/ 99.4%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot x}{\frac{\sin B}{\cos B}}} \]

      5. *-un-lft-identity 99.4%

         \[\leadsto 0 - \frac{\color{blue}{x}}{\frac{\sin B}{\cos B}} \]

      6. quot-tan 99.5%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   9. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   10. Step-by-step derivation

       1. neg-sub0 99.5%

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

       2. distribute-neg-frac 99.5%

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

   11. Simplified 99.5%

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

   if -4.6e12 < x < 52

   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 B around 0 98.2%

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

   if 52 < 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. Taylor expanded in x around 0 99.7%

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

      1. sub-div 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 97.5%

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

      1. associate-*r* 97.5%

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

      2. neg-mul-1 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4600000000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 52:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4600000000000 \lor \neg \left(x \leq 52\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -4600000000000.0) (not (<= x 52.0)))
   (/ (- x) (tan B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -4600000000000.0) || !(x <= 52.0)) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4600000000000.0d0)) .or. (.not. (x <= 52.0d0))) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -4600000000000.0) || !(x <= 52.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -4600000000000.0) or not (x <= 52.0):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -4600000000000.0) || !(x <= 52.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -4600000000000.0) || ~((x <= 52.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -4600000000000.0], N[Not[LessEqual[x, 52.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.6e12 or 52 < x

   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 inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. associate-*l/ 98.3%

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

      3. distribute-rgt-neg-in 98.3%

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

   7. Simplified 98.3%

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

      1. distribute-rgt-neg-out 98.3%

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

      2. neg-sub0 98.3%

         \[\leadsto \color{blue}{0 - \frac{\cos B}{\sin B} \cdot x} \]

      3. clear-num 98.3%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \cdot x \]

      4. associate-*l/ 98.3%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot x}{\frac{\sin B}{\cos B}}} \]

      5. *-un-lft-identity 98.3%

         \[\leadsto 0 - \frac{\color{blue}{x}}{\frac{\sin B}{\cos B}} \]

      6. quot-tan 98.4%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   9. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   10. Step-by-step derivation

       1. neg-sub0 98.4%

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

       2. distribute-neg-frac 98.4%

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

   11. Simplified 98.4%

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

   if -4.6e12 < x < 52

   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 B around 0 98.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4600000000000 \lor \neg \left(x \leq 52\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4800000000000 \lor \neg \left(x \leq 52\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -4800000000000.0) (not (<= x 52.0)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -4800000000000.0) || !(x <= 52.0)) {
		tmp = -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 <= (-4800000000000.0d0)) .or. (.not. (x <= 52.0d0))) then
        tmp = -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 <= -4800000000000.0) || !(x <= 52.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -4800000000000.0) or not (x <= 52.0):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -4800000000000.0) || !(x <= 52.0))
		tmp = Float64(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 <= -4800000000000.0) || ~((x <= 52.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -4800000000000.0], N[Not[LessEqual[x, 52.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8e12 or 52 < x

   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 inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. associate-*l/ 98.3%

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

      3. distribute-rgt-neg-in 98.3%

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

   7. Simplified 98.3%

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

      1. distribute-rgt-neg-out 98.3%

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

      2. neg-sub0 98.3%

         \[\leadsto \color{blue}{0 - \frac{\cos B}{\sin B} \cdot x} \]

      3. clear-num 98.3%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \cdot x \]

      4. associate-*l/ 98.3%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot x}{\frac{\sin B}{\cos B}}} \]

      5. *-un-lft-identity 98.3%

         \[\leadsto 0 - \frac{\color{blue}{x}}{\frac{\sin B}{\cos B}} \]

      6. quot-tan 98.4%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   9. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   10. Step-by-step derivation

       1. neg-sub0 98.4%

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

       2. distribute-neg-frac 98.4%

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

   11. Simplified 98.4%

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

   if -4.8e12 < x < 52

   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. Step-by-step derivation

      1. sub-div 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in B around 0 98.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4800000000000 \lor \neg \left(x \leq 52\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.35) (not (<= x 1.0))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.35) || !(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 <= (-1.35d0)) .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 <= -1.35) || !(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 <= -1.35) 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 <= -1.35) || !(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 <= -1.35) || ~((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, -1.35], 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 < -1.3500000000000001 or 1 < x

   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. 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 inf 96.3%

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

      1. mul-1-neg 96.3%

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

      2. associate-*l/ 96.3%

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

      3. distribute-rgt-neg-in 96.3%

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

   7. Simplified 96.3%

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

      1. distribute-rgt-neg-out 96.3%

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

      2. neg-sub0 96.3%

         \[\leadsto \color{blue}{0 - \frac{\cos B}{\sin B} \cdot x} \]

      3. clear-num 96.2%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \cdot x \]

      4. associate-*l/ 96.3%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot x}{\frac{\sin B}{\cos B}}} \]

      5. *-un-lft-identity 96.3%

         \[\leadsto 0 - \frac{\color{blue}{x}}{\frac{\sin B}{\cos B}} \]

      6. quot-tan 96.4%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   9. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   10. Step-by-step derivation

       1. neg-sub0 96.4%

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

       2. distribute-neg-frac 96.4%

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

   11. Simplified 96.4%

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

   if -1.3500000000000001 < 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 x around 0 98.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 74.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-14}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{elif}\;x \leq 780000:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -9.5e-14)
   (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B))
   (if (<= x 780000.0)
     (/ 1.0 (sin B))
     (- (* x (* B 0.3333333333333333)) (/ x B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -9.5e-14) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	} else if (x <= 780000.0) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (x * (B * 0.3333333333333333)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-9.5d-14)) then
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
    else if (x <= 780000.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (x * (b * 0.3333333333333333d0)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -9.5e-14) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	} else if (x <= 780000.0) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (x * (B * 0.3333333333333333)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -9.5e-14:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	elif x <= 780000.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (x * (B * 0.3333333333333333)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -9.5e-14)
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	elseif (x <= 780000.0)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(x * Float64(B * 0.3333333333333333)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -9.5e-14)
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	elseif (x <= 780000.0)
		tmp = 1.0 / sin(B);
	else
		tmp = (x * (B * 0.3333333333333333)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -9.5e-14], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 780000.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4999999999999999e-14

   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 67.0%

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

      1. +-commutative 67.0%

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

      2. mul-1-neg 67.0%

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

      3. sub-neg 67.0%

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

      4. associate--l+ 67.0%

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

      5. *-commutative 67.0%

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

      6. *-commutative 67.0%

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

      7. div-sub 67.0%

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

   6. Simplified 67.0%

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

   if -9.4999999999999999e-14 < x < 7.8e5

   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 97.7%

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

   if 7.8e5 < 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. 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 inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-*l/ 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 48.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + -1 \cdot \left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 22.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \frac{x}{B} + -1 \cdot \left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)\right)\right)} \]

      2. expm1-udef 22.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-1 \cdot \frac{x}{B} + -1 \cdot \left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)\right)} - 1} \]

      3. +-commutative 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot \left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right) + -1 \cdot \frac{x}{B}}\right)} - 1 \]

      4. *-commutative 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right) \cdot -1} + -1 \cdot \frac{x}{B}\right)} - 1 \]

      5. fma-def 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B, -1, -1 \cdot \frac{x}{B}\right)}\right)} - 1 \]

      6. *-commutative 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}, -1, -1 \cdot \frac{x}{B}\right)\right)} - 1 \]

      7. distribute-rgt-out-- 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}, -1, -1 \cdot \frac{x}{B}\right)\right)} - 1 \]

      8. metadata-eval 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right), -1, -1 \cdot \frac{x}{B}\right)\right)} - 1 \]

      9. mul-1-neg 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(B \cdot \left(x \cdot -0.3333333333333333\right), -1, \color{blue}{-\frac{x}{B}}\right)\right)} - 1 \]

   10. Applied egg-rr 22.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(B \cdot \left(x \cdot -0.3333333333333333\right), -1, -\frac{x}{B}\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 22.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(B \cdot \left(x \cdot -0.3333333333333333\right), -1, -\frac{x}{B}\right)\right)\right)} \]

       2. expm1-log1p 48.4%

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

       3. fma-neg 48.4%

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

       4. associate-*r* 48.4%

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

       5. associate-*l* 48.4%

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

       6. metadata-eval 48.4%

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

       7. *-commutative 48.4%

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

       8. *-commutative 48.4%

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

       9. *-commutative 48.4%

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

       10. associate-*l* 48.4%

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

   12. Simplified 48.4%

       \[\leadsto \color{blue}{x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-14}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{elif}\;x \leq 780000:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \end{array} \]

Alternative 7: 51.2% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B)))

C

double code(double B, double x) {
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
end function

Java

public static double code(double B, double x) {
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
}

Python

def code(B, x):
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)

Julia

function code(B, x)
	return Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B))
end

MATLAB

function tmp = code(B, x)
	tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
end

Wolfram

code[B_, x_] := N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $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. 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 54.2%

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

   1. +-commutative 54.2%

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

   2. mul-1-neg 54.2%

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

   3. sub-neg 54.2%

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

   4. associate--l+ 54.2%

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

   5. *-commutative 54.2%

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

   6. *-commutative 54.2%

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

   7. div-sub 54.2%

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

6. Simplified 54.2%

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

7. Final simplification 54.2%

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

Alternative 8: 49.0% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-5} \lor \neg \left(x \leq 6.2 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.35e-5) (not (<= x 6.2e+34))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.35e-5) || !(x <= 6.2e+34)) {
		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 <= (-2.35d-5)) .or. (.not. (x <= 6.2d+34))) 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 <= -2.35e-5) || !(x <= 6.2e+34)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.35e-5) or not (x <= 6.2e+34):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.35e-5) || !(x <= 6.2e+34))
		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 <= -2.35e-5) || ~((x <= 6.2e+34)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.35e-5], N[Not[LessEqual[x, 6.2e+34]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.34999999999999986e-5 or 6.19999999999999955e34 < 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 56.6%

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

      1. mul-1-neg 56.6%

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

      2. sub-neg 56.6%

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

   6. Simplified 56.6%

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

   7. Taylor expanded in x around inf 55.1%

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

      1. neg-mul-1 55.1%

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

      2. distribute-neg-frac 55.1%

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

   9. Simplified 55.1%

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

   if -2.34999999999999986e-5 < x < 6.19999999999999955e34

   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 51.5%

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

      1. mul-1-neg 51.5%

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

      2. sub-neg 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in x around 0 51.1%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-5} \lor \neg \left(x \leq 6.2 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 9: 51.1% 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 53.9%

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

   1. mul-1-neg 53.9%

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

   2. sub-neg 53.9%

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

6. Simplified 53.9%

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

7. Final simplification 53.9%

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

Alternative 10: 26.3% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. 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 53.9%

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

   1. mul-1-neg 53.9%

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

   2. sub-neg 53.9%

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

6. Simplified 53.9%

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

7. Taylor expanded in x around 0 28.9%

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

8. Final simplification 28.9%

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

Reproduce

herbie shell --seed 2023278 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))