VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.2s

Alternatives: 12

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. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*r/ 99.8%

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

   10. *-rgt-identity 99.8%

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

   11. tan-neg 99.8%

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

   12. distribute-neg-frac2 99.8%

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

   13. distribute-neg-frac 99.8%

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

   14. remove-double-neg 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. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+27} \lor \neg \left(x \leq 2000000\right):\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2e+27) (not (<= x 2000000.0)))
   (* (cos B) (/ x (- (sin B))))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2e+27) || !(x <= 2000000.0)) {
		tmp = cos(B) * (x / -sin(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 <= (-2d+27)) .or. (.not. (x <= 2000000.0d0))) then
        tmp = cos(b) * (x / -sin(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 <= -2e+27) || !(x <= 2000000.0)) {
		tmp = Math.cos(B) * (x / -Math.sin(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2e+27) or not (x <= 2000000.0):
		tmp = math.cos(B) * (x / -math.sin(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2e+27) || !(x <= 2000000.0))
		tmp = Float64(cos(B) * Float64(x / Float64(-sin(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 <= -2e+27) || ~((x <= 2000000.0)))
		tmp = cos(B) * (x / -sin(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2e+27], N[Not[LessEqual[x, 2000000.0]], $MachinePrecision]], N[(N[Cos[B], $MachinePrecision] * N[(x / (-N[Sin[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 < -2e27 or 2e6 < x

   1. Initial program 99.4%

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

      1. div-inv 99.8%

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

      2. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. associate-*l/ 98.6%

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

      3. distribute-lft-neg-in 98.6%

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

      4. *-commutative 98.6%

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

      5. distribute-neg-frac2 98.6%

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

   7. Simplified 98.6%

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

   if -2e27 < x < 2e6

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in B around inf 99.9%

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

      1. div-sub 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in B around 0 99.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+27} \lor \neg \left(x \leq 2000000\right):\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+27} \lor \neg \left(x \leq 2400000\right):\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2e+27) (not (<= x 2400000.0)))
   (* x (/ (cos B) (- (sin B))))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2e+27) || !(x <= 2400000.0)) {
		tmp = x * (cos(B) / -sin(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 <= (-2d+27)) .or. (.not. (x <= 2400000.0d0))) then
        tmp = x * (cos(b) / -sin(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 <= -2e+27) || !(x <= 2400000.0)) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2e+27) or not (x <= 2400000.0):
		tmp = x * (math.cos(B) / -math.sin(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2e+27) || !(x <= 2400000.0))
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(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 <= -2e+27) || ~((x <= 2400000.0)))
		tmp = x * (cos(B) / -sin(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2e+27], N[Not[LessEqual[x, 2400000.0]], $MachinePrecision]], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[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 < -2e27 or 2.4e6 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. associate-/l* 98.5%

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

      3. distribute-rgt-neg-in 98.5%

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

      4. distribute-neg-frac 98.5%

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

   5. Simplified 98.5%

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

   if -2e27 < x < 2.4e6

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in B around inf 99.9%

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

      1. div-sub 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in B around 0 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+27} \lor \neg \left(x \leq 2400000\right):\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin B\\ \mathbf{if}\;x \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{x \cdot \cos B}{t\_0}\\ \mathbf{elif}\;x \leq 650000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\cos B \cdot \frac{x}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (sin B))))
   (if (<= x -2e+27)
     (/ (* x (cos B)) t_0)
     (if (<= x 650000.0) (/ (- 1.0 x) (sin B)) (* (cos B) (/ x t_0))))))

C

double code(double B, double x) {
	double t_0 = -sin(B);
	double tmp;
	if (x <= -2e+27) {
		tmp = (x * cos(B)) / t_0;
	} else if (x <= 650000.0) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = cos(B) * (x / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sin(b)
    if (x <= (-2d+27)) then
        tmp = (x * cos(b)) / t_0
    else if (x <= 650000.0d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = cos(b) * (x / t_0)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -Math.sin(B);
	double tmp;
	if (x <= -2e+27) {
		tmp = (x * Math.cos(B)) / t_0;
	} else if (x <= 650000.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = Math.cos(B) * (x / t_0);
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -math.sin(B)
	tmp = 0
	if x <= -2e+27:
		tmp = (x * math.cos(B)) / t_0
	elif x <= 650000.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = math.cos(B) * (x / t_0)
	return tmp

Julia

function code(B, x)
	t_0 = Float64(-sin(B))
	tmp = 0.0
	if (x <= -2e+27)
		tmp = Float64(Float64(x * cos(B)) / t_0);
	elseif (x <= 650000.0)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = Float64(cos(B) * Float64(x / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -sin(B);
	tmp = 0.0;
	if (x <= -2e+27)
		tmp = (x * cos(B)) / t_0;
	elseif (x <= 650000.0)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = cos(B) * (x / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = (-N[Sin[B], $MachinePrecision])}, If[LessEqual[x, -2e+27], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, 650000.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[Cos[B], $MachinePrecision] * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e27

   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. *-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-frac-neg2 99.6%

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

      6. tan-neg 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 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. Step-by-step derivation

      1. div-inv 99.6%

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

      2. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-frac-neg2 99.7%

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

   9. Simplified 99.7%

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

   if -2e27 < x < 6.5e5

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in B around inf 99.9%

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

      1. div-sub 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in B around 0 99.6%

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

   if 6.5e5 < x

   1. Initial program 99.3%

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

      1. div-inv 99.7%

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

      2. clear-num 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around inf 97.4%

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

      1. mul-1-neg 97.4%

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

      2. associate-*l/ 97.4%

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

      3. distribute-lft-neg-in 97.4%

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

      4. *-commutative 97.4%

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

      5. distribute-neg-frac2 97.4%

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

   7. Simplified 97.4%

      \[\leadsto \color{blue}{\cos B \cdot \frac{x}{-\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.195:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} + x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.195)
   (+
    (* B 0.16666666666666666)
    (+ (/ 1.0 B) (* x (+ (* B 0.3333333333333333) (/ -1.0 B)))))
   (/ (+ 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.195) {
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / 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 (b <= 0.195d0) then
        tmp = (b * 0.16666666666666666d0) + ((1.0d0 / b) + (x * ((b * 0.3333333333333333d0) + ((-1.0d0) / 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 (B <= 0.195) {
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / B))));
	} else {
		tmp = (1.0 + x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.195:
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / B))))
	else:
		tmp = (1.0 + x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.195)
		tmp = Float64(Float64(B * 0.16666666666666666) + Float64(Float64(1.0 / B) + Float64(x * Float64(Float64(B * 0.3333333333333333) + Float64(-1.0 / 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 (B <= 0.195)
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / B))));
	else
		tmp = (1.0 + x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.195], N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / B), $MachinePrecision] + N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.19500000000000001

   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 68.5%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   4. Taylor expanded in x around 0 68.4%

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

   if 0.19500000000000001 < B

   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. div-inv 99.7%

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

      2. clear-num 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in B around inf 99.5%

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

      1. div-sub 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 45.7%

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

      1. div-sub 45.7%

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

      2. sub-neg 45.7%

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

      3. distribute-frac-neg 45.7%

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

      4. add-sqr-sqrt 21.3%

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

      5. sqrt-unprod 46.2%

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

      6. sqr-neg 46.2%

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

      7. sqrt-unprod 26.3%

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

      8. add-sqr-sqrt 49.4%

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

   10. Applied egg-rr 49.4%

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

       1. *-rgt-identity 49.4%

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

       2. *-lft-identity 49.4%

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

       3. associate-*l/ 49.4%

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

       4. distribute-lft-in 49.4%

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

       5. associate-*l/ 49.4%

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

       6. *-lft-identity 49.4%

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

       7. +-commutative 49.4%

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

   12. Simplified 49.4%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.195:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} + x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} + x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 3.6e+16)
   (+
    (* B 0.16666666666666666)
    (+ (/ 1.0 B) (* x (+ (* B 0.3333333333333333) (/ -1.0 B)))))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 3.6e+16) {
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / 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 <= 3.6d+16) then
        tmp = (b * 0.16666666666666666d0) + ((1.0d0 / b) + (x * ((b * 0.3333333333333333d0) + ((-1.0d0) / 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 <= 3.6e+16) {
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / B))));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 3.6e+16:
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / B))))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 3.6e+16)
		tmp = Float64(Float64(B * 0.16666666666666666) + Float64(Float64(1.0 / B) + Float64(x * Float64(Float64(B * 0.3333333333333333) + Float64(-1.0 / B)))));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 3.6e+16)
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / B))));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 3.6e+16], N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / B), $MachinePrecision] + N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.6e16

   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 67.3%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   4. Taylor expanded in x around 0 67.2%

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

   if 3.6e16 < 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 44.9%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} + x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.7% 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. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 99.8%

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

   2. clear-num 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in B around inf 99.7%

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

   1. div-sub 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in B around 0 73.5%

   \[\leadsto \frac{1 - \color{blue}{x}}{\sin B} \]
9. Add Preprocessing

Alternative 8: 51.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (b * 0.16666666666666666d0) + ((1.0d0 / b) + (x * ((b * 0.3333333333333333d0) + ((-1.0d0) / b))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / B), $MachinePrecision] + N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 53.9%

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

4. Taylor expanded in x around 0 53.9%

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

5. Final simplification 53.9%

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

Alternative 9: 50.1% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 48.9%

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

   4. Taylor expanded in x around inf 47.2%

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

      1. neg-mul-1 47.2%

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

      2. distribute-neg-frac2 47.2%

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

   6. Simplified 47.2%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in B around 0 58.6%

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

   4. Taylor expanded in x around 0 58.5%

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

4. Final simplification 52.2%

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

Alternative 10: 51.3% 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. Add Preprocessing

3. Taylor expanded in B around 0 53.9%

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

4. Taylor expanded in x around 0 53.9%

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

5. Taylor expanded in B around 0 53.5%

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

6. Final simplification 53.5%

   \[\leadsto B \cdot 0.16666666666666666 + \frac{1 - x}{B} \]
7. Add Preprocessing

Alternative 11: 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. Add Preprocessing

3. Taylor expanded in B around 0 53.2%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
4. Add Preprocessing

Alternative 12: 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 53.2%

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

4. Taylor expanded in x around 0 27.6%

   \[\leadsto \color{blue}{\frac{1}{B}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))