VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.2s

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: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.25) (not (<= x 1.05)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.25) || !(x <= 1.05)) {
		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 <= (-2.25d0)) .or. (.not. (x <= 1.05d0))) 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 <= -2.25) || !(x <= 1.05)) {
		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 <= -2.25) or not (x <= 1.05):
		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 <= -2.25) || !(x <= 1.05))
		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 <= -2.25) || ~((x <= 1.05)))
		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, -2.25], N[Not[LessEqual[x, 1.05]], $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 < -2.25 or 1.05000000000000004 < 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 B around 0 99.0%

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

   if -2.25 < x < 1.05000000000000004

   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. sub-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. /-rgt-identity 99.8%

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

      5. *-inverses 99.8%

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

      6. associate-/l* 99.8%

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

      7. div-sub 99.8%

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

      8. *-inverses 99.8%

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

      9. associate-/l* 99.8%

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

      10. *-inverses 99.8%

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

      11. /-rgt-identity 99.8%

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

      12. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 3: 73.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{B}\\ \mathbf{if}\;x \leq -6400:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) B)))
   (if (<= x -6400.0)
     (+ (* B (* x 0.3333333333333333)) t_0)
     (if (<= x 1.5e-6) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (x <= -6400.0) {
		tmp = (B * (x * 0.3333333333333333)) + t_0;
	} else if (x <= 1.5e-6) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = 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 = (1.0d0 - x) / b
    if (x <= (-6400.0d0)) then
        tmp = (b * (x * 0.3333333333333333d0)) + t_0
    else if (x <= 1.5d-6) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (x <= -6400.0) {
		tmp = (B * (x * 0.3333333333333333)) + t_0;
	} else if (x <= 1.5e-6) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 - x) / B
	tmp = 0
	if x <= -6400.0:
		tmp = (B * (x * 0.3333333333333333)) + t_0
	elif x <= 1.5e-6:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 - x) / B)
	tmp = 0.0
	if (x <= -6400.0)
		tmp = Float64(Float64(B * Float64(x * 0.3333333333333333)) + t_0);
	elseif (x <= 1.5e-6)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 - x) / B;
	tmp = 0.0;
	if (x <= -6400.0)
		tmp = (B * (x * 0.3333333333333333)) + t_0;
	elseif (x <= 1.5e-6)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[x, -6400.0], N[(N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x, 1.5e-6], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if x < -6400

   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 99.6%

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

      1. un-div-inv 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0 52.4%

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

      1. mul-1-neg 52.4%

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

      2. distribute-frac-neg 52.4%

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

      3. +-commutative 52.4%

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

      4. associate-*r* 52.4%

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

      5. *-commutative 52.4%

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

      6. associate-+l+ 52.4%

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

      7. associate-*r* 52.4%

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

      8. *-commutative 52.4%

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

      9. distribute-frac-neg 52.4%

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

      10. sub-neg 52.4%

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

      11. div-sub 52.4%

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

   9. Simplified 52.4%

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

   if -6400 < x < 1.5e-6

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 97.9%

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

   if 1.5e-6 < x

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in B around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. sub-neg 41.7%

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

   6. Simplified 41.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6400:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 4: 76.4% 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 99.8%

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

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. sub-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. /-rgt-identity 99.8%

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

   5. *-inverses 99.8%

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

   6. associate-/l* 99.7%

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

   7. div-sub 99.7%

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

   8. *-inverses 99.7%

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

   9. associate-/l* 99.8%

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

   10. *-inverses 99.8%

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

   11. /-rgt-identity 99.8%

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

   12. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in B around 0 75.5%

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

9. Final simplification 75.5%

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

Alternative 5: 51.4% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

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) + 0.16666666666666666d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] + 0.16666666666666666), $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 53.5%

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

   1. +-commutative 53.5%

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

   2. mul-1-neg 53.5%

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

   3. sub-neg 53.5%

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

   4. associate--l+ 53.5%

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

   5. *-commutative 53.5%

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

   6. *-commutative 53.5%

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

   7. div-sub 53.5%

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

6. Simplified 53.5%

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

7. Final simplification 53.5%

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

Alternative 6: 51.5% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(B * N[(x * 0.3333333333333333), $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 78.1%

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

   1. un-div-inv 78.2%

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

   2. neg-mul-1 78.2%

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

   3. associate-/l* 78.1%

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

6. Applied egg-rr 78.1%

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

7. Taylor expanded in B around 0 53.3%

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

   1. mul-1-neg 53.3%

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

   2. distribute-frac-neg 53.3%

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

   3. +-commutative 53.3%

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

   4. associate-*r* 53.3%

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

   5. *-commutative 53.3%

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

   6. associate-+l+ 53.3%

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

   7. associate-*r* 53.3%

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

   8. *-commutative 53.3%

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

   9. distribute-frac-neg 53.3%

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

   10. sub-neg 53.3%

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

   11. div-sub 53.3%

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

9. Simplified 53.3%

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

10. Final simplification 53.3%

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

Alternative 7: 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 53.2%

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

   1. mul-1-neg 53.2%

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

   2. sub-neg 53.2%

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

6. Simplified 53.2%

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

7. Final simplification 53.2%

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

Alternative 8: 3.1% accurate, 70.0× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (* B 0.16666666666666666))

C

double code(double B, double x) {
	return B * 0.16666666666666666;
}

Fortran

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

Java

public static double code(double B, double x) {
	return B * 0.16666666666666666;
}

Python

def code(B, x):
	return B * 0.16666666666666666

Julia

function code(B, x)
	return Float64(B * 0.16666666666666666)
end

MATLAB

function tmp = code(B, x)
	tmp = B * 0.16666666666666666;
end

Wolfram

code[B_, x_] := N[(B * 0.16666666666666666), $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 x around 0 53.6%

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

5. Taylor expanded in B around 0 32.8%

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

6. Taylor expanded in B around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot B} \]
7. Step-by-step derivation

   1. *-commutative 3.2%

      \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]

8. Simplified 3.2%

   \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]

9. Final simplification 3.2%

   \[\leadsto B \cdot 0.16666666666666666 \]

Alternative 9: 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. 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 78.1%

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

5. Taylor expanded in x around 0 32.5%

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

6. Final simplification 32.5%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))